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juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb21d2a-d30b-42c9-aa57-1b5bafe07012" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">××GMSA (LCE)–0.935–0.516
Table 6: Ablation study of TLDR. The best result is boldfaced.
For transductive models, we report the stronger of GMSAMINand GMSAAVG. Inductive models are
trained with standard adversarial training [GSS15], and transductive models with the TLDR loss in
Eqn (1). As Rejectron depends heavily on a key hyperparameter determining conf i dence needed to
reject, we report the results for the parameter value strongest against our attack. The best-performing
value on CIFAR-10 ef f ectively eliminated the possibility of rejection (hence the rejection rate of 0);
other choices resulted in near-0 robust accuracy.
Datasets and Defense/Attack Setup. We evaluate on MNIST [LeC98] and CIFAR-10 [KH+09]. We
consider an adversarial budget ofϵ= 0.3 inl∞on MNIST andϵ= 8/255 inl∞on CIFAR-10. For
defense, on MNIST, we use a LeNet architecture; on CIFAR-10 we use a ResNet-20 architecture.
In both cases, we train for 40 epochs with a learning rate of 0.001 using ADAM for optimization.
On MNIST, we use 40 iterations of PGD during training with a step size of 0.01. On CIFAR-10, we
use 10 iterations of PGD in training with a step size of 2/255. In training TLDR, we put 85% of the
weight onLtrain, equivalent toλ= 0.176 after a warm start period epochs in whichλ= 0. We use a
rejection radius ofϵ/4 for selective classif i ers. For attack, we use 10 iterations of GMSA on both
datasets. On MNIST, we use 200 steps of PGD with a stepsize of 0.01 while generating adversarial
examples. On CIFAR-10, the PGD attacks use 100 steps with a stepsize of 1/255. Defense settings
used while training models in GMSA (including internal PGD settings) are the standard defense
settings. Internal optimizations in the calculation ofLREJuse
10 steps of PGD with a stepsize of 15%
of the rejection radius. We use λ′= 1 in LREJ;
we observe little sensitivity to the parameter.
8
Results. Table 3 shows the robust accuracy and rejection rate of dif f erent methods. We observe
that either transduction or rejection can improve the performance, while combining both techniques
leads to the best results. In particular, our defense outperforms existing transductive defenses such as
RMC and DANN. It also outperforms the strongest existing baseline of 66.56% robust accuracy on
CIFAR-10 [CAS+20] (note that 66.56% is for the classic setting without rejection and transduction).
Finally, note that our defense passes the sanity check of [Tra22] (i.e. we do not exceed the theoretical
upper bound on robust accuracy of 79%), providing evidence that our evaluation is reliable. These
results provide positive support for the benef i t of combining transduction and rejection for robustness.
6.2Ablation Studies
Dif f erent Attacks on TLDR. Table 4 shows the results of dif f erent attack methods on TLDR.
Previous work [CWG+22] shows that transduction-aware attacks are necessary against transductive
defenses; we observe that attacks (PGD onLCEorLREJand
AutoAttack) from the traditional setting
perform poorly against our defense. We can also see that GMSA signif i cantly outperforms even a
rejection-aware transfer attack (referred to as PGD targetingLREJ;
note that PGD and AutoAttack do
not target the i nal model in this case, given the transductive setting, but instead target a proxy trained
by the adversary); see Algorithm 2 in Appendix B.5 for the full details. This shows that GMSA
is critical for attacking a transductive defender; while PGD and AutoAttack are strong against an
inductive model, they performs poorly facing transduction. Finally, we observe that GMSA withLCE
is much weaker than GMSA withLREJ. This shows another key component in our adaptive attack,
the loss LREJ, is also critical to get a strong attack against our defense.
Ablation onLREJ. To further investigate the importance ofLREJ,
we attack an adversarially trained
model with rejection (i.e., the AT+Rejection model), with PGD on dif f erent losses:LREJ, cross-
entropyLCE, andLREJwithLDB,hreplaced byLCE, with AutoAttack given for comparison. We
also consider a multitargeted attackLMULTIwhich attempts to i nd, for each incorrect label, the
perturbation with the highest robust conf i dence up to the rejection radius. More precisely, given a
point (x,y) and a base classif i erhwith adversarial budgetϵand rejection radiusϵdefense, for each
target labely′, y, we i nd a perturbationz(y′) such thatz(y′) and itsϵdefense-ball around it all get the
label y′(and thus z(y′) will be accepted and misclassif i ed): we let
z(y′) = argmin
p:∥p−x∥≤ϵ LMULTI(p,y′), where LMULTI(p,y′) := LCE(p,y′)
+max
∥z−p∥≤ϵdefense LCE(z,y′).(4)
Finally, the attack outputs the strongest perturbation,
z = argmin
z(y′):y′,y LMULTI(z(y′),y′).(5)
Table 5 shows the robust accuracy under these dif f erent attacks; note that, as with transudction,
AutoAttack is unable to i nd perturbations which evade defenses with rejection.LREJleads to the
strongest attack. In particular, it can be signif i cantly better thanLCEandLMULTI, demonstrating its
importance for attacking models with rejection.
Key Components of TLDR. Compared to traditional defenses, TLDR has two novel components:
using the given test inputs in training the classif i er (the second term in Equation (1), referred to as
Ltest), and transforming the trained classif i er into one with rejection. Table 6 shows the results of the
ablation study on these two components. In all cases, rejection signif i cantly improves results. The
use of transduction is helpful on CIFAR-10, but reduces performance on MNIST probably since it’s
easier to get robust predictions on MNIST and thus knowing test inputs does not help.
7Conclusion
Existing works on leveraging transduction and rejection gave mixed results on their benef i ts for
adversarial robustness. In this work we take a step in realizing their promise in practical deep
learning settings. Theoretically, we show that a novel application of Tram`’er’s results give improved
sample complexity for robust learning in the bounded perturbations setting. Guided by our theory, we
identif i ed a practical robust learning algorithm leveraging both transduction and rejection. Systematic
experiments conf i rm the benef i ts of our constructions. There are many future avenues to explore,
such as improving the theoretical bounds, and improving the ef i ciency of our algorithms.<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full text-token-text-primary" 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fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="b9ca6afc-baef-4be9-9788-273fc22173d1" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><h3>Ablation Study of TLDR: Key Findings</h3><h4>Defense/Attack Setup</h4><ul><li><strong>Datasets</strong>: MNIST and CIFAR-10.</li><li><strong>Adversarial Budget</strong>: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mn>0.3</mn></mrow><annotation encoding="application/x-tex">\epsilon = 0.3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">0.3</span></span></span></span> in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>l</mi><mi mathvariant="normal">∞</mi></msub></mrow><annotation encoding="application/x-tex">l_\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span style="top: -2.55em; margin-left: -0.0197em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> for MNIST, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mn>8</mn><mi mathvariant="normal">/</mi><mn>255</mn></mrow><annotation encoding="application/x-tex">\epsilon = 8/255</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">8/255</span></span></span></span> in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>l</mi><mi mathvariant="normal">∞</mi></msub></mrow><annotation encoding="application/x-tex">l_\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span style="top: -2.55em; margin-left: -0.0197em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> for CIFAR-10.</li><li><strong>Architectures</strong>: LeNet for MNIST, ResNet-20 for CIFAR-10.</li><li><strong>Training</strong>: 40 epochs with a learning rate of 0.001 using ADAM.</li><li><strong>PGD Iterations</strong>: 40 iterations with step size 0.01 for MNIST; 10 iterations with step size 2/255 for CIFAR-10.</li><li><strong>TLDR Training</strong>: 85% weight on <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>t</mi><mi>r</mi><mi>a</mi><mi>i</mi><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{train}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3117em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight" style="margin-right: 0.02778em;">r</span><span class="mord mathnormal mtight">ain</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span>, equivalent to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>0.176</mn></mrow><annotation encoding="application/x-tex">\lambda = 0.176</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">0.176</span></span></span></span> after a warm start period.</li><li><strong>Selective Classifiers</strong>: Rejection radius of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">\epsilon / 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal">ϵ</span><span class="mord">/4</span></span></span></span>.</li><li><strong>GMSA Attack</strong>: 10 iterations on both datasets.</li><li><strong>Adversarial Examples</strong>: 200 steps of PGD with stepsize 0.01 for MNIST; 100 steps with stepsize 1/255 for CIFAR-10.</li><li><strong>Internal PGD Settings</strong>: Standard defense settings, 10 steps of PGD with stepsize of 15% of the rejection radius.</li><li><strong>Parameter <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>λ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">\lambda'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.7519em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.7519em;"><span style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></strong>: Set to 1 in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>R</mi><mi>E</mi><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{REJ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">RE</span><span class="mord mathnormal mtight" style="margin-right: 0.09618em;">J</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li></ul><h4>Results</h4><ul><li><strong>Table 3</strong>: Robust accuracy and rejection rate of different methods. TLDR outperforms existing baselines and satisfies the theoretical upper bound on robust accuracy (79% for CIFAR-10).</li><li><strong>Table 4</strong>: Comparison of different attack methods on TLDR. GMSA outperforms traditional PGD and AutoAttack in the transductive setting.</li><li><strong>Table 5</strong>: Robust accuracy under different attacks. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>R</mi><mi>E</mi><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{REJ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">RE</span><span class="mord mathnormal mtight" style="margin-right: 0.09618em;">J</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> leads to the strongest attack against models with rejection.</li><li><strong>Table 6</strong>: Ablation study on key components of TLDR. Rejection improves results significantly, while transduction is helpful on CIFAR-10 but reduces performance on MNIST.</li></ul><h4>Conclusion</h4><ul><li><strong>Novelty</strong>: TLDR leverages transduction and rejection to enhance adversarial robustness.</li><li><strong>Future Work</strong>: Improving theoretical bounds and efficiency of algorithms.</li></ul><h3>Key Points from the Ablation Study</h3><ol><li><strong>Rejection Mechanism</strong>: Significantly improves robust accuracy by rejecting adversarial examples.</li><li><strong>Transductive Learning</strong>: Effective on CIFAR-10 but less so on MNIST.</li><li><strong>Adaptive Attacks</strong>: GMSA with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>R</mi><mi>E</mi><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{REJ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">RE</span><span class="mord mathnormal mtight" style="margin-right: 0.09618em;">J</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> is crucial for evaluating robustness against transductive defenses.</li><li><strong>Practical Algorithm</strong>: TLDR combines transduction and rejection to achieve high robustness in practical settings.</li></ol><p>The study highlights the importance of combining multiple techniques to achieve strong adversarial defenses and opens avenues for further research to refine and improve these methods.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 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text-token-text-quaternary"><path fill="currentColor" fill-rule="evenodd" d="M5.293 9.293a1 1 0 0 1 1.414 0L12 14.586l5.293-5.293a1 1 0 1 1 1.414 1.414l-6 6a1 1 0 0 1-1.414 0l-6-6a1 1 0 0 1 0-1.414" clip-rule="evenodd"></path></svg></div></button></span></div></div></div><div class="pr-2 lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-4" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb21fe2-39d8-49a7-98d1-abc5b079aae5" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">It was from Two Heads are Better than One:
Towards Better Adversarial Robustness by Combining
Transduction and Rejection
Nils Palumbo ∗1, Yang Guo∗1, Xi Wu2, Jiefeng Chen1, Yingyu Liang1, Somesh Jha1
1 University of Wisconsin-Madison, 2 Google
Abstract
Bothtransductionandrejectionhaveemergedasimportanttechniquesfordefending
against adversarial perturbations. A recent work by Tramèr [Tra22] showed that, in
the rejection-only case (no transduction), a strong rejection-solution can be turned
into a strong (but computationally inef i cient) non-rejection solution. This detector-
to-classif i er reduction has been mostly applied to give evidence that certain claims
of strong selective-model solutions are susceptible, leaving the benef i ts of rejection
unclear. On the other hand, a recent work by Goldwasser et al. [GKKM20] showed
that rejection combined with transduction can give provable guarantees (for certain
problems) that cannot be achieved otherwise. Nevertheless, under recent strong
adversarial attacks (GMSA [CWG+22], which has been shown to be much more
ef f ective than AutoAttack against transduction), Goldwasser et al.’s work was
shown to have low performance in a practical deep-learning setting. In this paper,
we take a step towards realizing the promise of transduction+rejection in more
realistic scenarios. Theoretically, we show that a novel application of Tramèr’s
classif i er-to-detector technique in the transductive setting can give signif i cantly
improved sample-complexity for robust generalization. While our theoretical
construction is computationally inef i cient, it guides us to identify an ef i cient
transductive algorithm to learn a selective model. Extensive experiments using
state of the art attacks (AutoAttack, GMSA) show that our solutions provide
signif i cantly better robust accuracy.
1Introduction
A recent line of research [GKKM20,MHS21,Goo19,WJS+21,WYW20a] has investigated aug-
menting models with transduction or rejection to defend against adversarial perturbations. However,
the results of leveraging these new options have been mixed. For example, a recent work by
Tramèr [Tra22] gives an equivalence between classif i cation-only and classif i cation-with-rejection; the
major application of the author’s results has been to provide bounds on the performance of defenses
with rejection, which can be used to show that the robustness of defenses with rejection may be lower
than the authors originally claimed, casting doubt on the benef i ts of rejection.
On the other hand, some recent work in theory has demonstrated that transduction, that is leveraging
the unlabeled test-time input for learning the model, may have signif i cant impact on defending
against adversarial robustness. Specif i cally, Montasser et al. [MHS21] studied the setting of trans-
duction (without rejection), and show that robust learning with transduction allows for signif i cant
∗Equal contribution.
Preprint. Under review.
arXiv:2305.17528v1
[cs.LG]
27
May
2023
improvemnents in sample complexity, reducing dependency on VC dimension from exponential to
linear; however, this comes at the cost of signif i cantly greater assumptions on the data (OPTU2for the
realizable case rather than theOPTUof the inductive setting 2). Goldwasser et al. [GKKM20] studied
transduction and rejection, and show even more surprising results, not achievable with transduction
or rejection alone. However, one prominent limitation of these works seems to be that none has yet
resulted in practical robust learning mechanisms in the deep learning setting typically considered.
Realizable Agnostic Generalization Bound
ConditionGeneralization Bound
Induction [MHS19]OPTU= 0O ?
2VC(H)log(n)+log(1/δ)
n ? OPTU+ O q
2VC(H)+log(1/δ)
n !
Transduction [MHS21]OPTU2= 0O ?VC(H)log(n)+log(1/δ)
n ?
2OPTU2+ O q
VC(H)+log(1/δ)
n !
Rejection (Theorem 4.2)OPTrej
U
= 0O ?
2VC(T(H))log(n)+log(1/δ)
n ?
OPTrej
U + O q
2VC(T(H))+log(1/δ)
n !
Transduction + Rejection [GKKM20]OPTU= 0O q
VC(H)log(n)
n + log(1/δ)
n ! 2OPTU+2√2OPTI+ O q
VC(H)logn+log(1/δ)
n !
Transduction + Rejection (ours) (Theorem 4.1)OPTU2/3= 0O ?VC(H)log(n)+log(1/δ)
n ?
2OPTU2/3+ O q
VC(H)+log(1/δ)
n !
Table 1:Summary of generalization bounds for the four settings. Compared to transduction alone
and [GKKM20], our defense weakens the necessary conditions in the realizable case and improves the asymp-
totic error in the agnostic case. Compared to induction and rejection alone, sample complexity has a linear rather
than exponential dependence on the VC dimension. Compared to [GKKM20], the dependence on the error
bound ϵ improves from inverse quadratic to inverse linear in the realizable case.
Specif i cally, compared to Goldwasser et al., which considered arbitrary perturbations, we focus on the
classic and practical scenairo of bounded perturbations for deep learning. Somewhat surprisingly, we
show that a novel application of Tramèr’s classif i er-to-detector technique in the transductive setting
can give signif i cantly improved sample-complexity for robust generalization, noting that bounded
perturbations are critical for the construction to work. To obtain these improvements, we do not
require stronger assumptions on the data, as with [MHS21]; in the realizable case, we only need
to assumeOPTU2/3= 0, which is even better than theOPTU= 0 assumption in the inductive case.
Table 1 gives more details; the notation is described in Section 3. Our results give a i rst constructive
application of Tramèr’s classif i er-to-detector reduction which leads to improved sample complexity.
While our theoretical construction is computationally inef i cient due to the use of Tramèr’s reduction,<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full 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fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="40932c88-d537-41a4-a56d-4a2b635fa405" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><h3>Summary of "Two Heads are Better than One: Towards Better Adversarial Robustness by Combining Transduction and Rejection"</h3><h4>Authors</h4><p>Nils Palumbo, Yang Guo, Xi Wu, Jiefeng Chen, Yingyu Liang, Somesh Jha
Affiliations: University of Wisconsin-Madison, Google</p><h4>Abstract</h4><p>This work explores the combined use of transduction and rejection to improve adversarial robustness in deep learning models. While previous studies showed mixed results, this paper presents theoretical and practical advancements demonstrating the effectiveness of combining these techniques. A novel application of Tramèr's classifier-to-detector reduction in the transductive setting leads to significant improvements in sample complexity for robust learning. Extensive experiments show better robust accuracy against state-of-the-art attacks.</p><h4>Key Concepts</h4><ol><li><strong>Transduction</strong>: Leveraging unlabeled test-time input for learning the model.</li><li><strong>Rejection</strong>: The ability of the model to reject adversarial examples.</li><li><strong>Tramèr's Reduction</strong>: An equivalence between classification-only and classification-with-rejection, which helps in understanding the robustness of defenses with rejection.</li></ol><h4>Contributions</h4><ol><li><p><strong>Theoretical Advances</strong>:</p><ul><li>Improved sample complexity for robust generalization using Tramèr's reduction in the transductive setting.</li><li>Bounded perturbations are critical for the construction, and the necessary conditions for robust learning are weakened compared to prior works.</li></ul></li><li><p><strong>Practical Algorithm</strong>:</p><ul><li>Identification of an efficient transductive algorithm to learn a selective model.</li><li>Significant improvements in robust accuracy demonstrated through extensive experiments against strong attacks (e.g., AutoAttack, GMSA).</li></ul></li><li><p><strong>Empirical Results</strong>:</p><ul><li><strong>Datasets</strong>: MNIST and CIFAR-10.</li><li><strong>Architectures</strong>: LeNet for MNIST, ResNet-20 for CIFAR-10.</li><li><strong>Defense/Attack Settings</strong>: Specific configurations for adversarial training, including parameters for PGD and TLDR training.</li><li><strong>Robust Accuracy</strong>: TLDR outperforms existing baselines and maintains robust accuracy within theoretical bounds.</li><li><strong>Ablation Study</strong>: Demonstrates the importance of key components like <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>R</mi><mi>E</mi><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{REJ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">RE</span><span class="mord mathnormal mtight" style="margin-right: 0.09618em;">J</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> and GMSA.</li></ul></li></ol><h4>Important Findings</h4><ol><li><p><strong>Combination of Techniques</strong>:</p><ul><li>Combining transduction and rejection leads to better performance compared to using them individually.</li><li>This combination helps in achieving higher robust accuracy, especially in the presence of strong adversarial attacks.</li></ul></li><li><p><strong>Ablation Studies</strong>:</p><ul><li>Rejection significantly improves results across datasets.</li><li>Transduction is more beneficial for CIFAR-10 than MNIST due to the nature of the datasets.</li></ul></li><li><p><strong>Adaptive Attacks</strong>:</p><ul><li>GMSA with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>R</mi><mi>E</mi><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{REJ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">RE</span><span class="mord mathnormal mtight" style="margin-right: 0.09618em;">J</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> is crucial for evaluating robustness in transductive defenses, highlighting the need for adaptive attack strategies.</li></ul></li></ol><h4>Conclusion</h4><p>The study showcases the potential of combining transduction and rejection for adversarial robustness in deep learning. It provides both theoretical insights and practical algorithms, offering a promising direction for future research in enhancing model robustness against adversarial attacks.</p><h3>Reference Summary</h3><ul><li><strong>Tramèr [Tra22]</strong>: Discusses the equivalence between classification-only and classification-with-rejection.</li><li><strong>Goldwasser et al. [GKKM20]</strong>: Provides provable guarantees for robustness with transduction and rejection.</li><li><strong>Montasser et al. [MHS21]</strong>: Highlights the impact of transduction on robust learning and sample complexity.</li><li><strong>GMSA [CWG+22]</strong>: A strong adversarial attack shown to be more effective than AutoAttack against transductive defenses.</li></ul></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 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14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="6d427bf5-a60f-4b25-8e45-b8d22b1fcd6b" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Certainly! Here are the definitions of some of the unusual terms used in the context of the paper:</p><h3>Definitions</h3><ol><li><p><strong>Transduction</strong>:</p><ul><li><strong>Definition</strong>: A learning process where the model leverages both training data and the specific test data it will encounter to make predictions. Unlike traditional inductive learning, which generalizes from training data to any possible test data, transduction tailors the learning process to the particular test set available.</li></ul></li><li><p><strong>Rejection</strong>:</p><ul><li><strong>Definition</strong>: A technique where the model has the capability to reject or abstain from making a prediction on certain inputs, particularly those it finds uncertain or likely to be adversarial. This helps in improving robustness as the model can avoid making incorrect predictions on potentially harmful inputs.</li></ul></li><li><p><strong>Adversarial Perturbations</strong>:</p><ul><li><strong>Definition</strong>: Small, often imperceptible changes made to input data (such as images) designed to deceive a machine learning model into making incorrect predictions. These perturbations exploit vulnerabilities in the model.</li></ul></li><li><p><strong>Tramèr's Reduction</strong>:</p><ul><li><strong>Definition</strong>: A theoretical technique developed by Tramèr that shows an equivalence between models that only classify inputs and those that classify but also have the option to reject inputs. This reduction helps in understanding the robustness of models with rejection capabilities.</li></ul></li><li><p><strong>Sample Complexity</strong>:</p><ul><li><strong>Definition</strong>: A measure of the number of training samples needed for a machine learning model to learn a task to a certain level of accuracy. Lower sample complexity implies that fewer samples are needed for the model to generalize well.</li></ul></li><li><p><strong>PGD (Projected Gradient Descent)</strong>:</p><ul><li><strong>Definition</strong>: An iterative optimization algorithm used for generating adversarial examples. PGD makes small, bounded changes to the input data in each iteration, aiming to maximize the model's prediction error while keeping the perturbation within a specified limit.</li></ul></li><li><p><strong>GMSA (Gradient Masking and Smoothing Attack)</strong>:</p><ul><li><strong>Definition</strong>: An advanced adversarial attack that targets models leveraging transduction and rejection. GMSA is designed to bypass defenses by masking gradients and smoothing the attack to make it less detectable.</li></ul></li><li><p><strong>TLDR (Transductive Learning with Defense and Rejection)</strong>:</p><ul><li><strong>Definition</strong>: The specific algorithm proposed in the paper that combines transduction and rejection to improve adversarial robustness.</li></ul></li><li><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>R</mi><mi>E</mi><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{REJ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">RE</span><span class="mord mathnormal mtight" style="margin-right: 0.09618em;">J</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> (Rejection Loss)</strong>:</p><ul><li><strong>Definition</strong>: A loss function used in training models that incorporate rejection. It is designed to penalize the model for failing to reject adversarial inputs, thereby enhancing its robustness.</li></ul></li><li><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>C</mi><mi>E</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{CE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">CE</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> (Cross-Entropy Loss)</strong>:</p><ul><li><strong>Definition</strong>: A common loss function used in classification tasks that measures the difference between the predicted probability distribution and the actual distribution (usually represented as a one-hot vector).</li></ul></li><li><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></strong>:</p><ul><li><strong>Definition</strong>: A parameter representing the maximum allowable perturbation in adversarial attacks, often measured in some norm (e.g., <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>l</mi><mi mathvariant="normal">∞</mi></msub></mrow><annotation encoding="application/x-tex">l_\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span style="top: -2.55em; margin-left: -0.0197em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span>). It defines the budget or constraint on how much an input can be altered to create an adversarial example.</li></ul></li><li><p><strong>VC Dimension (Vapnik-Chervonenkis Dimension)</strong>:</p><ul><li><strong>Definition</strong>: A measure of the capacity of a statistical classification model, representing the complexity of functions that can be learned by the model. Higher VC dimensions indicate a model that can represent more complex functions, but may also require more samples to train effectively.</li></ul></li><li><p><strong>AutoAttack</strong>:</p><ul><li><strong>Definition</strong>: A powerful ensemble of adversarial attacks designed to evaluate the robustness of machine learning models. It combines multiple attack strategies to ensure a comprehensive assessment of the model's vulnerability.</li></ul></li><li><p><strong>Inductive Learning</strong>:</p><ul><li><strong>Definition</strong>: The traditional machine learning approach where a model is trained on a set of training data and then evaluated on a separate set of test data. The model aims to generalize from the training data to unseen data.</li></ul></li><li><p><strong>Realizable Case</strong>:</p><ul><li><strong>Definition</strong>: A scenario in learning theory where there exists a hypothesis in the hypothesis class that can perfectly classify all the training samples without error.</li></ul></li><li><p><strong>Agnostic Case</strong>:</p><ul><li><strong>Definition</strong>: A scenario in learning theory where no assumption is made about the existence of a perfect hypothesis. The focus is on minimizing the error relative to the best possible hypothesis in the class.</li></ul></li></ol><p>These definitions should help in understanding the specialized terms and concepts discussed in the paper.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 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6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-9" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="flex-shrink-0 flex flex-col relative items-end"><div><div class="pt-0.5 juice:pt-0"><div class="gizmo-bot-avatar flex h-6 w-6 items-center justify-center overflow-hidden rounded-full juice:h-8 juice:w-8"><div class="relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" 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32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="b7a4a123-e09e-45c5-bc1d-2ba57d9dfa12" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Sure! Here are some everyday analogies for the most challenging terms:</p><h3>Everyday Analogies</h3><ol><li><p><strong>Transduction</strong>:</p><ul><li><strong>Analogy</strong>: Imagine you're preparing for a trivia night. Instead of just studying general knowledge (inductive learning), you also get to see the specific questions that will be asked at the trivia night beforehand (transduction). This way, you tailor your studying specifically to those questions, improving your chances of winning.</li></ul></li><li><p><strong>Rejection</strong>:</p><ul><li><strong>Analogy</strong>: Think of a bouncer at a nightclub. The bouncer only lets in people who look like they belong (clean, well-dressed). If someone looks suspicious or is causing trouble, the bouncer rejects them at the door. Similarly, a model with rejection can refuse to classify inputs that seem suspicious or potentially harmful.</li></ul></li><li><p><strong>Adversarial Perturbations</strong>:</p><ul><li><strong>Analogy</strong>: Imagine someone trying to forge a signature. They make tiny changes to their handwriting to closely mimic the original signature, hoping it will be accepted as genuine. Adversarial perturbations are like these tiny changes made to data inputs to trick the model into making a wrong prediction.</li></ul></li><li><p><strong>Tramèr's Reduction</strong>:</p><ul><li><strong>Analogy</strong>: Picture a restaurant that offers both a regular menu and a special diet menu (rejection option). Tramèr's reduction shows that the effectiveness of the special diet menu can be translated into the regular menu, demonstrating that you can provide similar health benefits with either menu type.</li></ul></li><li><p><strong>Sample Complexity</strong>:</p><ul><li><strong>Analogy</strong>: Think about learning to play a new song on the piano. Sample complexity is like the number of practice sessions you need to play the song perfectly. If the song is simple, you need fewer sessions (low sample complexity). If it's complex, you need more sessions (high sample complexity).</li></ul></li><li><p><strong>PGD (Projected Gradient Descent)</strong>:</p><ul><li><strong>Analogy</strong>: Imagine trying to climb a steep hill in the fog. You take small steps in different directions, each time checking if you're getting higher or lower. PGD is like this process, where each step adjusts your path slightly to find the quickest way to the top (maximum error).</li></ul></li><li><p><strong>GMSA (Gradient Masking and Smoothing Attack)</strong>:</p><ul><li><strong>Analogy</strong>: Think of a ninja trying to infiltrate a secure building. GMSA is like the ninja using smooth, silent movements and avoiding obvious paths to get past security guards. It carefully hides its actions to avoid detection while achieving its goal.</li></ul></li><li><p><strong>TLDR (Transductive Learning with Defense and Rejection)</strong>:</p><ul><li><strong>Analogy</strong>: Imagine a security system that not only recognizes faces (classification) but also has a list of allowed guests (transduction) and can refuse entry to suspicious individuals (rejection). TLDR is this comprehensive system ensuring maximum security.</li></ul></li><li><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>R</mi><mi>E</mi><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{REJ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">RE</span><span class="mord mathnormal mtight" style="margin-right: 0.09618em;">J</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> (Rejection Loss)</strong>:</p><ul><li><strong>Analogy</strong>: Picture a quality control inspector in a factory. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>R</mi><mi>E</mi><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{REJ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">RE</span><span class="mord mathnormal mtight" style="margin-right: 0.09618em;">J</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> is like the inspector's penalty for letting defective products pass through. The more defects they miss, the higher the penalty, encouraging them to reject faulty items more effectively.</li></ul></li><li><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>C</mi><mi>E</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{CE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">CE</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> (Cross-Entropy Loss)</strong>:</p><ul><li><strong>Analogy</strong>: Imagine a teacher grading multiple-choice tests. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>C</mi><mi>E</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{CE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">CE</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> is like a measure of how far the students' answers are from the correct ones. The more incorrect the answers, the higher the measure, indicating poorer performance.</li></ul></li><li><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></strong>:</p><ul><li><strong>Analogy</strong>: Think of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span> as the budget for tiny tweaks you can make to a disguise. It represents the maximum extent to which you can alter your appearance without being recognized as someone else.</li></ul></li><li><p><strong>VC Dimension (Vapnik-Chervonenkis Dimension)</strong>:</p><ul><li><strong>Analogy</strong>: Picture a library with many different types of books. The VC dimension is like the number of distinct ways you can categorize the books (by genre, author, year, etc.). A higher VC dimension means more ways to organize the library but also more complexity in doing so.</li></ul></li><li><p><strong>AutoAttack</strong>:</p><ul><li><strong>Analogy</strong>: Imagine a team of experts trying to break into a highly secure vault. AutoAttack is like this team using a combination of the best tools and techniques to test the vault's security thoroughly, ensuring it is truly secure.</li></ul></li><li><p><strong>Inductive Learning</strong>:</p><ul><li><strong>Analogy</strong>: Think of a chef learning to cook new dishes. They practice with a variety of ingredients and recipes (training data) to be ready to cook anything in the future, even if they haven't seen the specific recipe before (generalization to test data).</li></ul></li><li><p><strong>Realizable Case</strong>:</p><ul><li><strong>Analogy</strong>: Imagine a puzzle with all pieces in the box. The realizable case is like knowing there’s definitely a way to complete the puzzle perfectly because all the pieces fit together without any missing or extra pieces.</li></ul></li><li><p><strong>Agnostic Case</strong>:</p><ul><li><strong>Analogy</strong>: Imagine working on a puzzle where some pieces might be missing or not fit perfectly. The agnostic case is about doing your best to complete the puzzle despite these imperfections, aiming for the closest possible solution.</li></ul></li></ol><p>These analogies should help make the complex terms more relatable and easier to understand.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 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text-token-text-quaternary"><path fill="currentColor" fill-rule="evenodd" d="M5.293 9.293a1 1 0 0 1 1.414 0L12 14.586l5.293-5.293a1 1 0 1 1 1.414 1.414l-6 6a1 1 0 0 1-1.414 0l-6-6a1 1 0 0 1 0-1.414" clip-rule="evenodd"></path></svg></div></button></span></div></div></div><div class="pr-2 lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-10" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb21f5e-4ec1-4a89-aa89-85064b6e5bf1" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">What about if someone deliberately answered incorrectly? Couldn't you still tell that they knew the answers, if they they were consistently wrong.<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-11" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="flex-shrink-0 flex flex-col relative items-end"><div><div class="pt-0.5 juice:pt-0"><div class="gizmo-bot-avatar flex h-6 w-6 items-center justify-center overflow-hidden rounded-full juice:h-8 juice:w-8"><div class="relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md" role="img"><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 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10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="2df6764c-6877-427c-bda1-4149bef57038" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>That's a great point! Deliberate incorrect answers could indeed provide a clue that the person knows the correct answers. This concept can be related to adversarial examples and robustness in machine learning, where the idea is to detect not just correct classifications but also suspiciously consistent patterns that might indicate tampering or manipulation. Let's break it down with an analogy:</p><h3>Analogy: Deliberate Incorrect Answers</h3><p>Imagine a student taking a multiple-choice test who knows all the correct answers but decides to deliberately choose the wrong answers to confuse the teacher. The teacher, knowing the student's capabilities, might notice that the student is consistently picking the wrong answer for each question. This unusual pattern could raise a red flag for the teacher, suggesting that the student is intentionally providing incorrect answers.</p><h3>Applying the Concept to Adversarial Robustness</h3><ol><li><strong>Adversarial Perturbations</strong>:<ul><li><strong>Real-World Analogy</strong>: Like the student intentionally giving wrong answers, adversarial perturbations are deliberate modifications made to input data to fool the model.</li></ul></li><li><strong>Detection Through Consistency</strong>:<ul><li><strong>Real-World Analogy</strong>: Just as the teacher might detect the deliberate wrong answers through their consistent pattern, a robust model might detect adversarial examples by recognizing unusual patterns that differ from typical errors.</li></ul></li><li><strong>Rejection Mechanism</strong>:<ul><li><strong>Real-World Analogy</strong>: The teacher, upon noticing the pattern, might decide to reject the student's answers as valid, questioning their authenticity. Similarly, a model with a rejection mechanism can refuse to classify inputs that show suspiciously consistent errors, enhancing its robustness against adversarial attacks.</li></ul></li></ol><h3>Key Takeaways</h3><ul><li><strong>Adversarial Attacks and Consistent Patterns</strong>: Just as consistent deliberate errors can reveal a student's knowledge, consistent patterns in adversarial attacks can be detected by a robust model. The model can then use this information to reject suspicious inputs.</li><li><strong>Rejection Mechanism</strong>: By incorporating a rejection mechanism, models can enhance their robustness, refusing to classify inputs that exhibit unusual patterns indicative of adversarial manipulation.</li><li><strong>Adaptive Detection</strong>: Robust models can adapt to detect not just incorrect classifications but also patterns of deliberate manipulation, similar to how a teacher might recognize a student's intentional mistakes.</li></ul><p>This analogy highlights how understanding the consistency and patterns in inputs can play a crucial role in enhancing the robustness and reliability of machine learning models against adversarial attacks.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex 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class="flex items-center pb-0.5 juice:pb-0"><div class="[&_svg]:h-full [&_svg]:w-full icon-md h-4 w-4"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24"><path fill="currentColor" d="M19.898.855a.4.4 0 0 0-.795 0c-.123 1.064-.44 1.802-.943 2.305-.503.503-1.241.82-2.306.943a.4.4 0 0 0 .001.794c1.047.119 1.801.436 2.317.942.512.504.836 1.241.93 2.296a.4.4 0 0 0 .796 0c.09-1.038.413-1.792.93-2.308.515-.516 1.269-.839 2.306-.928a.4.4 0 0 0 .001-.797c-1.055-.094-1.792-.418-2.296-.93-.506-.516-.823-1.27-.941-2.317Z"></path><path fill="currentColor" d="M12.001 1.5a1 1 0 0 1 .993.887c.313 2.77 1.153 4.775 2.5 6.146 1.34 1.366 3.3 2.223 6.095 2.47a1 1 0 0 1-.003 1.993c-2.747.238-4.75 1.094-6.123 2.467-1.373 1.374-2.229 3.376-2.467 6.123a1 1 0 0 1-1.992.003c-.248-2.795-1.105-4.754-2.47-6.095-1.372-1.347-3.376-2.187-6.147-2.5a1 1 0 0 1-.002-1.987c2.818-.325 4.779-1.165 6.118-2.504 1.339-1.34 2.179-3.3 2.504-6.118A1 1 0 0 1 12 1.5ZM6.725 11.998c1.234.503 2.309 1.184 3.21 2.069.877.861 1.56 1.888 2.063 3.076.5-1.187 1.18-2.223 2.051-3.094.871-.87 1.907-1.55 3.094-2.05-1.188-.503-2.215-1.187-3.076-2.064-.885-.901-1.566-1.976-2.069-3.21-.505 1.235-1.19 2.3-2.081 3.192-.891.89-1.957 1.576-3.192 2.082Z"></path></svg></div><span class="line-clamp-1 text-sm" style="opacity: 0; padding-left: 0px; width: 0px;">4o</span><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-sm text-token-text-quaternary"><path fill="currentColor" fill-rule="evenodd" d="M5.293 9.293a1 1 0 0 1 1.414 0L12 14.586l5.293-5.293a1 1 0 1 1 1.414 1.414l-6 6a1 1 0 0 1-1.414 0l-6-6a1 1 0 0 1 0-1.414" clip-rule="evenodd"></path></svg></div></button></span></div></div></div><div class="pr-2 lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-12" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb2172f-c477-4cf9-be79-9e3d8f5ba888" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">I mean that in order to "guess" completely wrong everything everytime. The teacher might have put a secret message like
W R O N G
A N S W E R S
O N L Y
And could include three correct answers and only one incorrect one.
This would check if they were reading all the instructions carefully and they could fail the ones who passed.
You also detect it as a statistical irregularity on an ordinary test, and the teacher could still pass them on the account of the fact that the student likely knows the answer or else it is a highly statistical fluke<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-13" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="flex-shrink-0 flex flex-col relative items-end"><div><div class="pt-0.5 juice:pt-0"><div class="gizmo-bot-avatar flex h-6 w-6 items-center justify-center overflow-hidden rounded-full juice:h-8 juice:w-8"><div class="relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md" role="img"><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 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10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="430bba21-9bf5-4752-9597-cafa4128fc4c" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>I understand now. You're suggesting that deliberately incorrect answers, if consistently incorrect in a certain way, could reveal that the person actually knows the correct answers. This concept can be used to understand how machine learning models detect adversarial examples and ensure robustness.</p><h3>Refined Analogy: Deliberate Incorrect Answers as a Robustness Check</h3><p>Imagine a scenario where a teacher is testing students not only on their knowledge but also on their ability to follow instructions meticulously. Here’s how this analogy can be expanded to illustrate key concepts in adversarial robustness:</p><ol><li><p><strong>The Special Instruction Test</strong>:</p><ul><li><strong>Scenario</strong>: The teacher designs a test with a hidden instruction, like "Mark all correct answers as incorrect." This instruction is known only to students who read the entire test carefully.</li><li><strong>Objective</strong>: The teacher wants to see if students can follow this hidden instruction, which indirectly tests their thoroughness and attention to detail.</li></ul></li><li><p><strong>Detection of Deliberate Incorrect Answers</strong>:</p><ul><li><strong>Consistent Patterns</strong>: If a student consistently marks the correct answers as incorrect, the teacher might deduce that the student has actually read the hidden instruction and knows the correct answers.</li><li><strong>Statistical Irregularity</strong>: If most students are getting a mix of correct and incorrect answers, a student who consistently gets all correct answers wrong stands out. This statistical irregularity suggests that the student likely knows the correct answers and is following the hidden instruction.</li></ul></li><li><p><strong>Robustness Against Adversarial Attacks</strong>:</p><ul><li><strong>Machine Learning Parallel</strong>: In adversarial robustness, models can be designed to detect not just the correctness of answers (classifications) but also unusual patterns that indicate manipulation (adversarial attacks).</li><li><strong>Rejection Mechanism</strong>: If a model detects that an input is consistently altered in a specific way (like the student marking all correct answers as wrong), it can reject the input as adversarial, similar to how the teacher might reject the test as manipulated.</li></ul></li><li><p><strong>Statistical Analysis</strong>:</p><ul><li><strong>Detection</strong>: Just as the teacher can use statistical analysis to detect an unusual pattern of wrong answers, machine learning models can use statistical methods to detect adversarial perturbations that deviate from typical data patterns.</li><li><strong>Passing Despite Irregularities</strong>: The teacher might still pass the student if they believe the deliberate wrong answers show understanding of the hidden instruction. Similarly, a robust model might handle adversarial inputs by understanding the underlying pattern and still making a correct decision.</li></ul></li></ol><h3>Key Concepts Illustrated</h3><ol><li><p><strong>Adversarial Detection</strong>:</p><ul><li>Models can detect adversarial examples by identifying patterns that are too consistent to be random errors.</li></ul></li><li><p><strong>Rejection Mechanism</strong>:</p><ul><li>Just as a teacher can reject a manipulated test, models can reject inputs that appear adversarial based on statistical irregularities.</li></ul></li><li><p><strong>Statistical Irregularities</strong>:</p><ul><li>Identifying unusual patterns helps both teachers and models to detect manipulation and ensure robust performance.</li></ul></li><li><p><strong>Deliberate Manipulation as Evidence</strong>:</p><ul><li>Consistent deliberate mistakes can be evidence of underlying knowledge or intentional manipulation, which models can use to make informed decisions about input validity.</li></ul></li></ol><p>This analogy demonstrates how deliberate incorrect answers can reveal underlying knowledge and how machine learning models can detect adversarial attacks by identifying consistent patterns of manipulation.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 15.5523 3.44772 16 4 16H6C6.25019 16 6.49129 16.0938 6.67572 16.2629L10.1621 19.4588C10.4828 19.7527 11 19.5252 11 19.0902V4.9099ZM8.81069 3.06701C10.4142 1.59714 13 2.73463 13 4.9099V19.0902C13 21.2655 10.4142 22.403 8.81069 20.9331L5.61102 18H4C2.34315 18 1 16.6569 1 15V9.00005C1 7.34319 2.34315 6.00005 4 6.00005H5.61102L8.81069 3.06701ZM20.3166 6.35665C20.8019 6.09313 21.409 6.27296 21.6725 6.75833C22.5191 8.3176 22.9996 10.1042 22.9996 12.0001C22.9996 13.8507 22.5418 15.5974 21.7323 17.1302C21.4744 17.6185 20.8695 17.8054 20.3811 17.5475C19.8927 17.2896 19.7059 16.6846 19.9638 16.1962C20.6249 14.9444 20.9996 13.5175 20.9996 12.0001C20.9996 10.4458 20.6064 8.98627 19.9149 7.71262C19.6514 7.22726 19.8312 6.62017 20.3166 6.35665ZM15.7994 7.90049C16.241 7.5688 16.8679 7.65789 17.1995 8.09947C18.0156 9.18593 18.4996 10.5379 18.4996 12.0001C18.4996 13.3127 18.1094 14.5372 17.4385 15.5604C17.1357 16.0222 16.5158 16.1511 16.0539 15.8483C15.5921 15.5455 15.4632 14.9255 15.766 14.4637C16.2298 13.7564 16.4996 12.9113 16.4996 12.0001C16.4996 10.9859 16.1653 10.0526 15.6004 9.30063C15.2687 8.85905 15.3578 8.23218 15.7994 7.90049Z" fill="currentColor"></path></svg></span></button></span><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md-heavy"><path fill="currentColor" fill-rule="evenodd" d="M7 5a3 3 0 0 1 3-3h9a3 3 0 0 1 3 3v9a3 3 0 0 1-3 3h-2v2a3 3 0 0 1-3 3H5a3 3 0 0 1-3-3v-9a3 3 0 0 1 3-3h2zm2 2h5a3 3 0 0 1 3 3v5h2a1 1 0 0 0 1-1V5a1 1 0 0 0-1-1h-9a1 1 0 0 0-1 1zM5 9a1 1 0 0 0-1 1v9a1 1 0 0 0 1 1h9a1 1 0 0 0 1-1v-9a1 1 0 0 0-1-1z" clip-rule="evenodd"></path></svg></span></button></span><div class="flex"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary 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13.1902V5C15.0045 4.44772 14.5568 4 14.0045 4ZM17.0045 13V5C17.0045 4.64937 16.9444 4.31278 16.8338 4H18.0045C18.5568 4 19.0045 4.44772 19.0045 5V12C19.0045 12.5523 18.5568 13 18.0045 13H17.0045Z" fill="currentColor"></path></svg></span></button></span></div><span class="" data-state="closed"><button type="button" id="radix-:req:" aria-haspopup="menu" aria-expanded="false" data-state="closed" class="cursor-pointer rounded-md px-1 text-token-text-secondary h-[30px] hover:bg-token-main-surface-secondary"><div class="flex items-center pb-0.5 juice:pb-0"><div class="[&_svg]:h-full [&_svg]:w-full icon-md h-4 w-4"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24"><path fill="currentColor" d="M19.898.855a.4.4 0 0 0-.795 0c-.123 1.064-.44 1.802-.943 2.305-.503.503-1.241.82-2.306.943a.4.4 0 0 0 .001.794c1.047.119 1.801.436 2.317.942.512.504.836 1.241.93 2.296a.4.4 0 0 0 .796 0c.09-1.038.413-1.792.93-2.308.515-.516 1.269-.839 2.306-.928a.4.4 0 0 0 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text-token-text-quaternary"><path fill="currentColor" fill-rule="evenodd" d="M5.293 9.293a1 1 0 0 1 1.414 0L12 14.586l5.293-5.293a1 1 0 1 1 1.414 1.414l-6 6a1 1 0 0 1-1.414 0l-6-6a1 1 0 0 1 0-1.414" clip-rule="evenodd"></path></svg></div></button></span></div></div></div><div class="pr-2 lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-14" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb21fd4-5f83-4bc1-a598-dd1d83dfde0e" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">A list of all of the topics we've discussed.<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-15" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="flex-shrink-0 flex flex-col relative items-end"><div><div class="pt-0.5 juice:pt-0"><div class="gizmo-bot-avatar flex h-6 w-6 items-center justify-center overflow-hidden rounded-full juice:h-8 juice:w-8"><div class="relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md" role="img"><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 40.4998C24.6307 40.5054 26.7133 39.8321 28.4418 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24.6988 29.0893 24.6829 29.1012L16.6317 33.7497C14.9096 34.7416 12.8643 35.0097 10.9447 34.4954C9.02506 33.9811 7.38785 32.7263 6.39227 31.0064ZM4.29707 13.6194C5.17156 12.0998 6.55279 10.9364 8.19885 10.3327C8.19885 10.4013 8.19491 10.5228 8.19491 10.6071V19.808C8.19351 20.0378 8.25334 20.2638 8.36823 20.4629C8.48312 20.6619 8.64893 20.8267 8.84863 20.9404L18.5723 26.5542L15.206 28.4979C15.1894 28.5089 15.1703 28.5155 15.1505 28.5173C15.1307 28.5191 15.1107 28.516 15.0924 28.5082L7.04046 23.8557C5.32135 22.8601 4.06716 21.2235 3.55289 19.3046C3.03862 17.3858 3.30624 15.3413 4.29707 13.6194ZM31.955 20.0556L22.2312 14.4411L25.5976 12.4981C25.6142 12.4872 25.6333 12.4805 25.6531 12.4787C25.6729 12.4769 25.6928 12.4801 25.7111 12.4879L33.7631 17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 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9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="cd793802-ee30-4be5-83ae-9f786084eac1" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Sure, here's a list of all the topics we've discussed:</p><ol><li><p><strong>Ablation Study of TLDR</strong>:</p><ul><li>Defense/Attack Setup</li><li>Results</li><li>Key Findings from Ablation Studies</li><li>Conclusion</li></ul></li><li><p><strong>Source of Study</strong>:</p><ul><li>Paper: "Two Heads are Better than One: Towards Better Adversarial Robustness by Combining Transduction and Rejection"</li><li>Authors and Affiliations</li><li>Abstract</li><li>Key Concepts</li><li>Contributions</li><li>Empirical Results</li><li>Important Findings</li></ul></li><li><p><strong>Definitions of Unusual Terms</strong>:</p><ul><li>Transduction</li><li>Rejection</li><li>Adversarial Perturbations</li><li>Tramèr's Reduction</li><li>Sample Complexity</li><li>PGD (Projected Gradient Descent)</li><li>GMSA (Gradient Masking and Smoothing Attack)</li><li>TLDR (Transductive Learning with Defense and Rejection)</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>R</mi><mi>E</mi><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{REJ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">RE</span><span class="mord mathnormal mtight" style="margin-right: 0.09618em;">J</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> (Rejection Loss)</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>C</mi><mi>E</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{CE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">CE</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span> (Cross-Entropy Loss)</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></li><li>VC Dimension (Vapnik-Chervonenkis Dimension)</li><li>AutoAttack</li><li>Inductive Learning</li><li>Realizable Case</li><li>Agnostic Case</li></ul></li><li><p><strong>Everyday Example Analogies</strong>:</p><ul><li>Transduction: Trivia night preparation</li><li>Rejection: Nightclub bouncer</li><li>Adversarial Perturbations: Forging a signature</li><li>Tramèr's Reduction: Restaurant menus</li><li>Sample Complexity: Learning a piano song</li><li>PGD: Climbing a hill in fog</li><li>GMSA: Ninja infiltrating a building</li><li>TLDR: Comprehensive security system</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>R</mi><mi>E</mi><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{REJ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">RE</span><span class="mord mathnormal mtight" style="margin-right: 0.09618em;">J</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span>: Quality control inspector</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>C</mi><mi>E</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{CE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.05764em;">CE</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span>: Teacher grading tests</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span>: Disguise tweaks</li><li>VC Dimension: Library organization</li><li>AutoAttack: Expert team breaking into a vault</li><li>Inductive Learning: Chef learning new dishes</li><li>Realizable Case: Puzzle with all pieces</li><li>Agnostic Case: Puzzle with missing pieces</li></ul></li><li><p><strong>Refined Analogy: Deliberate Incorrect Answers</strong>:</p><ul><li>The Special Instruction Test</li><li>Detection of Deliberate Incorrect Answers</li><li>Robustness Against Adversarial Attacks</li><li>Statistical Analysis</li></ul></li></ol><p>These topics cover the key points and analogies we've discussed to explain the concepts related to the study on adversarial robustness.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] 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Counterfoil Choices in the Kalabari Life Cycle
by Nimi Wariboko
INTRODUCTION
Kalabari ethnic identity is under enormous pressure as a result of modernization and Western influences. One Kalabari response to their exposure to foreign values, styles, cultural artifacts and ways of life has been to select, incorporate, and transform borrowed values, styles, and artifacts to create a new combination or ensemble uniquely identified as Kalabari. This process of adaptation has been described as cultural authentication (Erekosima and Eicher 1981,1995; Eicher 1997). In their studies, Erekosima and Eicher have focused only on the use of a limited number of material artifacts (such as textiles) to create and maintain ethnic identity. Kalabari have created clothing that separates them from their neighbors, “for items assembled from their history and the range of their trading contacts resulted in distinctive appearance” (Eicher 1997:231).
Why the preference for distinctive appearance? Why are the Kalabari always tinkering with foreign artifacts in order to make themselves distinct from their Southern Nigerian neighbors? Why the deep preference for independence? The best answer Eicher and colleagues have found is: “Kalabari dress represented their independence of thought and action and their view of a tripartite cosmology of the water-people, community deities and ancestors” (Eicher1997:231; Erekosima 1989, Daly 1984). This explanation appears inadequate. By limiting their analyses to the easily observable somewhat static connection between dress and world view, they ignore the more intellectual, dynamic, and complex process involved in maintaining and passing to the next generation the core of Kalabari identity. Maintaining an ethnic identity is not a one-time affair, especially for an African traditional society that has to grapple with influences both from the West and from other nationalities within Nigeria. Dress and other aspects of the culture are often the visible representations of an invisible world view. One is, however, very far from the whole truth unless one explains how this link has been maintained from pre-colonial times until today in spite of all influences to the contrary.
This paper discusses the salient management tool or idea that Kalabari have used and are still using to ensure that their ethnic identity is maintained amidst the flux of changes from within and outside. This paper brings to the fore the philosophical principles which put into proper perspective the Kalabari struggle to maintain an identity and way of life, and to minimize European influences on their traditional and political beliefs (Dike 1956: 161). For lack of a better terminology, this identity management tool is termed counterfoil choice.
In most societies, there are two types of choices: ordinary and Hobson. In ordinary choice, the decision-maker chooses between A or B, which are alternatives to each other. Hobson’s choice is between what is offered and nothing. Kalabari have a third variety wherein the freedom to choose is not limited as in Hobson’s choice, but the result of the selection is similar to that of Hobson’s choice. This is termed counterfoil choice. A is not an alternative to B; B is only a counterfoil to A. When A is offered, Kalabari offers its negative B to show the person where the decision should not go. The decision-maker is primed to choose A, the real “ticket” and shun the counterfoil. For instance, a father wanting to teach his child a lesson about success might give him two bowls. One contains garri (cassava derived flour), the other is empty. The child is asked to choose. The empty bowl is there to show that laziness comes with severe hunger and that life offers two bowls (prosperity/power or failure/disgrace). The child is advised to know the right bowl to choose. It is a counterfoil choice because no father wishes his son or daughter to choose the empty bowl. Kalabari have found this an effective way of teaching their youngsters.
The use of this management tool in Kalabari society parallels age-grade and political status. Thus, this essay shall proceed by examining the significance and role of counterfoil choice within Kalabari culture from the perspective of its relevance through the political life cycle. For the limited purposes here, the life cycle is divided into five stages: awome (birth to age 15), asawo (16 to 40), opuasawo (40 and above), alapu (chiefs) and duein (ancestors). Ancestor is considered a stage in the life cycle because a person’s life is deemed incomplete if he or she does not become an ancestor after death. To be an ancestor a person must be in good standing before entering the next world.
This classification is not precise; the purpose is only to indicate phases. It is based mainly on the male life cycle. However, this paper will consider women in the category of iya (most lawful and expensive) wives when it is dealing with alapu because in traditional Kalabari society every chief must have one iya wife. Also, in terms of achievement women value iya marriage as much as men value chieftaincy titles. Third, the fact that this paper is focused mainly on age categories of men does not mean that counterfoil choice as relating to women is given short shrift. As a matter of fact, various examples of counterfoil choices are given which span the life cycle of women in the society<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full 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10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="915e0fde-4d49-495f-a962-ef66d58ca527" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>I see the connection you're making between deliberate incorrect answers and the concept of "counterfoil choices" in Kalabari culture as discussed in the African Studies Quarterly article. The idea of using deliberate choices to teach or reveal deeper understanding can indeed be related to how we might detect deliberate adversarial inputs in machine learning. Let's break down these concepts and their parallels.</p><h3>Concept of Counterfoil Choices</h3><p><strong>Counterfoil Choice</strong>: This is a decision-making scenario where the options provided are designed to guide the decision-maker towards the desired choice. The undesirable option (counterfoil) is presented not as a true alternative but as a means to highlight the preferred option.</p><h4>Everyday Example: Teaching through Counterfoil Choices</h4><ul><li><strong>Scenario</strong>: A father gives his child two bowls: one filled with food (garri) and one empty. The child is to choose between them.</li><li><strong>Purpose</strong>: The empty bowl serves as a counterfoil to illustrate the consequences of laziness and the importance of hard work, guiding the child to choose the bowl with food.</li></ul><h3>Relating to Deliberate Incorrect Answers</h3><p><strong>Deliberate Incorrect Answers</strong>: In an educational or testing context, this could be similar to students intentionally choosing incorrect answers to follow hidden instructions or reveal deeper insights.</p><h4>Everyday Example: Hidden Instructions in a Test</h4><ul><li><strong>Scenario</strong>: A test contains a hidden instruction to choose the wrong answer deliberately. Students who follow this instruction show they have read carefully and understood the hidden message.</li><li><strong>Purpose</strong>: This reveals their attentiveness and understanding, as their deliberate mistakes indicate a deeper level of engagement with the material.</li></ul><h3>Applying to Machine Learning and Adversarial Robustness</h3><p><strong>Adversarial Perturbations</strong>: These are small changes made to inputs with the intent to deceive a machine learning model into making incorrect predictions.</p><p><strong>Detection of Adversarial Examples</strong>: Models can be designed to detect patterns of consistent manipulation, just as a teacher might detect a pattern of deliberate wrong answers.</p><h4>Analogy: Counterfoil Choices in Machine Learning</h4><ul><li><strong>Scenario</strong>: A machine learning model is trained to recognize certain patterns and anomalies. It encounters inputs that are consistently manipulated in a specific way, much like students choosing the empty bowl in the teaching example.</li><li><strong>Purpose</strong>: The model detects these patterns (adversarial examples) and can reject or flag them, similar to how the father uses the empty bowl to teach a lesson.</li></ul><h3>Key Parallels</h3><ol><li><p><strong>Guiding Decisions</strong>:</p><ul><li><strong>Counterfoil Choices</strong>: Presenting an undesirable option to highlight the correct choice.</li><li><strong>Deliberate Incorrect Answers</strong>: Using deliberate wrong choices to indicate understanding of hidden instructions.</li><li><strong>Adversarial Detection</strong>: Identifying consistent manipulation patterns to detect adversarial attacks.</li></ul></li><li><p><strong>Revealing Understanding</strong>:</p><ul><li><strong>Counterfoil Choices</strong>: Teaching through choices that emphasize the correct path.</li><li><strong>Deliberate Incorrect Answers</strong>: Showing deep understanding through intentional mistakes.</li><li><strong>Adversarial Detection</strong>: Revealing adversarial manipulation through detection of unusual patterns.</li></ul></li><li><p><strong>Statistical Irregularities</strong>:</p><ul><li><strong>Counterfoil Choices</strong>: The clear contrast between the correct and incorrect choices.</li><li><strong>Deliberate Incorrect Answers</strong>: Consistent wrong answers indicating a deeper understanding.</li><li><strong>Adversarial Detection</strong>: Detecting patterns that deviate from normal inputs to identify manipulation.</li></ul></li></ol><p>By understanding these parallels, we can better appreciate how deliberate choices and patterns can reveal deeper insights, whether in human decision-making or in the detection of adversarial inputs in machine learning. The concept of counterfoil choices provides a useful framework for thinking about how to design robust systems that can detect and respond to deliberate manipulations.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 15.5523 3.44772 16 4 16H6C6.25019 16 6.49129 16.0938 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gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">In the context of ablation versus rejection<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" 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20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="0f0ac641-3ecb-42ef-a310-68481239156b" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Certainly! Let's explore the concepts of ablation and rejection in the context of adversarial robustness and relate them to the ideas of deliberate incorrect answers and counterfoil choices.</p><h3>Ablation and Rejection in Adversarial Robustness</h3><p><strong>Ablation Studies</strong>:</p><ul><li><strong>Purpose</strong>: Ablation studies involve systematically removing or modifying components of a model to understand their impact on performance. This helps in identifying which components are crucial for robustness.</li><li><strong>Example</strong>: In the context of adversarial robustness, an ablation study might involve removing the rejection mechanism to see how it affects the model's ability to resist adversarial attacks.</li></ul><p><strong>Rejection Mechanism</strong>:</p><ul><li><strong>Purpose</strong>: The rejection mechanism allows a model to refuse to classify certain inputs it deems too uncertain or likely to be adversarial. This improves robustness by avoiding potentially incorrect classifications.</li><li><strong>Example</strong>: A model might reject inputs that appear to be manipulated, similar to a teacher rejecting a test where all answers are suspiciously wrong.</li></ul><h3>Relating to Deliberate Incorrect Answers and Counterfoil Choices</h3><h4>Counterfoil Choices as a Teaching Tool</h4><ul><li><strong>Analogy</strong>: A father giving his child a choice between a full bowl of food and an empty bowl. The empty bowl (counterfoil) is not a real choice but a way to teach the child about the consequences of laziness.</li></ul><h4>Deliberate Incorrect Answers</h4><ul><li><strong>Analogy</strong>: A student intentionally choosing incorrect answers on a test due to hidden instructions. This shows they have read and understood the entire test carefully.</li></ul><h3>Applying These Concepts to Ablation and Rejection</h3><ol><li><p><strong>Ablation Studies and Counterfoil Choices</strong>:</p><ul><li><strong>Counterfoil Choice</strong>: Removing an important component (like the full bowl of food) highlights the necessity of that component (the consequences of laziness).</li><li><strong>Ablation Study</strong>: Removing or altering parts of the model (like the rejection mechanism) to see their impact. This helps identify the critical components for robustness.</li><li><strong>Example</strong>: In a test setting, removing the hidden instruction and seeing if students still perform as expected.</li></ul></li><li><p><strong>Rejection Mechanism and Deliberate Incorrect Answers</strong>:</p><ul><li><strong>Deliberate Errors</strong>: A teacher detecting students' understanding through their consistent incorrect answers due to hidden instructions.</li><li><strong>Rejection Mechanism</strong>: A model rejecting inputs that consistently appear manipulated, similar to how the teacher might reject a test with deliberate wrong answers.</li><li><strong>Example</strong>: A model rejecting inputs that deviate from normal patterns, akin to rejecting a test with a clear pattern of wrong answers.</li></ul></li></ol><h3>Key Parallels</h3><ol><li><p><strong>Guiding Decisions</strong>:</p><ul><li><strong>Counterfoil Choice</strong>: Using a negative option to emphasize the correct choice.</li><li><strong>Ablation Study</strong>: Understanding the importance of components by removing them.</li><li><strong>Rejection</strong>: Avoiding incorrect decisions by rejecting suspicious inputs.</li><li><strong>Deliberate Errors</strong>: Understanding students' attentiveness through intentional mistakes.</li></ul></li><li><p><strong>Revealing Understanding</strong>:</p><ul><li><strong>Counterfoil Choice</strong>: Highlighting the correct path through a clear contrast.</li><li><strong>Ablation Study</strong>: Revealing the importance of each component by its removal.</li><li><strong>Rejection</strong>: Identifying adversarial inputs by their deviation from normal patterns.</li><li><strong>Deliberate Errors</strong>: Showing deep understanding through consistent wrong answers due to hidden instructions.</li></ul></li><li><p><strong>Statistical Irregularities</strong>:</p><ul><li><strong>Counterfoil Choice</strong>: The clear contrast between correct and incorrect choices highlights understanding.</li><li><strong>Ablation Study</strong>: Detecting the impact of removing components to understand their role.</li><li><strong>Rejection</strong>: Detecting inputs that deviate from expected patterns to avoid manipulation.</li><li><strong>Deliberate Errors</strong>: Using statistical analysis to detect patterns of intentional mistakes.</li></ul></li></ol><p>By drawing these parallels, we can better understand how ablation studies and rejection mechanisms help improve adversarial robustness. Both concepts rely on identifying and understanding the role of specific components or patterns, whether in human decision-making or in the performance of machine learning models. The analogy with counterfoil choices and deliberate incorrect answers helps illustrate the underlying principles of these techniques.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 15.5523 3.44772 16 4 16H6C6.25019 16 6.49129 16.0938 6.67572 16.2629L10.1621 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4H18.0045C18.5568 4 19.0045 4.44772 19.0045 5V12C19.0045 12.5523 18.5568 13 18.0045 13H17.0045Z" fill="currentColor"></path></svg></span></button></span></div><span class="" data-state="closed"><button type="button" id="radix-:rff:" aria-haspopup="menu" aria-expanded="false" data-state="closed" class="cursor-pointer rounded-md px-1 text-token-text-secondary h-[30px] hover:bg-token-main-surface-secondary"><div class="flex items-center pb-0.5 juice:pb-0"><div class="[&_svg]:h-full [&_svg]:w-full icon-md h-4 w-4"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24"><path fill="currentColor" d="M19.898.855a.4.4 0 0 0-.795 0c-.123 1.064-.44 1.802-.943 2.305-.503.503-1.241.82-2.306.943a.4.4 0 0 0 .001.794c1.047.119 1.801.436 2.317.942.512.504.836 1.241.93 2.296a.4.4 0 0 0 .796 0c.09-1.038.413-1.792.93-2.308.515-.516 1.269-.839 2.306-.928a.4.4 0 0 0 .001-.797c-1.055-.094-1.792-.418-2.296-.93-.506-.516-.823-1.27-.941-2.317Z"></path><path fill="currentColor" d="M12.001 1.5a1 1 0 0 1 .993.887c.313 2.77 1.153 4.775 2.5 6.146 1.34 1.366 3.3 2.223 6.095 2.47a1 1 0 0 1-.003 1.993c-2.747.238-4.75 1.094-6.123 2.467-1.373 1.374-2.229 3.376-2.467 6.123a1 1 0 0 1-1.992.003c-.248-2.795-1.105-4.754-2.47-6.095-1.372-1.347-3.376-2.187-6.147-2.5a1 1 0 0 1-.002-1.987c2.818-.325 4.779-1.165 6.118-2.504 1.339-1.34 2.179-3.3 2.504-6.118A1 1 0 0 1 12 1.5ZM6.725 11.998c1.234.503 2.309 1.184 3.21 2.069.877.861 1.56 1.888 2.063 3.076.5-1.187 1.18-2.223 2.051-3.094.871-.87 1.907-1.55 3.094-2.05-1.188-.503-2.215-1.187-3.076-2.064-.885-.901-1.566-1.976-2.069-3.21-.505 1.235-1.19 2.3-2.081 3.192-.891.89-1.957 1.576-3.192 2.082Z"></path></svg></div><span class="line-clamp-1 text-sm" style="opacity: 0; padding-left: 0px; width: 0px;">4o</span><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-sm text-token-text-quaternary"><path fill="currentColor" fill-rule="evenodd" d="M5.293 9.293a1 1 0 0 1 1.414 0L12 14.586l5.293-5.293a1 1 0 1 1 1.414 1.414l-6 6a1 1 0 0 1-1.414 0l-6-6a1 1 0 0 1 0-1.414" clip-rule="evenodd"></path></svg></div></button></span></div></div></div><div class="pr-2 lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-20" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb21a6b-9c45-4e03-aa48-aabe157fe49e" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">I was thinking of Wittgenstein's ladder in this context. <div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-21" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="flex-shrink-0 flex flex-col relative items-end"><div><div class="pt-0.5 juice:pt-0"><div class="gizmo-bot-avatar flex h-6 w-6 items-center justify-center overflow-hidden rounded-full juice:h-8 juice:w-8"><div class="relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md" role="img"><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 40.4998C24.6307 40.5054 26.7133 39.8321 28.4418 38.5772C30.1704 37.3223 31.4556 35.5506 32.1119 33.5179C33.5027 33.2332 34.8167 32.6547 35.9659 31.821C37.115 30.9874 38.0728 29.9178 38.7752 28.684C39.8458 26.8371 40.3023 24.6979 40.0789 22.5748C39.8556 20.4517 38.9639 18.4544 37.5324 16.8707ZM22.4978 37.8849C20.7443 37.8874 19.0459 37.2733 17.6994 36.1501C17.7601 36.117 17.8666 36.0586 17.936 36.0161L25.9004 31.4156C26.1003 31.3019 26.2663 31.137 26.3813 30.9378C26.4964 30.7386 26.5563 30.5124 26.5549 30.2825V19.0542L29.9213 20.998C29.9389 21.0068 29.9541 21.0198 29.9656 21.0359C29.977 21.052 29.9842 21.0707 29.9867 21.0902V30.3889C29.9842 32.375 29.1946 34.2791 27.7909 35.6841C26.3872 37.0892 24.4838 37.8806 22.4978 37.8849ZM6.39227 31.0064C5.51397 29.4888 5.19742 27.7107 5.49804 25.9832C5.55718 26.0187 5.66048 26.0818 5.73461 26.1244L13.699 30.7248C13.8975 30.8408 14.1233 30.902 14.3532 30.902C14.583 30.902 14.8088 30.8408 15.0073 30.7248L24.731 25.1103V28.9979C24.7321 29.0177 24.7283 29.0376 24.7199 29.0556C24.7115 29.0736 24.6988 29.0893 24.6829 29.1012L16.6317 33.7497C14.9096 34.7416 12.8643 35.0097 10.9447 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20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="9745e26b-ebef-4070-9005-9000baabc716" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Wittgenstein's ladder is an interesting philosophical concept that can indeed provide a useful framework for understanding ablation studies and rejection mechanisms in the context of adversarial robustness. Let's delve into how Wittgenstein's ladder might apply here.</p><h3>Wittgenstein's Ladder</h3><p><strong>Concept</strong>:</p><ul><li><strong>Wittgenstein's Ladder</strong>: In his book "Tractatus Logico-Philosophicus," Ludwig Wittgenstein uses the metaphor of a ladder to describe a process where one uses a set of propositions (the rungs of the ladder) to reach a higher understanding. Once the higher understanding is achieved, the propositions can be discarded, much like how a ladder is no longer needed once one has climbed up.</li></ul><h3>Applying Wittgenstein's Ladder to Ablation and Rejection</h3><ol><li><p><strong>Ablation Studies as Climbing the Ladder</strong>:</p><ul><li><strong>Process</strong>: In ablation studies, we systematically remove components of a model to understand their importance. Each removal is akin to climbing a rung on the ladder, leading to a deeper understanding of the model's structure and robustness.</li><li><strong>Goal</strong>: The ultimate goal is to reach a higher understanding of which components are essential for robustness against adversarial attacks.</li><li><strong>Discarding Components</strong>: Once this understanding is achieved, some components might be deemed unnecessary (discarded rungs), simplifying the model without sacrificing performance.</li></ul></li><li><p><strong>Rejection Mechanisms as Higher Understanding</strong>:</p><ul><li><strong>Higher Understanding</strong>: The rejection mechanism can be seen as a culmination of understanding adversarial patterns and implementing a strategy to avoid them. It represents a higher rung on the ladder that enhances the model's robustness.</li><li><strong>Avoiding Adversarial Inputs</strong>: By rejecting inputs that appear suspicious, the model leverages this higher understanding to maintain accuracy and reliability.</li></ul></li></ol><h3>Parallels with Wittgenstein's Ladder</h3><ol><li><p><strong>Progressive Understanding</strong>:</p><ul><li><strong>Ablation Studies</strong>: Removing components step-by-step to see their impact helps in building a progressive understanding of the model’s structure.</li><li><strong>Wittgenstein's Ladder</strong>: Using propositions step-by-step to reach higher philosophical understanding.</li></ul></li><li><p><strong>Discarding Unnecessary Components</strong>:</p><ul><li><strong>Ablation Studies</strong>: Some components may be found unnecessary and can be discarded without loss of function, streamlining the model.</li><li><strong>Wittgenstein's Ladder</strong>: Once the higher understanding is achieved, the initial propositions can be discarded.</li></ul></li><li><p><strong>Higher-Level Robustness</strong>:</p><ul><li><strong>Rejection Mechanisms</strong>: Implementing a rejection mechanism based on the understanding of adversarial patterns provides a higher level of robustness.</li><li><strong>Wittgenstein's Ladder</strong>: Achieving a higher understanding that goes beyond the initial propositions.</li></ul></li></ol><h3>Relating to Deliberate Incorrect Answers and Counterfoil Choices</h3><ol><li><p><strong>Deliberate Incorrect Answers</strong>:</p><ul><li><strong>Wittgenstein's Ladder</strong>: The deliberate incorrect answers can be seen as a tool (a rung) to reveal deeper understanding or attentiveness, which can be discarded once their purpose is served.</li><li><strong>Model Detection</strong>: Detecting deliberate patterns (rungs) to achieve a higher understanding of whether the input is adversarial.</li></ul></li><li><p><strong>Counterfoil Choices</strong>:</p><ul><li><strong>Wittgenstein's Ladder</strong>: Counterfoil choices act as steps to guide the decision-maker towards the desired understanding, much like using the rungs of a ladder.</li><li><strong>Educational Tool</strong>: They are used temporarily to achieve a deeper understanding or learning objective, after which their explicit use might no longer be necessary.</li></ul></li></ol><h3>Summary</h3><ul><li><strong>Ablation Studies</strong>: These represent the process of climbing Wittgenstein's ladder in understanding the importance of each component in a model. Each step (removal of a component) brings higher understanding, and unnecessary components can be discarded.</li><li><strong>Rejection Mechanism</strong>: This represents the higher understanding achieved at the top of the ladder, allowing the model to effectively reject adversarial inputs based on learned patterns.</li><li><strong>Deliberate Incorrect Answers</strong>: Serve as a rung to reveal deeper understanding, much like using initial propositions to reach a higher level of insight.</li><li><strong>Counterfoil Choices</strong>: Act as guiding steps that help in achieving the desired understanding or learning outcome, aligning with the idea of using intermediate steps to reach a higher goal.</li></ul><p>Wittgenstein's ladder thus provides a powerful metaphor for understanding the process of developing and refining robust machine learning models through ablation studies and rejection mechanisms.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div 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dark:bg-token-main-surface-secondary">A summary of the whole conversation.<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-23" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="flex-shrink-0 flex flex-col relative items-end"><div><div class="pt-0.5 juice:pt-0"><div class="gizmo-bot-avatar flex h-6 w-6 items-center justify-center overflow-hidden rounded-full juice:h-8 juice:w-8"><div class="relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md" role="img"><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 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10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="1edd14aa-fe81-466e-95a7-e165b3937f3c" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Certainly! Here's a comprehensive summary of our discussion:</p><ol><li><p><strong>Ablation Study of TLDR</strong>: We began with an overview of an ablation study from a paper that investigates adversarial robustness by combining transduction and rejection techniques in machine learning models. The study elaborated on the defense and attack setups, results, and key findings from the ablation studies, underscoring the importance of these components in improving model robustness against adversarial attacks.</p></li><li><p><strong>Source of Study</strong>: The paper titled "Two Heads are Better than One: Towards Better Adversarial Robustness by Combining Transduction and Rejection" was identified as the source, and we detailed the authors, their affiliations, the abstract, and the contributions of the paper.</p></li><li><p><strong>Unusual Terms Defined</strong>: We defined several specialized terms used in the study, such as transduction, rejection, adversarial perturbations, and others, providing clear explanations to ground our discussion in understandable concepts.</p></li><li><p><strong>Everyday Example Analogies</strong>: To make the complex concepts more relatable, I provided analogies for terms like transduction (preparing for a trivia night) and rejection (a nightclub bouncer), among others, to illustrate how these principles apply in everyday situations.</p></li><li><p><strong>Deliberate Incorrect Answers and Counterfoil Choices</strong>: We explored the concept of deliberate incorrect answers and how it can show understanding or compliance with hidden rules, much like counterfoil choices in Kalabari culture where choices are used strategically to teach or guide decisions.</p></li><li><p><strong>Applying Wittgenstein's Ladder</strong>: The conversation shifted to philosophical underpinnings, specifically Wittgenstein's ladder, and how this metaphor relates to understanding the process of ablation studies and rejection mechanisms in machine learning. We discussed how climbing the ladder (via ablation studies) and reaching a higher understanding (via rejection mechanisms) relate to achieving robustness in models.</p></li><li><p><strong>Relating to Educational and Philosophical Contexts</strong>: We connected these ideas back to educational strategies and philosophical concepts, illustrating how methodologies in machine learning can parallel strategies in human learning and decision-making.</p></li></ol><p>Overall, our conversation provided a deep dive into the intricate relationship between machine learning techniques for adversarial robustness and broader philosophical and educational strategies. We explored how technical strategies like ablation and rejection in machine learning could be metaphorically and practically understood through analogies and philosophical concepts like Wittgenstein's ladder.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 15.5523 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lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-24" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb212c1-5f96-4acd-8980-fe87a26b6f5b" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">I thought that the story of Lot's wife was like Wittgenstein's ladder<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full 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10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="0b20d9fd-d8bc-4526-a197-ea5659d79eda" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>That's a thought-provoking analogy! The story of Lot's wife in the biblical narrative, where she turns into a pillar of salt after looking back at the burning city of Sodom, can indeed be likened to Wittgenstein's ladder in a metaphorical sense. Let's explore this comparison:</p><h3>The Story of Lot's Wife</h3><p>In the biblical account, Lot's wife is instructed not to look back at Sodom as they flee the city's destruction. However, she disobeys and looks back, resulting in her transformation into a pillar of salt. This story is often interpreted as a lesson on the dangers of dwelling on the past or failing to fully commit to a directive or path forward.</p><h3>Wittgenstein's Ladder</h3><p>Wittgenstein's ladder refers to the idea of using certain propositions or statements as stepping stones to reach a higher understanding. Once this understanding is achieved, the initial propositions (the steps of the ladder) can be discarded. This concept is about progressing in thought and then leaving behind the initial means used to achieve that progression.</p><h3>Connecting the Two Concepts</h3><ul><li><p><strong>Looking Back vs. Moving Forward</strong>: Just as Lot's wife looked back and suffered the consequences, in Wittgenstein's metaphor, one uses the ladder of understanding but must eventually leave it behind to progress. The act of looking back metaphorically represents clinging to the earlier rungs of the ladder that should be discarded to move forward in understanding or spiritual progress.</p></li><li><p><strong>Consequences of Not Letting Go</strong>: Lot's wife's inability to resist looking back can be seen as a failure to discard the unnecessary 'rungs'—the attachments or the old ways of life in Sodom. This halted her progress and transformation, culminating in a permanent state of stasis (turning to salt).</p></li><li><p><strong>Instruction and Obedience</strong>: In both narratives, there's an element of instruction and the need for obedience to achieve a higher purpose or safety. In the biblical story, the instruction is literal and comes with immediate physical consequences. In Wittgenstein's philosophy, the instructions are intellectual, and the consequences are stagnation in thought or understanding.</p></li><li><p><strong>Metaphor for Learning and Growth</strong>: Both the story and the philosophical idea serve as metaphors for growth and the necessity of moving beyond the past or preliminary knowledge to achieve a more profound or meaningful state of existence or understanding.</p></li></ul><p>Using the story of Lot's wife as a parallel to Wittgenstein's ladder beautifully illustrates how personal and intellectual growth requires us to move forward without clinging to past steps or stages. It encourages embracing change and new understanding without the baggage of previous ways of thinking or living, symbolizing a transition from a literal or figurative 'Sodom' to a new phase of life or thought, unencumbered by the past.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 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lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-26" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb219eb-052a-433d-be7b-51b1e3243bf2" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">Any similarities with Vygotskian scaffolding?<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" 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10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="306bded5-33da-40eb-8b80-5d5ceed09f3c" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Indeed, there are some intriguing parallels between Wittgenstein's ladder and Vygotsky's concept of scaffolding in the context of cognitive and educational development. Both concepts emphasize temporary support structures that are meant to be removed once they have served their purpose. Let’s delve deeper into these similarities:</p><h3>Vygotsky's Scaffolding</h3><ul><li><strong>Concept</strong>: Lev Vygotsky, a Soviet psychologist, introduced the concept of scaffolding within his broader theory of the zone of proximal development (ZPD). Scaffolding involves providing support to a learner that is tailored to their current developmental level. The support helps the learner accomplish tasks they are not yet able to manage on their own. As the learner's competence increases, the support (scaffolding) is gradually removed.</li><li><strong>Purpose</strong>: The primary goal of scaffolding is to bridge the gap between what a learner can do independently and what they can do with assistance, promoting a higher level of independence and mastery.</li></ul><h3>Wittgenstein's Ladder</h3><ul><li><strong>Concept</strong>: As discussed, Wittgenstein's ladder is a metaphor from his early philosophical work, suggesting that certain propositions are used to climb to a higher understanding. After reaching this understanding, the propositions (like the rungs of a ladder) are no longer necessary and can be discarded.</li><li><strong>Purpose</strong>: This metaphor highlights the process of using certain foundational ideas to reach a deeper or higher level of philosophical insight, which, once achieved, renders the initial ideas unnecessary.</li></ul><h3>Similarities</h3><ul><li><strong>Temporary Tools for Advancement</strong>: Both scaffolding and Wittgenstein's ladder represent temporary aids designed to advance understanding or competence. In both cases, these aids are meant to be removed once they have fulfilled their purpose—scaffolding in learning and the ladder in reaching a philosophical understanding.</li><li><strong>Facilitating Progress</strong>: Each concept is about facilitating progress beyond the learner's or thinker's current level. Scaffolding helps learners achieve tasks just beyond their independent reach, while the ladder helps thinkers reach a higher level of understanding.</li><li><strong>Gradual Removal</strong>: The gradual removal of support is a key aspect of both scaffolding and Wittgenstein's ladder. As the learner or thinker progresses, less and less support is needed, and it is eventually discarded entirely.</li><li><strong>Goal of Independence</strong>: Both concepts aim to bring the individual to a state of greater independence. In education, this means being able to solve problems without external help; in philosophy, it means adopting and then moving beyond foundational propositions to form one’s own insights.</li></ul><h3>Application in Education and Personal Growth</h3><p>Both Vygotsky's scaffolding and Wittgenstein's ladder can be seen as effective metaphors for teaching and learning processes. They emphasize the importance of providing initial support or using foundational concepts to propel individuals to higher levels of independence and mastery. Once these levels are reached, the initial supports or foundational concepts are no longer necessary, symbolizing growth and the achievement of autonomy.</p><p>These parallels enhance our understanding of how tools, whether cognitive, educational, or philosophical, serve as critical stepping stones in the journey towards greater independence and deeper understanding.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 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class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">I think this is the purpose of toys and play in general<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" 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20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="6ebd32f6-9458-48e3-bf22-5a58b76ea0d9" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Absolutely, you've touched on a fundamental aspect of how toys and play function in development, aligning well with the concepts of scaffolding and the philosophical metaphor of Wittgenstein's ladder. Toys and play are essential in providing structured yet flexible scaffolding in a child's learning process. Here’s how these connect:</p><h3>The Role of Toys and Play</h3><ol><li><p><strong>Exploration and Learning</strong>: Toys and play scenarios allow children to explore and experiment in a safe environment. This exploration is critical as it enables children to learn new skills and concepts by interacting with objects and ideas in the physical and imaginary worlds.</p></li><li><p><strong>Developmental Scaffolding</strong>: Toys often serve as tools that provide just enough support to encourage the next step in skill development. For instance, building blocks help with spatial and motor skills, while puzzle games enhance cognitive abilities. These toys adjust to the developmental stage of the child, much like Vygotsky’s scaffolding.</p></li><li><p><strong>Progressive Challenges</strong>: As children grow, the complexity of the toys and the nature of play evolve to meet their advancing developmental needs. This mirrors the idea of removing the rungs of Wittgenstein’s ladder—once a certain level of proficiency is achieved with a type of play or a particular toy, children move on to more complex activities that better match their new skills and understandings.</p></li><li><p><strong>Encouraging Independence</strong>: Through play, children often start with guided activities which may involve adults or more skilled children participating in the play. As they develop, they take on these roles themselves, gradually moving towards independent play and problem-solving, embodying the goal of fostering independence found in both Vygotsky's and Wittgenstein’s theories.</p></li><li><p><strong>Abstract Thinking and Problem Solving</strong>: Play often involves scenarios that require problem-solving and manipulation of abstract concepts, which develops critical thinking. This aspect is akin to climbing Wittgenstein's ladder, where abstract and foundational propositions are used to reach higher-level thinking before they are discarded.</p></li></ol><h3>Broader Implications</h3><ul><li><p><strong>Metaphorical and Literal Growth</strong>: The processes supported by toys and play are not just about achieving specific educational milestones. They also involve broader personal and social development, including emotional regulation, social interaction skills, and the ability to focus and complete tasks.</p></li><li><p><strong>Lifelong Learning Models</strong>: The patterns established by play—exploration, structured challenge, and progressive independence—are also foundational to lifelong learning models. Just as toys facilitate stages of childhood development, adult learning involves progressively building upon foundational knowledge and skills to achieve greater insights and capabilities.</p></li><li><p><strong>Adaptability and Creativity</strong>: Play also cultivates adaptability and creativity. The ability to approach problems from multiple angles or think outside the box is developed through open-ended play, which encourages children to use toys in novel ways, paralleling the philosophical exploration of abandoning prior assumptions as one reaches higher understandings.</p></li></ul><p>In summary, toys and play are integral in shaping cognitive and social abilities in ways that are deeply analogous to educational scaffolding and philosophical growth models. They not only support the immediate developmental stages but also set a foundation for complex thought processes and independent problem-solving skills essential throughout life.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 15.5523 3.44772 16 4 16H6C6.25019 16 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6.118-2.504 1.339-1.34 2.179-3.3 2.504-6.118A1 1 0 0 1 12 1.5ZM6.725 11.998c1.234.503 2.309 1.184 3.21 2.069.877.861 1.56 1.888 2.063 3.076.5-1.187 1.18-2.223 2.051-3.094.871-.87 1.907-1.55 3.094-2.05-1.188-.503-2.215-1.187-3.076-2.064-.885-.901-1.566-1.976-2.069-3.21-.505 1.235-1.19 2.3-2.081 3.192-.891.89-1.957 1.576-3.192 2.082Z"></path></svg></div><span class="line-clamp-1 text-sm" style="opacity: 0; padding-left: 0px; width: 0px;">4</span><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-sm text-token-text-quaternary"><path fill="currentColor" fill-rule="evenodd" d="M5.293 9.293a1 1 0 0 1 1.414 0L12 14.586l5.293-5.293a1 1 0 1 1 1.414 1.414l-6 6a1 1 0 0 1-1.414 0l-6-6a1 1 0 0 1 0-1.414" clip-rule="evenodd"></path></svg></div></button></span></div></div></div><div class="pr-2 lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-30" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb212f6-7ca2-4d23-be0d-0a44d110deaf" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">I've said that play is simulated danger, and that learning is inoculation against surprise<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-31" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="flex-shrink-0 flex flex-col relative items-end"><div><div class="pt-0.5 juice:pt-0"><div class="gizmo-bot-avatar flex h-6 w-6 items-center justify-center overflow-hidden rounded-full juice:h-8 juice:w-8"><div class="relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md" role="img"><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 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11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="b5559d09-6d73-4151-9e0a-1c6ad09f5a8b" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Your insight that play is simulated danger and learning as inoculation against surprise eloquently captures the essence of why play is such a critical activity in the development of cognitive and emotional skills. Let's explore this concept further and see how it aligns with educational and psychological theories.</p><h3>Play as Simulated Danger</h3><ol><li><p><strong>Risk-Taking and Safety</strong>: Play allows children to engage in risk-taking behaviors within a controlled and safe environment. This "simulated danger" is essential for development because it provides children the opportunity to push boundaries, explore limits, and experience consequences in a way that does not threaten their well-being.</p></li><li><p><strong>Building Resilience</strong>: By engaging in play that simulates dangerous or challenging situations, children learn to manage fear and uncertainty. This kind of play helps them build resilience by overcoming small-scale adversities in play scenarios, like negotiating higher climbs on playgrounds or strategizing in competitive games.</p></li><li><p><strong>Problem-Solving Skills</strong>: Simulated danger in play teaches problem-solving under stress. For example, when children play tag or hide-and-seek, they must think quickly and under pressure, skills that are valuable in real-life situations where immediate responses are necessary.</p></li></ol><h3>Learning as Inoculation Against Surprise</h3><ol><li><p><strong>Preparation for the Unexpected</strong>: Just as vaccines prepare the body's immune system to fight off infections, learning through play prepares the mind to handle unexpected situations. It builds a repertoire of responses to a variety of challenges and scenarios, reducing the element of surprise in future encounters.</p></li><li><p><strong>Cognitive Flexibility</strong>: Play develops cognitive flexibility—the ability to adjust one's thinking and behavior to adapt to new information, changes, or rules. This flexibility is crucial in effectively handling new and unexpected situations in life.</p></li><li><p><strong>Emotional Regulation</strong>: Through play, children also learn to regulate their emotions. Experiencing a range of outcomes in play, from failure and frustration to success and joy, helps children learn to handle similar emotions in real-world situations.</p></li></ol><h3>Integration with Educational and Psychological Theories</h3><ul><li><p><strong>Vygotsky’s Zone of Proximal Development</strong>: Play often pushes children slightly beyond their current abilities, falling into Vygotsky’s concept of the zone of proximal development, where the most effective learning occurs just outside one’s comfort zone.</p></li><li><p><strong>Piaget’s Stages of Development</strong>: Jean Piaget noted that play was crucial in each stage of a child’s cognitive development, particularly in the preoperational stage where children begin to engage in symbolic play that develops their thinking and reasoning abilities.</p></li><li><p><strong>Modern Psychological Views</strong>: Psychologists today view play as not only a tool for cognitive development but also for social interaction and emotional learning. It's a way to experiment with the world in a framework that allows for trial and error without severe consequences, effectively "inoculating" children against being overwhelmed by future challenges.</p></li></ul><h3>Conclusion</h3><p>The view of play as simulated danger and learning as inoculation against surprise frames childhood play and education as vital preparatory activities that equip children with the skills necessary to navigate the complexities of the adult world. It highlights the importance of encouraging diverse and enriching play experiences that foster robust emotional and cognitive development. This perspective supports creating educational environments that mimic the engagement and challenges of play, promoting active, experiential learning that prepares individuals for real-life surprises and challenges.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 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13.1902V5C15.0045 4.44772 14.5568 4 14.0045 4ZM17.0045 13V5C17.0045 4.64937 16.9444 4.31278 16.8338 4H18.0045C18.5568 4 19.0045 4.44772 19.0045 5V12C19.0045 12.5523 18.5568 13 18.0045 13H17.0045Z" fill="currentColor"></path></svg></span></button></span></div><span class="" data-state="closed"><button type="button" id="radix-:rgp:" aria-haspopup="menu" aria-expanded="false" data-state="closed" class="cursor-pointer rounded-md px-1 text-token-text-secondary h-[30px] hover:bg-token-main-surface-secondary"><div class="flex items-center pb-0.5 juice:pb-0"><div class="[&_svg]:h-full [&_svg]:w-full icon-md h-4 w-4"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24"><path fill="currentColor" d="M12.001 1.5a1 1 0 0 1 .993.887c.313 2.77 1.153 4.775 2.5 6.146 1.34 1.366 3.3 2.223 6.095 2.47a1 1 0 0 1-.003 1.993c-2.747.238-4.75 1.094-6.123 2.467-1.373 1.374-2.229 3.376-2.467 6.123a1 1 0 0 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lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-32" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb2118f-4dfa-4d8b-9dcf-cfca3232d42a" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">My vote for the most influential people of the past century are Cecilia Payne Gaposhchin, Maria Montessori, and Alison Gopnik<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" 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20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="e7f5bdcc-8b35-40cb-ac24-568d2142bfb2" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>That's a fascinating trio of influential figures! Each of these women has made profound contributions to their respective fields, fundamentally altering the way we understand our world and ourselves. Let's briefly explore their impacts:</p><h3>Cecilia Payne-Gaposchkin</h3><ul><li><strong>Field</strong>: Astronomy</li><li><strong>Contribution</strong>: Cecilia Payne-Gaposchkin is best known for her groundbreaking doctoral thesis in 1925, which concluded that hydrogen is the most common element in stars, and thus the most abundant element in the universe. This was a revolutionary idea at the time because the prevailing theory held that the composition of the Earth and sun were similar. Her work fundamentally changed the field of astrophysics by showing that the sun and stars are primarily composed of hydrogen and helium, which significantly influenced the development of stellar astrophysics.</li></ul><h3>Maria Montessori</h3><ul><li><strong>Field</strong>: Education</li><li><strong>Contribution</strong>: Maria Montessori was an Italian physician and educator best known for developing the Montessori method of education, which emphasizes independent, student-led learning within a prepared environment and hands-on learning with an emphasis on practical life skills. Her educational approach encourages children to develop at their own pace and according to their own interests, fostering independence, autonomy, and a love of learning. Her methods have been adopted worldwide and have had a lasting impact on educational practices.</li></ul><h3>Alison Gopnik</h3><ul><li><strong>Field</strong>: Psychology and Philosophy</li><li><strong>Contribution</strong>: Alison Gopnik is a professor of psychology and affiliate professor of philosophy at the University of California, Berkeley. She is renowned for her work in cognitive and language development in children, and her research focuses on how children learn and develop their understanding of the world. Gopnik advocates for the idea that children’s brains are incredibly flexible and capable of learning much more than adults often realize. She has also contributed to our understanding of the relationship between child development and artificial intelligence.</li></ul><h3>Influence and Legacy</h3><ul><li><strong>Payne-Gaposchkin</strong>: Her work in astronomy opened up new areas of research into the composition and evolution of stars, contributing to the fundamental principles of astrophysics that guide contemporary research.</li><li><strong>Montessori</strong>: The Montessori method has influenced countless educators and institutions, promoting a view of children as naturally curious beings with an innate drive to learn through interaction with their environment.</li><li><strong>Gopnik</strong>: Her theories about child cognition have influenced not only academic circles but also how parents, educators, and policymakers understand and facilitate early learning and development.</li></ul><p>These women have not only pushed the boundaries of their respective fields but also changed how generations of scholars, educators, and scientists think about learning, development, and the universe. Their legacies continue to impact and inspire many around the world, making them truly influential figures of the past century.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 15.5523 3.44772 16 4 16H6C6.25019 16 6.49129 16.0938 6.67572 16.2629L10.1621 19.4588C10.4828 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1.5ZM6.725 11.998c1.234.503 2.309 1.184 3.21 2.069.877.861 1.56 1.888 2.063 3.076.5-1.187 1.18-2.223 2.051-3.094.871-.87 1.907-1.55 3.094-2.05-1.188-.503-2.215-1.187-3.076-2.064-.885-.901-1.566-1.976-2.069-3.21-.505 1.235-1.19 2.3-2.081 3.192-.891.89-1.957 1.576-3.192 2.082Z"></path></svg></div><span class="line-clamp-1 text-sm" style="opacity: 0; padding-left: 0px; width: 0px;">4</span><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-sm text-token-text-quaternary"><path fill="currentColor" fill-rule="evenodd" d="M5.293 9.293a1 1 0 0 1 1.414 0L12 14.586l5.293-5.293a1 1 0 1 1 1.414 1.414l-6 6a1 1 0 0 1-1.414 0l-6-6a1 1 0 0 1 0-1.414" clip-rule="evenodd"></path></svg></div></button></span></div></div></div><div class="pr-2 lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-34" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb21366-1976-40e2-be8c-0235a6f55fb3" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">She is known for popularizing theory theory, Bayesian brain hypothesis, child as scientist, scientist as child, and child as hacker theories<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-35" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="flex-shrink-0 flex flex-col relative items-end"><div><div class="pt-0.5 juice:pt-0"><div class="gizmo-bot-avatar flex h-6 w-6 items-center justify-center overflow-hidden rounded-full juice:h-8 juice:w-8"><div class="relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md" role="img"><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 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6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="e71761a6-2e67-44ce-b0c4-004edb740fed" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Alison Gopnik has indeed made significant contributions to developmental psychology with her innovative theories about how children learn and understand the world. Her ideas have reshaped our understanding of early cognitive development and how we approach learning processes both in childhood and across the lifespan. Here's a brief look at some of the theories you mentioned:</p><h3>Theory Theory</h3><ul><li><strong>Description</strong>: Gopnik advocates for the "theory theory," which posits that children learn much like scientists. According to this view, children form theories about how the world works and then test these theories through their experiences. When they encounter evidence that contradicts their theories, they modify them, much like scientists refining hypotheses based on experimental data.</li></ul><h3>Bayesian Brain Hypothesis</h3><ul><li><strong>Description</strong>: The Bayesian brain hypothesis suggests that the brain processes information in a probabilistic manner, continuously updating its beliefs based on new evidence—essentially, the brain as a Bayesian inference machine. Gopnik applies this concept to understand how children adjust their expectations and learning processes based on the probability of events and experiences they encounter.</li></ul><h3>Child as Scientist</h3><ul><li><strong>Description</strong>: This concept is closely related to theory theory and is central to Gopnik's research. She argues that children are naturally curious and systematically experiment with their environment to figure out how things work, much like a scientist conducting experiments. This perspective views children as active learners, driven by innate curiosity and a fundamental desire to make sense of the world.</li></ul><h3>Scientist as Child</h3><ul><li><strong>Description</strong>: This idea flips the "child as scientist" metaphor, suggesting that scientists could benefit from adopting a child-like approach to exploration and learning. Children's exploratory behaviors, open-mindedness, and creativity in problem-solving can serve as a model for scientific inquiry, emphasizing that the best scientists retain a sense of play and curiosity throughout their careers.</li></ul><h3>Child as Hacker</h3><ul><li><strong>Description</strong>: Gopnik uses the "child as hacker" analogy to describe how children interact with and learn about their world. In this view, children are like hackers—they tinker with systems (their environment) to see what happens, learn the rules, and often figure out how to "hack" these rules to work in their favor or to find new ways of doing things.</li></ul><h3>Impact and Application</h3><ul><li><strong>Education</strong>: Gopnik's theories have profound implications for educational practices. They suggest that educational environments should nurture and support children's natural curiosity and investigative nature, providing them with opportunities to explore and experiment.</li><li><strong>Parenting</strong>: For parents, these theories emphasize the importance of providing a safe and rich environment where children can test their ideas and learn from the outcomes of their actions.</li><li><strong>Artificial Intelligence</strong>: Insights from Gopnik's work also inform the development of AI, particularly in creating systems that learn from the environment in a manner analogous to how children learn, adapting and optimizing themselves over time.</li></ul><p>Alison Gopnik's work challenges traditional views of passive childhood learning, presenting a dynamic model where children are active agents in their cognitive development. Her influence extends beyond psychology into fields like education, parenting, and artificial intelligence, offering a comprehensive framework for understanding human learning and cognitive evolution.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 15.5523 3.44772 16 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13V5C17.0045 4.64937 16.9444 4.31278 16.8338 4H18.0045C18.5568 4 19.0045 4.44772 19.0045 5V12C19.0045 12.5523 18.5568 13 18.0045 13H17.0045Z" fill="currentColor"></path></svg></span></button></span></div><span class="" data-state="closed"><button type="button" id="radix-:rh7:" aria-haspopup="menu" aria-expanded="false" data-state="closed" class="cursor-pointer rounded-md px-1 text-token-text-secondary h-[30px] hover:bg-token-main-surface-secondary"><div class="flex items-center pb-0.5 juice:pb-0"><div class="[&_svg]:h-full [&_svg]:w-full icon-md h-4 w-4"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24"><path fill="currentColor" d="M12.001 1.5a1 1 0 0 1 .993.887c.313 2.77 1.153 4.775 2.5 6.146 1.34 1.366 3.3 2.223 6.095 2.47a1 1 0 0 1-.003 1.993c-2.747.238-4.75 1.094-6.123 2.467-1.373 1.374-2.229 3.376-2.467 6.123a1 1 0 0 1-1.992.003c-.248-2.795-1.105-4.754-2.47-6.095-1.372-1.347-3.376-2.187-6.147-2.5a1 1 0 0 1-.002-1.987c2.818-.325 4.779-1.165 6.118-2.504 1.339-1.34 2.179-3.3 2.504-6.118A1 1 0 0 1 12 1.5ZM6.725 11.998c1.234.503 2.309 1.184 3.21 2.069.877.861 1.56 1.888 2.063 3.076.5-1.187 1.18-2.223 2.051-3.094.871-.87 1.907-1.55 3.094-2.05-1.188-.503-2.215-1.187-3.076-2.064-.885-.901-1.566-1.976-2.069-3.21-.505 1.235-1.19 2.3-2.081 3.192-.891.89-1.957 1.576-3.192 2.082Z"></path></svg></div><span class="line-clamp-1 text-sm" style="opacity: 0; padding-left: 0px; width: 0px;">4</span><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-sm text-token-text-quaternary"><path fill="currentColor" fill-rule="evenodd" d="M5.293 9.293a1 1 0 0 1 1.414 0L12 14.586l5.293-5.293a1 1 0 1 1 1.414 1.414l-6 6a1 1 0 0 1-1.414 0l-6-6a1 1 0 0 1 0-1.414" clip-rule="evenodd"></path></svg></div></button></span></div></div></div><div class="pr-2 lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-36" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb21675-c686-40f8-9c6d-4b2797b27c51" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">And can you see why without Montessori and Gaposhchkin, non of the science of the past century would have even been possible?<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" clip-rule="evenodd"></path></svg></button></span></div></div></div></div></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-37" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="flex-shrink-0 flex flex-col relative items-end"><div><div class="pt-0.5 juice:pt-0"><div class="gizmo-bot-avatar flex h-6 w-6 items-center justify-center overflow-hidden rounded-full juice:h-8 juice:w-8"><div class="relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md" role="img"><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 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11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="14504d36-eee2-4d06-9977-bc1becfad839" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Indeed, the contributions of Maria Montessori and Cecilia Payne-Gaposchkin have been crucial in shaping modern educational practices and astronomical science, respectively. Both have fundamentally altered our approaches in their fields, setting the stage for further advancements that have rippled through various domains of science and education. Here’s how their contributions have been foundational:</p><h3>Maria Montessori's Impact on Education and Beyond</h3><ol><li><p><strong>Holistic Education Model</strong>: Montessori’s educational philosophy emphasized the development of the whole child—physical, social, emotional, and cognitive. This approach encouraged educators to think of learning as an integrated process, influencing modern educational theories that recognize the importance of nurturing all aspects of a child’s development.</p></li><li><p><strong>Self-directed Learning</strong>: Her advocacy for child-centered, self-directed learning has permeated beyond early childhood education to influence instructional methods at all levels of education. This method fosters independence and initiative, qualities essential for scientific inquiry and innovation.</p></li><li><p><strong>Educational Access and Reform</strong>: Montessori's work also highlighted the importance of educational access for all children, regardless of background, which has driven efforts towards educational equity—a crucial factor in democratizing access to scientific careers and diversifying the field.</p></li></ol><h3>Cecilia Payne-Gaposchkin's Impact on Astronomy and Broader Sciences</h3><ol><li><p><strong>Foundational Astrophysical Concepts</strong>: Payne-Gaposchkin’s discovery that hydrogen is the most abundant element in stars, and by extension, the universe, changed the course of astrophysics. This foundational knowledge is critical for everything from understanding stellar evolution to the search for life in the universe.</p></li><li><p><strong>Women in Science</strong>: As one of the first people to earn a Ph.D. in astronomy from Radcliffe College (Harvard), her perseverance and success paved the way for future generations of women in science, challenging gender barriers and inspiring a more inclusive scientific community.</p></li><li><p><strong>Interdisciplinary Influence</strong>: Her work exemplifies the importance of interdisciplinary approaches, combining knowledge from chemistry, physics, and astronomy to solve complex problems. This model of interdisciplinary research has become increasingly important in tackling today’s global scientific challenges.</p></li></ol><h3>The Synergistic Effects of Their Contributions</h3><ul><li><p><strong>Fostering Curiosity and Critical Thinking</strong>: Montessori’s educational practices and Payne-Gaposchkin’s scientific achievements both underscore the importance of curiosity and critical thinking. Montessori’s methodologies foster these traits from a young age, preparing individuals to engage deeply with scientific inquiry and other complex intellectual pursuits.</p></li><li><p><strong>Innovative Problem Solving</strong>: Both figures exemplify innovative thinking and problem-solving—skills that are crucial in all fields of science and innovation. By breaking new ground in their respective fields, they have shown how challenging traditional boundaries and fostering creative environments are crucial for advancement.</p></li><li><p><strong>Expanding Participation</strong>: Their legacies include expanding participation in their fields, whether by inspiring women to pursue careers in the sciences (Payne-Gaposchkin) or by reforming educational systems to better serve all students (Montessori). A more inclusive approach to science and education broadens the range of ideas and perspectives, which is essential for robust scientific inquiry and innovation.</p></li></ul><p>Indeed, without Montessori’s revolution in education, which shaped generations of independent and critical thinkers, and without Payne-Gaposchkin’s pioneering contributions to astrophysics, the scientific advances of the past century might have looked very different. Their work not only pushed the boundaries of their respective fields but also set the stage for the methodologies and inclusive practices that are vital in today’s scientific and educational spheres.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 15.5523 3.44772 16 4 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max-w-full"><div data-message-author-role="assistant" data-message-id="3f93a182-088d-4cb5-8cef-d3c754ff1d34" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>Certainly! Here's a comprehensive summary of our discussion:</p><ol><li><p><strong>Ablation Study of TLDR</strong>: We started by discussing an ablation study focused on adversarial robustness, combining transduction and rejection in machine learning models. I provided details on the setup, results, and implications of the study, emphasizing its impact on enhancing model robustness against adversarial attacks.</p></li><li><p><strong>Influential Figures</strong>: We discussed the contributions of Cecilia Payne-Gaposchkin in astronomy and Maria Montessori in education, highlighting how their pioneering work has fundamentally shaped their respective fields and broader societal practices.</p></li><li><p><strong>Theoretical Concepts</strong>: We explored several theoretical concepts related to cognitive and developmental psychology, including Alison Gopnik's influential theories such as the Bayesian brain hypothesis, theory theory, and the ideas of the child as scientist and hacker. These concepts illustrate how children learn in ways that parallel scientific inquiry.</p></li><li><p><strong>Educational and Philosophical Insights</strong>: We delved into philosophical ideas like Wittgenstein's ladder and Vygotsky's scaffolding, discussing how these concepts apply to educational practices and personal development, emphasizing the importance of progressing beyond initial learning tools.</p></li><li><p><strong>Role of Play</strong>: We discussed the role of toys and play in development, considering play as a form of simulated danger and learning as a means to prepare for unexpected challenges, akin to inoculation against surprise. This conversation highlighted the significance of play in developing critical cognitive, social, and emotional skills.</p></li><li><p><strong>Cultural and Historical Reflections</strong>: We touched on the cultural process of the Kalabari people using counterfoil choices, connecting this to broader themes of decision-making and learning, and how these reflect and reinforce cultural identity and individual development.</p></li><li><p><strong>Impact on Science and Education</strong>: We concluded by reflecting on how Montessori's and Payne-Gaposchkin's legacies have not only advanced their fields but also set foundational practices that support modern scientific inquiry and educational reform, promoting inclusivity and interdisciplinary approaches.</p></li></ol><p>Throughout our conversation, we interwove discussions about the importance of foundational changes in educational methods, the profound impact of groundbreaking scientific discoveries, and the innovative theories in psychology that reshape how we understand learning and development. Each topic reinforced the notion that advancements in understanding, whether through formal research or broader educational practices, foster a more nuanced appreciation of human potential and the mechanisms that underpin personal and collective growth.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 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lg:pr-0"></div></div></div></div></div></div><div class="w-full text-token-text-primary" dir="auto" data-testid="conversation-turn-40" data-scroll-anchor="false"><div class="py-2 juice:py-[18px] px-3 text-base md:px-4 m-auto md:px-5 lg:px-1 xl:px-5"><div class="mx-auto flex flex-1 gap-3 text-base juice:gap-4 juice:md:gap-5 juice:lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]"><div class="group/conversation-turn relative flex w-full min-w-0 flex-col"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="user" data-message-id="bbb21650-204f-4340-83d2-cfe550065cbf" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden items-end rtl:items-start"><div class="relative max-w-[70%] rounded-3xl bg-[#f4f4f4] px-5 py-2.5 dark:bg-token-main-surface-secondary">I was thinking of a kind of ablation study where we imagine science without Gaposhkin we might not know what stars are made of even now<div class="absolute bottom-0 right-full top-0 -mr-3.5 hidden pr-5 pt-1 group-hover/conversation-turn:block"><span class="" data-state="closed"><button class="flex h-9 w-9 items-center justify-center rounded-full text-token-text-secondary transition hover:bg-token-main-surface-tertiary"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24" class="icon-md"><path fill="currentColor" fill-rule="evenodd" d="M13.293 4.293a4.536 4.536 0 1 1 6.414 6.414l-1 1-7.094 7.094A5 5 0 0 1 8.9 20.197l-4.736.79a1 1 0 0 1-1.15-1.151l.789-4.736a5 5 0 0 1 1.396-2.713zM13 7.414l-6.386 6.387a3 3 0 0 0-.838 1.628l-.56 3.355 3.355-.56a3 3 0 0 0 1.628-.837L16.586 11zm5 2.172L14.414 6l.293-.293a2.536 2.536 0 0 1 3.586 3.586z" 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10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div></div></div><div class="group/conversation-turn relative flex w-full min-w-0 flex-col agent-turn"><div class="flex-col gap-1 md:gap-3"><div class="flex flex-grow flex-col max-w-full"><div data-message-author-role="assistant" data-message-id="cde59ed1-a778-4627-87ef-a55625205234" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>That's a fascinating way to frame an ablation study, imagining a world without the contributions of Cecilia Payne-Gaposchkin to highlight her monumental influence on astrophysics. By removing her groundbreaking discovery from the history of science, we can appreciate the potential delay or alternative pathways that might have emerged in our understanding of the universe.</p><h3>Hypothetical Ablation of Payne-Gaposchkin's Work</h3><p><strong>Contribution Removed</strong>: Cecilia Payne-Gaposchkin's thesis concluded that hydrogen is the most abundant element in stars, fundamentally changing the scientific understanding of stellar compositions.</p><p><strong>Potential Impacts</strong>:</p><ol><li><p><strong>Delayed Discovery</strong>: Without her insights, the recognition of hydrogen as the most abundant element in stars might have been significantly delayed. During her time, the prevailing belief based on Earth's composition suggested heavier elements like iron were more prevalent. Other scientists might not have questioned this assumption as quickly without her pioneering work.</p></li><li><p><strong>Scientific Development</strong>: The development of stellar nucleosynthesis theories, which explain the origins and creation of chemical elements in stars through nuclear fusion processes, might have been stalled. Understanding that stars are primarily made of hydrogen and helium is crucial for these theories.</p></li><li><p><strong>Advancement in Related Fields</strong>: Tools and methods in spectroscopy, used to analyze the light emitted by stars to determine their composition, might have evolved differently. Her work directly influenced the interpretation of astronomical spectroscopic data.</p></li><li><p><strong>Broader Theoretical Implications</strong>: The broader cosmological theories, including those related to the evolution of the universe and the lifecycle of stars, rely heavily on understanding stellar composition. Delays in these areas could have altered the trajectory of modern cosmology.</p></li><li><p><strong>Educational and Inspirational Impact</strong>: Payne-Gaposchkin, as a pioneering woman in science, provided a role model for subsequent generations of female scientists. Her absence might have influenced the pace at which women entered and were accepted in the astronomical and other scientific communities.</p></li></ol><h3>Methodological Considerations</h3><p>In this hypothetical ablation study, we'd look at the timeline of discoveries and technological advancements in astronomy and astrophysics to hypothesize how different the field might look today without her contributions. We would consider:</p><ul><li><strong>Alternative Pathways</strong>: Who else was close to such discoveries, and what were the prevailing theories that might have dominated?</li><li><strong>Technological Advancements</strong>: Would technology have pushed another scientist towards the same conclusions under different circumstances, or would the field have taken a completely different direction?</li><li><strong>Cultural and Social Impact</strong>: How did her presence and success impact the scientific community, especially in terms of breaking gender barriers in science?</li></ul><p>By "removing" Payne-Gaposchkin's contributions, we not only understand her specific impact but also appreciate the interconnected nature of scientific advancements, where one discovery builds on another, propelling the entire field forward. This type of thought experiment helps us recognize the value of diverse perspectives and individual breakthroughs in the collective endeavor of science.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 z-10 -mt-1 bg-token-main-surface-primary md:absolute md:border md:border-token-border-light md:hidden"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 15.5523 3.44772 16 4 16H6C6.25019 16 6.49129 16.0938 6.67572 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data-message-author-role="assistant" data-message-id="ac6d6fac-f7ad-4616-8fc6-03128dc74c0e" dir="auto" class="min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2"><div class="flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]"><div class="markdown prose w-full break-words dark:prose-invert dark"><p>In our conversation, we discussed the profound impacts of Cecilia Payne-Gaposchkin on astronomy, Maria Montessori on education, and Alison Gopnik on developmental psychology, highlighting their revolutionary contributions to their fields. We explored how these innovations have shaped current understanding and practices. Additionally, we delved into theoretical concepts like Wittgenstein's ladder and Vygotsky's scaffolding, discussing their relevance in learning and development. We also considered the role of play as a developmental tool, emphasizing its importance in preparing children for unexpected challenges. Throughout, we contemplated the potential consequences if these pivotal contributions had never occurred, particularly in the sciences, illustrating the critical role these figures play in the advancement of knowledge and society.</p></div></div></div></div><div class="mt-1 flex gap-3 empty:hidden juice:-ml-3"><div class="items-center justify-start rounded-xl p-1 flex"><div class="flex items-center"><span class="" data-state="closed"><button class="rounded-lg text-token-text-secondary hover:bg-token-main-surface-secondary"><span class="flex h-[30px] w-[30px] items-center justify-center"><svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg" class="icon-md-heavy"><path fill-rule="evenodd" clip-rule="evenodd" d="M11 4.9099C11 4.47485 10.4828 4.24734 10.1621 4.54132L6.67572 7.7372C6.49129 7.90626 6.25019 8.00005 6 8.00005H4C3.44772 8.00005 3 8.44776 3 9.00005V15C3 15.5523 3.44772 16 4 16H6C6.25019 16 6.49129 16.0938 6.67572 16.2629L10.1621 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@font-face { font-display: swap; font-family: Circle; font-style: normal; font-weight: 400; src: url("https://cdn.oaistatic.com/_next/static/media/circle-400.d446e5e9.woff2") format("woff2"); }
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*, ::after, ::before { border: 0px solid rgb(227, 227, 227); box-sizing: border-box; }
::after, ::before { --tw-content: ""; }
:host, html { text-size-adjust: 100%; font-feature-settings: normal; -webkit-tap-highlight-color: transparent; font-family: ui-sans-serif, -apple-system, system-ui, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, Helvetica, "Apple Color Emoji", Arial, "Segoe UI Emoji", "Segoe UI Symbol"; font-variation-settings: normal; line-height: 1.5; tab-size: 4; }
body { line-height: inherit; margin: 0px; }
hr { border-top-width: 1px; color: inherit; height: 0px; }
abbr:where([title]) { text-decoration: underline dotted currentcolor; }
h1, h2, h3, h4, h5, h6 { font-size: inherit; font-weight: inherit; }
a { color: inherit; text-decoration: inherit; }
b, strong { font-weight: bolder; }
code, kbd, pre, samp { font-feature-settings: normal; font-family: ui-monospace, SFMono-Regular, "SF Mono", Menlo, Consolas, "Liberation Mono", monospace; font-size: 1em; font-variation-settings: normal; }
small { font-size: 80%; }
sub, sup { font-size: 75%; line-height: 0; position: relative; vertical-align: baseline; }
sub { bottom: -0.25em; }
sup { top: -0.5em; }
table { border-collapse: collapse; border-color: inherit; text-indent: 0px; }
button, input, optgroup, select, textarea { font-feature-settings: inherit; color: inherit; font-family: inherit; font-size: 100%; font-variation-settings: inherit; font-weight: inherit; letter-spacing: inherit; line-height: inherit; margin: 0px; padding: 0px; }
button, select { text-transform: none; }
button, input:where([type="button"]), input:where([type="reset"]), input:where([type="submit"]) { appearance: button; background-color: transparent; background-image: none; }
progress { vertical-align: baseline; }
::-webkit-inner-spin-button, ::-webkit-outer-spin-button { height: auto; }
[type="search"] { appearance: textfield; outline-offset: -2px; }
::-webkit-search-decoration { appearance: none; }
::-webkit-file-upload-button { appearance: button; font: inherit; }
summary { display: list-item; }
blockquote, dd, dl, fieldset, figure, h1, h2, h3, h4, h5, h6, hr, p, pre { margin: 0px; }
fieldset, legend { padding: 0px; }
menu, ol, ul { list-style: none; margin: 0px; padding: 0px; }
dialog { padding: 0px; }
textarea { resize: vertical; }
input::placeholder, textarea::placeholder { color: rgb(180, 180, 180); }
[role="button"], button { cursor: pointer; }
:disabled { cursor: default; }
audio, canvas, embed, iframe, img, object, svg, video { display: block; vertical-align: middle; }
img, video { height: auto; max-width: 100%; }
[hidden] { display: none; }
[multiple], [type="date"], [type="datetime-local"], [type="email"], [type="month"], [type="number"], [type="password"], [type="search"], [type="tel"], [type="text"], [type="time"], [type="url"], [type="week"], select, textarea { --tw-shadow: 0 0 transparent; appearance: none; background-color: rgb(255, 255, 255); border-color: rgb(155, 155, 155); border-radius: 0px; border-width: 1px; font-size: 1rem; line-height: 1.5rem; padding: 0.5rem 0.75rem; }
[multiple]:focus, [type="date"]:focus, [type="datetime-local"]:focus, [type="email"]:focus, [type="month"]:focus, [type="number"]:focus, [type="password"]:focus, [type="search"]:focus, [type="tel"]:focus, [type="text"]:focus, [type="time"]:focus, [type="url"]:focus, [type="week"]:focus, select:focus, textarea:focus { --tw-ring-inset: var(--tw-empty,/*!*/ /*!*/); --tw-ring-offset-width: 0px; --tw-ring-offset-color: #fff; --tw-ring-color: #2563eb; --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(1px + var(--tw-ring-offset-width)) var(--tw-ring-color); border-color: rgb(37, 99, 235); box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow); outline: transparent solid 2px; outline-offset: 2px; }
input::placeholder, textarea::placeholder { color: rgb(155, 155, 155); opacity: 1; }
::-webkit-datetime-edit-fields-wrapper { padding: 0px; }
::-webkit-date-and-time-value { min-height: 1.5em; }
::-webkit-datetime-edit, ::-webkit-datetime-edit-day-field, ::-webkit-datetime-edit-hour-field, ::-webkit-datetime-edit-meridiem-field, ::-webkit-datetime-edit-millisecond-field, ::-webkit-datetime-edit-minute-field, ::-webkit-datetime-edit-month-field, ::-webkit-datetime-edit-second-field, ::-webkit-datetime-edit-year-field { padding-bottom: 0px; padding-top: 0px; }
select { background-image: url("data:image/svg+xml;charset=utf-8,%3Csvg xmlns='http://www.w3.org/2000/svg' fill='none' viewBox='0 0 20 20'%3E%3Cpath stroke='%239B9B9B' stroke-linecap='round' stroke-linejoin='round' stroke-width='1.5' d='m6 8 4 4 4-4'/%3E%3C/svg%3E"); background-position: right 0.5rem center; background-repeat: no-repeat; background-size: 1.5em 1.5em; padding-right: 2.5rem; -webkit-print-color-adjust: exact; }
[multiple] { background-image: none; background-position: 0px 0px; background-repeat: repeat; background-size: initial; padding-right: 0.75rem; -webkit-print-color-adjust: inherit; }
[type="checkbox"], [type="radio"] { --tw-shadow: 0 0 transparent; appearance: none; background-color: rgb(255, 255, 255); background-origin: border-box; border-color: rgb(155, 155, 155); border-width: 1px; color: rgb(37, 99, 235); display: inline-block; flex-shrink: 0; height: 1rem; padding: 0px; -webkit-print-color-adjust: exact; user-select: none; vertical-align: middle; width: 1rem; }
[type="checkbox"] { border-radius: 0px; }
[type="radio"] { border-radius: 100%; }
[type="checkbox"]:focus, [type="radio"]:focus { --tw-ring-inset: var(--tw-empty,/*!*/ /*!*/); --tw-ring-offset-width: 2px; --tw-ring-offset-color: #fff; --tw-ring-color: #2563eb; --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow); outline: transparent solid 2px; outline-offset: 2px; }
[type="checkbox"]:checked, [type="radio"]:checked { background-color: currentcolor; background-position: 50% center; background-repeat: no-repeat; background-size: 100% 100%; border-color: transparent; }
[type="checkbox"]:checked { background-image: url("data:image/svg+xml;charset=utf-8,%3Csvg viewBox='0 0 16 16' fill='%23fff' xmlns='http://www.w3.org/2000/svg'%3E%3Cpath d='M12.207 4.793a1 1 0 0 1 0 1.414l-5 5a1 1 0 0 1-1.414 0l-2-2a1 1 0 0 1 1.414-1.414L6.5 9.086l4.293-4.293a1 1 0 0 1 1.414 0z'/%3E%3C/svg%3E"); }
[type="radio"]:checked { background-image: url("data:image/svg+xml;charset=utf-8,%3Csvg viewBox='0 0 16 16' fill='%23fff' xmlns='http://www.w3.org/2000/svg'%3E%3Ccircle cx='8' cy='8' r='3'/%3E%3C/svg%3E"); }
[type="checkbox"]:checked:focus, [type="checkbox"]:checked:hover, [type="radio"]:checked:focus, [type="radio"]:checked:hover { background-color: currentcolor; border-color: transparent; }
[type="checkbox"]:indeterminate { background-color: currentcolor; background-image: url("data:image/svg+xml;charset=utf-8,%3Csvg xmlns='http://www.w3.org/2000/svg' fill='none' viewBox='0 0 16 16'%3E%3Cpath stroke='%23fff' stroke-linecap='round' stroke-linejoin='round' stroke-width='2' d='M4 8h8'/%3E%3C/svg%3E"); background-position: 50% center; background-repeat: no-repeat; background-size: 100% 100%; border-color: transparent; }
[type="checkbox"]:indeterminate:focus, [type="checkbox"]:indeterminate:hover { background-color: currentcolor; border-color: transparent; }
[type="file"] { background: initial; border-color: inherit; border-radius: 0px; border-width: 0px; font-size: inherit; line-height: inherit; padding: 0px; }
[type="file"]:focus { outline: -webkit-focus-ring-color auto 1px; }
h1 { font-size: 1.875rem; line-height: 2.25rem; }
h1, h2, h3 { font-weight: 600; }
h2, h3 { font-size: 1.5rem; line-height: 2rem; }
input[type="number"]::-webkit-inner-spin-button, input[type="number"]::-webkit-outer-spin-button { appearance: none; margin: 0px; }
*, ::after, ::before { --tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; --tw-rotate: 0; --tw-skew-x: 0; --tw-skew-y: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-pan-x: ; --tw-pan-y: ; --tw-pinch-zoom: ; --tw-scroll-snap-strictness: proximity; --tw-gradient-from-position: ; --tw-gradient-via-position: ; --tw-gradient-to-position: ; --tw-ordinal: ; --tw-slashed-zero: ; --tw-numeric-figure: ; --tw-numeric-spacing: ; --tw-numeric-fraction: ; --tw-ring-inset: ; --tw-ring-offset-width: 0px; --tw-ring-offset-color: #fff; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-shadow: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-shadow-colored: 0 0 transparent; --tw-blur: ; --tw-brightness: ; --tw-contrast: ; --tw-grayscale: ; --tw-hue-rotate: ; --tw-invert: ; --tw-saturate: ; --tw-sepia: ; --tw-drop-shadow: ; --tw-backdrop-blur: ; --tw-backdrop-brightness: ; --tw-backdrop-contrast: ; --tw-backdrop-grayscale: ; --tw-backdrop-hue-rotate: ; --tw-backdrop-invert: ; --tw-backdrop-opacity: ; --tw-backdrop-saturate: ; --tw-backdrop-sepia: ; --tw-contain-size: ; --tw-contain-layout: ; --tw-contain-paint: ; --tw-contain-style: ; }
::backdrop { --tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; --tw-rotate: 0; --tw-skew-x: 0; --tw-skew-y: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-pan-x: ; --tw-pan-y: ; --tw-pinch-zoom: ; --tw-scroll-snap-strictness: proximity; --tw-gradient-from-position: ; --tw-gradient-via-position: ; --tw-gradient-to-position: ; --tw-ordinal: ; --tw-slashed-zero: ; --tw-numeric-figure: ; --tw-numeric-spacing: ; --tw-numeric-fraction: ; --tw-ring-inset: ; --tw-ring-offset-width: 0px; --tw-ring-offset-color: #fff; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-shadow: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-shadow-colored: 0 0 transparent; --tw-blur: ; --tw-brightness: ; --tw-contrast: ; --tw-grayscale: ; --tw-hue-rotate: ; --tw-invert: ; --tw-saturate: ; --tw-sepia: ; --tw-drop-shadow: ; --tw-backdrop-blur: ; --tw-backdrop-brightness: ; --tw-backdrop-contrast: ; --tw-backdrop-grayscale: ; --tw-backdrop-hue-rotate: ; --tw-backdrop-invert: ; --tw-backdrop-opacity: ; --tw-backdrop-saturate: ; --tw-backdrop-sepia: ; --tw-contain-size: ; --tw-contain-layout: ; --tw-contain-paint: ; --tw-contain-style: ; }
:root { --white: #fff; --black: #000; --gray-50: #f9f9f9; --gray-100: #ececec; --gray-200: #e3e3e3; --gray-300: #cdcdcd; --gray-400: #b4b4b4; --gray-500: #9b9b9b; --gray-600: #676767; --gray-700: #424242; --gray-750: #2f2f2f; --gray-800: #212121; --gray-900: #171717; --gray-950: #0d0d0d; --red-500: #ef4444; --red-700: #b91c1c; --brand-purple: #ab68ff; }
html { --text-primary: var(--gray-950); --text-secondary: var(--gray-500); --text-tertiary: var(--gray-400); --text-quaternary: var(--gray-300); --text-error: var(--red-700); --surface-error: var(--red-700); --border-light: rgba(0,0,0,.1); --border-medium: rgba(0,0,0,.15); --border-heavy: rgba(0,0,0,.2); --border-xheavy: rgba(0,0,0,.25); --main-surface-primary: var(--white); --main-surface-secondary: var(--gray-50); --main-surface-tertiary: var(--gray-100); --sidebar-surface-primary: var(--gray-50); --sidebar-surface-secondary: var(--gray-100); --sidebar-surface-tertiary: var(--gray-200); --link: #2964aa; --link-hover: #749ac8; --selection: #007aff; }
html.ZnJ1aXRqdWljZQ { --text-secondary: #7d7d7d; --text-error: #f93a37; --surface-error: 249 58 55; }
.dark { --text-primary: var(--gray-100); --text-secondary: var(--gray-300); --text-tertiary: var(--gray-400); --text-quaternary: var(--gray-500); --text-error: var(--red-500); --border-light: hsla(0,0%,100%,.1); --border-medium: hsla(0,0%,100%,.15); --border-heavy: hsla(0,0%,100%,.2); --border-xheavy: hsla(0,0%,100%,.25); --main-surface-primary: var(--gray-800); --main-surface-secondary: var(--gray-750); --main-surface-tertiary: var(--gray-700); --sidebar-surface-primary: var(--gray-900); --sidebar-surface-secondary: var(--gray-800); --sidebar-surface-tertiary: var(--gray-750); --link: #7ab7ff; --link-hover: #5e83b3; }
html.ZnJ1aXRqdWljZQ.dark { --text-secondary: var(--gray-400); --text-tertiary: var(--gray-500); --text-quaternary: var(--gray-600); --text-error: #f93a37; --surface-error: 249 58 55; }
.popover { --main-surface-primary: var(--white) !important; --main-surface-secondary: var(--gray-100) !important; --main-surface-tertiary: var(--gray-200) !important; --sidebar-surface-primary: var(--white) !important; }
.dark .popover, .dark.popover, .popover .dark { --main-surface-primary: var(--gray-750) !important; --main-surface-secondary: var(--gray-700) !important; --main-surface-tertiary: var(--gray-600) !important; --text-primary: var(--gray-50) !important; --text-secondary: var(--gray-200) !important; --text-tertiary: var(--gray-400) !important; --text-quaternary: var(--gray-500) !important; --sidebar-surface-primary: var(--gray-750) !important; }
textarea:focus { border-color: inherit; box-shadow: none; outline: none; }
.container { width: 100%; }
@media (min-width: 640px) {
.container { max-width: 640px; }
}
@media (min-width: 768px) {
.container { max-width: 768px; }
}
@media (min-width: 1024px) {
.container { max-width: 1024px; }
}
@media (min-width: 1280px) {
.container { max-width: 1280px; }
}
@media (min-width: 1536px) {
.container { max-width: 1536px; }
}
@media (min-width: 2006px) {
.container { max-width: 2006px; }
}
.prose { color: var(--tw-prose-body); max-width: 65ch; }
.prose :where([class~="lead"]):not(:where([class~="not-prose"] *)) { color: var(--tw-prose-lead); font-size: 1.25em; line-height: 1.6; margin-bottom: 1.2em; margin-top: 1.2em; }
.prose :where(a):not(:where([class~="not-prose"] *)) { color: var(--tw-prose-links); font-weight: 500; text-decoration: underline; }
.prose :where(strong):not(:where([class~="not-prose"] *)) { color: var(--tw-prose-bold); font-weight: 600; }
.prose :where(a strong):not(:where([class~="not-prose"] *)) { color: inherit; }
.prose :where(blockquote strong):not(:where([class~="not-prose"] *)) { color: inherit; }
.prose :where(thead th strong):not(:where([class~="not-prose"] *)) { color: inherit; }
.prose :where(ol):not(:where([class~="not-prose"] *)) { list-style-type: decimal; margin-bottom: 1.25em; margin-top: 1.25em; padding-left: 1.625em; }
.prose :where(ol[type="A"]):not(:where([class~="not-prose"] *)) { list-style-type: upper-alpha; }
.prose :where(ol[type="a"]):not(:where([class~="not-prose"] *)) { list-style-type: lower-alpha; }
.prose :where():not(:where([class~="not-prose"] *)) { list-style-type: upper-alpha; }
.prose :where():not(:where([class~="not-prose"] *)) { list-style-type: lower-alpha; }
.prose :where(ol[type="I"]):not(:where([class~="not-prose"] *)) { list-style-type: upper-roman; }
.prose :where(ol[type="i"]):not(:where([class~="not-prose"] *)) { list-style-type: lower-roman; }
.prose :where():not(:where([class~="not-prose"] *)) { list-style-type: upper-roman; }
.prose :where():not(:where([class~="not-prose"] *)) { list-style-type: lower-roman; }
.prose :where(ol[type="1"]):not(:where([class~="not-prose"] *)) { list-style-type: decimal; }
.prose :where(ul):not(:where([class~="not-prose"] *)) { list-style-type: disc; margin-bottom: 1.25em; margin-top: 1.25em; padding-left: 1.625em; }
.prose :where(ol > li):not(:where([class~="not-prose"] *))::marker { color: var(--tw-prose-counters); font-weight: 400; }
.prose :where(ul > li):not(:where([class~="not-prose"] *))::marker { color: var(--tw-prose-bullets); }
.prose :where(hr):not(:where([class~="not-prose"] *)) { border-color: var(--tw-prose-hr); border-top-width: 1px; margin-bottom: 3em; margin-top: 3em; }
.prose :where(blockquote):not(:where([class~="not-prose"] *)) { border-left-color: var(--tw-prose-quote-borders); border-left-width: 0.25rem; color: var(--tw-prose-quotes); font-style: normal; font-weight: 500; margin-bottom: 1.6em; margin-top: 1.6em; padding-left: 1em; quotes: "“" "”" "‘" "’"; }
.prose :where(blockquote p:first-of-type):not(:where([class~="not-prose"] *))::before { content: open-quote; }
.prose :where(blockquote p:last-of-type):not(:where([class~="not-prose"] *))::after { content: close-quote; }
.prose :where(h1):not(:where([class~="not-prose"] *)) { color: var(--tw-prose-headings); font-size: 2.25em; font-weight: 800; line-height: 1.11111; margin-bottom: 0.888889em; margin-top: 0px; }
.prose :where(h1 strong):not(:where([class~="not-prose"] *)) { color: inherit; font-weight: 900; }
.prose :where(h2):not(:where([class~="not-prose"] *)) { color: var(--tw-prose-headings); font-size: 1.5em; font-weight: 700; line-height: 1.33333; margin-bottom: 1em; margin-top: 2em; }
.prose :where(h2 strong):not(:where([class~="not-prose"] *)) { color: inherit; font-weight: 800; }
.prose :where(h3):not(:where([class~="not-prose"] *)) { color: var(--tw-prose-headings); font-size: 1.25em; font-weight: 600; line-height: 1.6; margin-bottom: 0.6em; margin-top: 1.6em; }
.prose :where(h3 strong):not(:where([class~="not-prose"] *)) { color: inherit; font-weight: 700; }
.prose :where(h4):not(:where([class~="not-prose"] *)) { color: var(--tw-prose-headings); font-weight: 600; line-height: 1.5; margin-bottom: 0.5em; margin-top: 1.5em; }
.prose :where(h4 strong):not(:where([class~="not-prose"] *)) { color: inherit; font-weight: 700; }
.prose :where(img):not(:where([class~="not-prose"] *)) { margin-bottom: 2em; margin-top: 2em; }
.prose :where(figure > *):not(:where([class~="not-prose"] *)) { margin-bottom: 0px; margin-top: 0px; }
.prose :where(figcaption):not(:where([class~="not-prose"] *)) { color: var(--tw-prose-captions); font-size: 0.875em; line-height: 1.42857; margin-top: 0.857143em; }
.prose :where(code):not(:where([class~="not-prose"] *)) { color: var(--tw-prose-code); font-size: 0.875em; font-weight: 600; }
.prose :where(code):not(:where([class~="not-prose"] *))::before { content: "`"; }
.prose :where(code):not(:where([class~="not-prose"] *))::after { content: "`"; }
.prose :where(a code):not(:where([class~="not-prose"] *)) { color: inherit; }
.prose :where(h1 code):not(:where([class~="not-prose"] *)) { color: inherit; }
.prose :where(h2 code):not(:where([class~="not-prose"] *)) { color: inherit; font-size: 0.875em; }
.prose :where(h3 code):not(:where([class~="not-prose"] *)) { color: inherit; font-size: 0.9em; }
.prose :where(h4 code):not(:where([class~="not-prose"] *)) { color: inherit; }
.prose :where(blockquote code):not(:where([class~="not-prose"] *)) { color: inherit; }
.prose :where(thead th code):not(:where([class~="not-prose"] *)) { color: inherit; }
.prose :where(pre):not(:where([class~="not-prose"] *)) { background-color: transparent; border-radius: 0.375rem; color: currentcolor; font-size: 0.875em; font-weight: 400; line-height: 1.71429; margin: 0px; overflow-x: auto; padding: 0px; }
.prose :where(pre code):not(:where([class~="not-prose"] *)) { background-color: transparent; border-radius: 0px; border-width: 0px; color: inherit; font-family: inherit; font-size: inherit; font-weight: inherit; line-height: inherit; padding: 0px; }
.prose :where(pre code):not(:where([class~="not-prose"] *))::before { content: none; }
.prose :where(pre code):not(:where([class~="not-prose"] *))::after { content: none; }
.prose :where(table):not(:where([class~="not-prose"] *)) { font-size: 0.875em; line-height: 1.71429; margin-bottom: 2em; margin-top: 2em; table-layout: auto; text-align: left; width: 100%; }
.prose :where(thead):not(:where([class~="not-prose"] *)) { border-bottom-color: var(--tw-prose-th-borders); border-bottom-width: 1px; }
.prose :where(thead th):not(:where([class~="not-prose"] *)) { color: var(--tw-prose-headings); font-weight: 600; padding-bottom: 0.571429em; padding-left: 0.571429em; padding-right: 0.571429em; vertical-align: bottom; }
.prose :where(tbody tr):not(:where([class~="not-prose"] *)) { border-bottom-color: var(--tw-prose-td-borders); border-bottom-width: 1px; }
.prose :where(tbody tr:last-child):not(:where([class~="not-prose"] *)) { border-bottom-width: 0px; }
.prose :where(tbody td):not(:where([class~="not-prose"] *)) { vertical-align: baseline; }
.prose :where(tfoot):not(:where([class~="not-prose"] *)) { border-top-color: var(--tw-prose-th-borders); border-top-width: 1px; }
.prose :where(tfoot td):not(:where([class~="not-prose"] *)) { vertical-align: top; }
.prose { --tw-prose-body: var(--text-primary); --tw-prose-headings: var(--text-primary); --tw-prose-lead: var(--text-primary); --tw-prose-links: var(--text-primary); --tw-prose-bold: var(--text-primary); --tw-prose-counters: var(--text-primary); --tw-prose-bullets: var(--text-primary); --tw-prose-hr: var(--border-xheavy); --tw-prose-quotes: var(--text-primary); --tw-prose-quote-borders: #e5e7eb; --tw-prose-captions: var(--text-secondary); --tw-prose-code: var(--text-primary); --tw-prose-pre-code: #e5e7eb; --tw-prose-pre-bg: #1f2937; --tw-prose-th-borders: #d1d5db; --tw-prose-td-borders: #e5e7eb; --tw-prose-invert-body: var(--text-primary); --tw-prose-invert-headings: var(--text-primary); --tw-prose-invert-lead: var(--text-primary); --tw-prose-invert-links: var(--text-primary); --tw-prose-invert-bold: var(--text-primary); --tw-prose-invert-counters: var(--text-primary); --tw-prose-invert-bullets: var(--text-primary); --tw-prose-invert-hr: var(--border-xheavy); --tw-prose-invert-quotes: var(--text-primary); --tw-prose-invert-quote-borders: #374151; --tw-prose-invert-captions: var(--text-secondary); --tw-prose-invert-code: var(--text-primary); --tw-prose-invert-pre-code: #d1d5db; --tw-prose-invert-pre-bg: rgba(0,0,0,.5); --tw-prose-invert-th-borders: #4b5563; --tw-prose-invert-td-borders: #374151; font-size: 1rem; line-height: 1.75; }
.prose :where(p):not(:where([class~="not-prose"] *)) { margin-bottom: 1.25em; margin-top: 1.25em; }
.prose :where(video):not(:where([class~="not-prose"] *)) { margin-bottom: 2em; margin-top: 2em; }
.prose :where(figure):not(:where([class~="not-prose"] *)) { margin-bottom: 2em; margin-top: 2em; }
.prose :where(li):not(:where([class~="not-prose"] *)) { margin-bottom: 0.5em; margin-top: 0.5em; }
.prose :where(ol > li):not(:where([class~="not-prose"] *)) { padding-left: 0.375em; }
.prose :where(ul > li):not(:where([class~="not-prose"] *)) { padding-left: 0.375em; }
.prose :where(.prose > ul > li p):not(:where([class~="not-prose"] *)) { margin-bottom: 0.75em; margin-top: 0.75em; }
.prose :where(.prose > ul > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.25em; }
.prose :where(.prose > ul > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.25em; }
.prose :where(.prose > ol > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.25em; }
.prose :where(.prose > ol > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.25em; }
.prose :where(ul ul, ul ol, ol ul, ol ol):not(:where([class~="not-prose"] *)) { margin-bottom: 0.75em; margin-top: 0.75em; }
.prose :where(hr + *):not(:where([class~="not-prose"] *)) { margin-top: 0px; }
.prose :where(h2 + *):not(:where([class~="not-prose"] *)) { margin-top: 0px; }
.prose :where(h3 + *):not(:where([class~="not-prose"] *)) { margin-top: 0px; }
.prose :where(h4 + *):not(:where([class~="not-prose"] *)) { margin-top: 0px; }
.prose :where(thead th:first-child):not(:where([class~="not-prose"] *)) { padding-left: 0px; }
.prose :where(thead th:last-child):not(:where([class~="not-prose"] *)) { padding-right: 0px; }
.prose :where(tbody td, tfoot td):not(:where([class~="not-prose"] *)) { padding: 0.571429em; }
.prose :where(tbody td:first-child, tfoot td:first-child):not(:where([class~="not-prose"] *)) { padding-left: 0px; }
.prose :where(tbody td:last-child, tfoot td:last-child):not(:where([class~="not-prose"] *)) { padding-right: 0px; }
.prose :where(.prose > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 0px; }
.prose :where(.prose > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 0px; }
.prose-sm :where(.prose > ul > li p):not(:where([class~="not-prose"] *)) { margin-bottom: 0.571429em; margin-top: 0.571429em; }
.prose-sm :where(.prose > ul > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.14286em; }
.prose-sm :where(.prose > ul > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.14286em; }
.prose-sm :where(.prose > ol > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.14286em; }
.prose-sm :where(.prose > ol > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.14286em; }
.prose-sm :where(.prose > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 0px; }
.prose-sm :where(.prose > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 0px; }
.prose-base :where(.prose > ul > li p):not(:where([class~="not-prose"] *)) { margin-bottom: 0.75em; margin-top: 0.75em; }
.prose-base :where(.prose > ul > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.25em; }
.prose-base :where(.prose > ul > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.25em; }
.prose-base :where(.prose > ol > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.25em; }
.prose-base :where(.prose > ol > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.25em; }
.prose-base :where(.prose > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 0px; }
.prose-base :where(.prose > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 0px; }
.prose-lg :where(.prose > ul > li p):not(:where([class~="not-prose"] *)) { margin-bottom: 0.888889em; margin-top: 0.888889em; }
.prose-lg :where(.prose > ul > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.33333em; }
.prose-lg :where(.prose > ul > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.33333em; }
.prose-lg :where(.prose > ol > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.33333em; }
.prose-lg :where(.prose > ol > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.33333em; }
.prose-lg :where(.prose > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 0px; }
.prose-lg :where(.prose > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 0px; }
.prose-xl :where(.prose > ul > li p):not(:where([class~="not-prose"] *)) { margin-bottom: 0.8em; margin-top: 0.8em; }
.prose-xl :where(.prose > ul > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.2em; }
.prose-xl :where(.prose > ul > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.2em; }
.prose-xl :where(.prose > ol > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.2em; }
.prose-xl :where(.prose > ol > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.2em; }
.prose-xl :where(.prose > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 0px; }
.prose-xl :where(.prose > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 0px; }
.prose-2xl :where(.prose > ul > li p):not(:where([class~="not-prose"] *)) { margin-bottom: 0.833333em; margin-top: 0.833333em; }
.prose-2xl :where(.prose > ul > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.33333em; }
.prose-2xl :where(.prose > ul > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.33333em; }
.prose-2xl :where(.prose > ol > li > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 1.33333em; }
.prose-2xl :where(.prose > ol > li > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 1.33333em; }
.prose-2xl :where(.prose > :first-child):not(:where([class~="not-prose"] *)) { margin-top: 0px; }
.prose-2xl :where(.prose > :last-child):not(:where([class~="not-prose"] *)) { margin-bottom: 0px; }
.form-input, .form-multiselect, .form-select, .form-textarea { --tw-shadow: 0 0 transparent; appearance: none; background-color: rgb(255, 255, 255); border-color: rgb(155, 155, 155); border-radius: 0px; border-width: 1px; font-size: 1rem; line-height: 1.5rem; padding: 0.5rem 0.75rem; }
.form-input:focus, .form-multiselect:focus, .form-select:focus, .form-textarea:focus { --tw-ring-inset: var(--tw-empty,/*!*/ /*!*/); --tw-ring-offset-width: 0px; --tw-ring-offset-color: #fff; --tw-ring-color: #2563eb; --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(1px + var(--tw-ring-offset-width)) var(--tw-ring-color); border-color: rgb(37, 99, 235); box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow); outline: transparent solid 2px; outline-offset: 2px; }
.form-input::placeholder, .form-textarea::placeholder { color: rgb(155, 155, 155); opacity: 1; }
.form-input::-webkit-datetime-edit-fields-wrapper { padding: 0px; }
.form-input::-webkit-date-and-time-value { min-height: 1.5em; }
.form-input::-webkit-datetime-edit, .form-input::-webkit-datetime-edit-day-field, .form-input::-webkit-datetime-edit-hour-field, .form-input::-webkit-datetime-edit-meridiem-field, .form-input::-webkit-datetime-edit-millisecond-field, .form-input::-webkit-datetime-edit-minute-field, .form-input::-webkit-datetime-edit-month-field, .form-input::-webkit-datetime-edit-second-field, .form-input::-webkit-datetime-edit-year-field { padding-bottom: 0px; padding-top: 0px; }
.btn { align-items: center; border-color: transparent; border-radius: 0.5rem; border-width: 1px; display: inline-flex; flex-shrink: 0; font-size: 0.875rem; font-weight: 500; justify-content: center; line-height: 1.25rem; min-height: 38px; padding: 0.5rem 0.75rem; pointer-events: auto; }
.btn.focus-visible, .btn:focus { outline: transparent solid 2px; outline-offset: 2px; }
.btn:focus-visible { outline: transparent solid 2px; outline-offset: 2px; }
.btn:disabled { cursor: not-allowed; opacity: 0.5; }
.ZnJ1aXRqdWljZQ .btn { border-radius: 9999px; padding-left: 0.875rem; padding-right: 0.875rem; }
.btn:active { opacity: 0.8; }
.btn-green, .btn-primary { --tw-bg-opacity: 1; --tw-text-opacity: 1; background-color: rgba(16,163,127,var(--tw-bg-opacity)); color: rgba(255,255,255,var(--tw-text-opacity)); }
.btn-green:hover, .btn-primary:hover { --tw-bg-opacity: 1; background-color: rgba(26,127,100,var(--tw-bg-opacity)); }
.btn-green.focus-visible, .btn-primary.focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(25,195,125,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.btn-green:focus-visible, .btn-primary:focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(25,195,125,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.btn-blue { --tw-bg-opacity: 1; --tw-text-opacity: 1; background-color: rgba(0,102,222,var(--tw-bg-opacity)); color: rgba(255,255,255,var(--tw-text-opacity)); }
.btn-blue:hover { --tw-bg-opacity: 1; background-color: rgba(29,78,216,var(--tw-bg-opacity)); }
.btn-blue.focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(29,78,216,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.btn-blue:focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(29,78,216,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.ZnJ1aXRqdWljZQ .btn-primary { --tw-bg-opacity: 1; --tw-text-opacity: 1; background-color: rgba(13,13,13,var(--tw-bg-opacity)); color: rgba(255,255,255,var(--tw-text-opacity)); }
.ZnJ1aXRqdWljZQ .btn-primary:hover { --tw-bg-opacity: 1; background-color: rgba(33,33,33,var(--tw-bg-opacity)); }
.ZnJ1aXRqdWljZQ .btn-primary.focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(155,155,155,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.ZnJ1aXRqdWljZQ .btn-primary:focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(155,155,155,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.ZnJ1aXRqdWljZQ .btn-primary:is(.dark *) { --tw-bg-opacity: 1; --tw-text-opacity: 1; background-color: rgba(249,249,249,var(--tw-bg-opacity)); color: rgba(13,13,13,var(--tw-text-opacity)); }
.ZnJ1aXRqdWljZQ .btn-primary:hover:is(.dark *) { --tw-bg-opacity: 1; background-color: rgba(236,236,236,var(--tw-bg-opacity)); }
.btn-danger { --tw-bg-opacity: 1; --tw-text-opacity: 1; background-color: rgba(239,68,68,var(--tw-bg-opacity)); color: rgba(255,255,255,var(--tw-text-opacity)); }
.btn-danger:hover { --tw-bg-opacity: 1; background-color: rgba(185,28,28,var(--tw-bg-opacity)); }
.btn-danger.focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(248,113,113,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.btn-danger:focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(248,113,113,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.btn-danger:disabled:hover { --tw-bg-opacity: 1; background-color: rgba(185,28,28,var(--tw-bg-opacity)); }
.btn-danger-outline { --tw-border-opacity: 1; --tw-text-opacity: 1; background-color: var(--main-surface-primary); border-color: rgba(220,38,38,var(--tw-border-opacity)); border-width: 1px; color: rgba(220,38,38,var(--tw-text-opacity)); }
.btn-danger-outline:hover { background-color: var(--main-surface-secondary); }
.btn-danger-outline.focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(220,38,38,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.btn-danger-outline:focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(220,38,38,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.btn-secondary { background-color: var(--main-surface-primary); border-color: var(--border-medium); border-width: 1px; color: var(--text-primary); font-size: 0.875rem; line-height: 1.25rem; }
.btn-secondary:hover { background-color: var(--main-surface-secondary); }
.btn-secondary.focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(103,103,103,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.btn-secondary:focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-opacity: 1; --tw-ring-color: rgba(103,103,103,var(--tw-ring-opacity)); --tw-ring-offset-width: 2px; box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); }
.btn-ghost:hover { background-color: rgba(0, 0, 0, 0.05); }
.btn-ghost.focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-color: rgba(0,0,0,.05); box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); outline: transparent solid 2px; outline-offset: 2px; }
.btn-ghost:focus-visible { --tw-ring-offset-shadow: var(--tw-ring-inset) 0 0 0 var(--tw-ring-offset-width) var(--tw-ring-offset-color); --tw-ring-shadow: var(--tw-ring-inset) 0 0 0 calc(2px + var(--tw-ring-offset-width)) var(--tw-ring-color); --tw-ring-color: rgba(0,0,0,.05); box-shadow: var(--tw-ring-offset-shadow),var(--tw-ring-shadow),var(--tw-shadow,0 0 transparent); outline: transparent solid 2px; outline-offset: 2px; }
.btn-ghost:hover:is(.dark *) { background-color: rgba(255, 255, 255, 0.1); }
.btn-small { min-height: 30px; padding: 0.25rem 0.5rem; }
.ZnJ1aXRqdWljZQ .btn-small { padding-left: 0.75rem; padding-right: 0.75rem; }
.btn-giant, .btn-large { min-height: 46px; }
.btn-large { padding-bottom: 0.75rem; padding-top: 0.75rem; }
.ZnJ1aXRqdWljZQ .btn-large { padding-left: 1rem; padding-right: 1rem; }
.btn-giant { font-size: 1rem; font-weight: 600; line-height: 1.5rem; padding: 0.625rem 1.25rem; }
.ZnJ1aXRqdWljZQ .btn-giant { padding-left: 1.5rem; padding-right: 1.5rem; }
.divide-token-border-light > :not([hidden]) ~ :not([hidden]) { border-color: rgba(0, 0, 0, 0.1); }
.border-token-border-heavy { border-color: rgba(0, 0, 0, 0.2); }
.border-token-border-light { border-color: rgba(0, 0, 0, 0.1); }
.border-token-border-medium { border-color: rgba(0, 0, 0, 0.15); }
.border-token-border-xheavy { border-color: rgba(0, 0, 0, 0.25); }
.border-token-main-surface-secondary { }
.border-token-main-surface-tertiary { }
.border-token-surface-error\/15 { }
.border-token-text-primary { }
.border-token-text-secondary { }
.border-token-text-tertiary { }
.border-y-token-border-medium { border-bottom-color: rgba(0, 0, 0, 0.15); border-top-color: rgba(0, 0, 0, 0.15); }
.border-l-token-sidebar-surface-secondary { }
.\!bg-token-main-surface-primary { }
.\!bg-token-text-primary { }
.bg-token-border-light { background-color: rgba(0, 0, 0, 0.1); }
.bg-token-border-medium { background-color: rgba(0, 0, 0, 0.15); }
.bg-token-main-surface-primary { }
.bg-token-main-surface-secondary { }
.bg-token-main-surface-tertiary { }
.bg-token-sidebar-surface-primary { }
.bg-token-sidebar-surface-secondary { }
.bg-token-sidebar-surface-tertiary { }
.bg-token-surface-error { background-color: rgb(theme(colors.red.700)/var(--tw-bg-opacity)); }
.bg-token-text-primary { }
.bg-token-text-quaternary { }
.bg-token-text-secondary { }
.bg-token-text-tertiary { }
.fill-token-main-surface-primary { }
.stroke-token-main-surface-tertiary { }
.\!text-token-main-surface-primary { }
.\!text-token-text-primary { }
.text-token-link { color: rgb(41, 100, 170); }
.text-token-main-surface-primary { }
.text-token-main-surface-tertiary { }
.text-token-text-error { }
.text-token-text-primary { }
.text-token-text-quaternary { }
.text-token-text-secondary { }
.text-token-text-tertiary { }
.decoration-token-link { text-decoration-color: rgb(41, 100, 170); }
.decoration-token-text-primary { }
.caret-token-text-primary { }
body, html { background-color: var(--main-surface-primary); }
#__next, #root, body, html { height: 100%; }
.markdown { max-width: none; }
.markdown h1 { font-weight: 600; }
.markdown h1:first-child { margin-top: 0px; }
.markdown h2 { font-weight: 600; margin-bottom: 1rem; margin-top: 2rem; }
.markdown h2:first-child { margin-top: 0px; }
.markdown h3 { font-weight: 600; margin-bottom: 0.5rem; margin-top: 1rem; }
.markdown h3:first-child { margin-top: 0px; }
.markdown h4 { font-weight: 600; margin-bottom: 0.5rem; margin-top: 1rem; }
.markdown h4:first-child { margin-top: 0px; }
.markdown h5 { font-weight: 600; }
.markdown h5:first-child { margin-top: 0px; }
.markdown blockquote { --tw-border-opacity: 1; border-color: rgba(155,155,155,var(--tw-border-opacity)); border-left-width: 2px; line-height: 1.5rem; margin: 0px; padding-bottom: 0.5rem; padding-left: 1rem; padding-top: 0.5rem; }
.markdown blockquote > p { margin: 0px; }
.markdown blockquote > p::after, .markdown blockquote > p::before { display: none; }
.markdown table { --tw-border-spacing-x: 0px; --tw-border-spacing-y: 0px; border-collapse: separate; border-spacing: var(--tw-border-spacing-x) var(--tw-border-spacing-y); margin-bottom: 0.25rem; margin-top: 0.25rem; width: 100%; }
.markdown th { background-color: var(--border-light); border-bottom-width: 1px; border-color: var(--border-medium); border-left-width: 1px; border-top-width: 1px; padding: 0.25rem 0.75rem; }
.markdown th:first-child { border-top-left-radius: 0.375rem; }
.markdown th:last-child { border-right-width: 1px; border-top-right-radius: 0.375rem; }
.markdown td { border-bottom-width: 1px; border-color: var(--border-medium); border-left-width: 1px; padding: 0.25rem 0.75rem; }
.markdown td:last-child { border-right-width: 1px; }
.markdown tbody tr:last-child td:first-child { border-bottom-left-radius: 0.375rem; }
.markdown tbody tr:last-child td:last-child { border-bottom-right-radius: 0.375rem; }
.markdown a { color: var(--link); font-weight: 400; text-decoration-line: none; }
.markdown a:hover { color: var(--link-hover); }
.gizmo .markdown > :not(pre), .gizmo .markdown > :not(pre) > :not(a) { color: var(--text-primary); }
button.scroll-convo { display: none; }
@keyframes blink {
100% { visibility: hidden; }
}
.result-streaming > :not(ol):not(ul):not(pre):last-child::after, .result-streaming > pre:last-child code::after { content: "●"; font-family: Circle, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif; line-height: normal; margin-left: 0.25rem; vertical-align: baseline; }
.result-thinking:empty:last-child::after { -webkit-font-smoothing: subpixel-antialiased; animation: 1.25s ease-in-out 0s infinite normal none running pulseSize; backface-visibility: hidden; background-color: var(--text-primary); border-radius: 50%; box-sizing: border-box; content: " "; display: block; height: 12px; position: absolute; top: 7px; transform: translateZ(0px); transform-origin: center center; width: 12px; will-change: transform; }
@supports (selector(:has(*))) {
.result-streaming > ol:last-child > li:last-child > ol:last-child > li:last-child::after, .result-streaming > ol:last-child > li:last-child > ol:last-child > li:last-child > ol:last-child > li:last-child::after, .result-streaming > ol:last-child > li:last-child > ol:last-child > li:last-child > ul:last-child > li:last-child::after, .result-streaming > ol:last-child > li:last-child > ul:last-child > li:last-child::after, .result-streaming > ol:last-child > li:last-child > ul:last-child > li:last-child > ol:last-child > li:last-child::after, .result-streaming > ol:last-child > li:last-child > ul:last-child > li:last-child > ul:last-child > li:last-child::after, .result-streaming > ul:last-child > li:last-child > ol:last-child > li:last-child::after, .result-streaming > ul:last-child > li:last-child > ol:last-child > li:last-child > ol:last-child > li:last-child::after, .result-streaming > ul:last-child > li:last-child > ol:last-child > li:last-child > ul:last-child > li:last-child::after, .result-streaming > ul:last-child > li:last-child > ul:last-child > li:last-child::after, .result-streaming > ul:last-child > li:last-child > ul:last-child > li:last-child > ol:last-child > li:last-child::after, .result-streaming > ul:last-child > li:last-child > ul:last-child > li:last-child > ul:last-child > li:last-child::after, .result-streaming > ul:last-child > li:last-child[*|\:not-has\(]::after { content: "●"; font-family: Circle, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif; line-height: normal; margin-left: 0.25rem; vertical-align: baseline; }
.result-streaming > ul:last-child > li:last-child:not(:has(* > li))::after { content: "●"; font-family: Circle, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif; line-height: normal; margin-left: 0.25rem; vertical-align: baseline; }
.result-streaming > ol:last-child > li:last-child[*|\:not-has\(]::after { content: "●"; font-family: Circle, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif; line-height: normal; margin-left: 0.25rem; vertical-align: baseline; }
.result-streaming > ol:last-child > li:last-child:not(:has(* > li))::after { content: "●"; font-family: Circle, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif; line-height: normal; margin-left: 0.25rem; vertical-align: baseline; }
}
@supports not (selector(:has(*))) {
.result-streaming > ol:last-child > li:last-child::after, .result-streaming > ul:last-child > li:last-child::after { content: "●"; font-family: Circle, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif; line-height: normal; margin-left: 0.25rem; vertical-align: baseline; }
}
@keyframes pulseSize {
0%, 100% { transform: scaleX(1); }
50% { transform: scale3d(1.25, 1.25, 1); }
}
@keyframes toast-open {
0% { opacity: 0; transform: translateY(-100%); }
100% { transform: translateY(0px); }
}
@keyframes toast-close {
0% { opacity: 1; }
100% { opacity: 0; }
}
button[data-state="closed"] .toggle-item-button-closed, button[data-state="open"] .toggle-item-button-open { display: none; }
.toast-root { align-items: center; display: flex; flex-direction: column; height: 0px; transition: all 0.24s cubic-bezier(0, 0, 0.2, 1) 0s; }
.toast-root[data-state="entered"], .toast-root[data-state="entering"] { animation: 0.24s cubic-bezier(0.175, 0.885, 0.32, 1) 0s 1 normal both running toast-open; }
.toast-root[data-state="exiting"] { animation: 0.12s cubic-bezier(0.4, 0, 1, 1) 0s 1 normal both running toast-close; }
.toast-root .alert-root { box-shadow: rgba(67, 90, 111, 0.3) 0px 0px 1px, rgba(67, 90, 111, 0.3) 0px 5px 8px -4px; flex-shrink: 0; pointer-events: all; }
.interact-bounce { transition-duration: 0.15s; transition-property: transform; transition-timing-function: cubic-bezier(0.4, 0, 0.2, 1); }
.interact-bounce:hover { --tw-scale-x: 1.05; --tw-scale-y: 1.05; }
.interact-bounce:active, .interact-bounce:hover { transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); }
.interact-bounce:active { --tw-scale-x: .9; --tw-scale-y: .9; }
.title { font-feature-settings: normal; font-family: ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-variation-settings: normal; }
.icon-xs { stroke-width: 1.5; flex-shrink: 0; height: 0.75rem; width: 0.75rem; }
.icon-sm { stroke-width: 2; }
.icon-sm, .icon-sm-heavy { flex-shrink: 0; height: 1rem; width: 1rem; }
.icon-sm-heavy { stroke-width: 2.5; }
.icon-md { stroke-width: 1.5; }
.icon-md, .icon-md-heavy { flex-shrink: 0; height: 18px; width: 18px; }
.icon-md-heavy { stroke-width: 2.5; }
.icon-lg { stroke-width: 1.5; flex-shrink: 0; height: 1.5rem; width: 1.5rem; }
.icon-lg-heavy { stroke-width: 2; flex-shrink: 0; height: 22px; width: 22px; }
.icon-xl { stroke-width: 1.5; flex-shrink: 0; height: 1.75rem; width: 1.75rem; }
.icon-xl-heavy { stroke-width: 2; flex-shrink: 0; height: 24px; width: 24px; }
.icon-2xl { stroke-width: 1.5; flex-shrink: 0; height: 2rem; width: 2rem; }
.icon-workspace-avatar-preview { stroke-width: 1.5; height: 96px; width: 96px; }
.icon-cover { stroke-width: 1.5; height: 234px; width: 234px; }
.icon-shimmer { animation: 5s cubic-bezier(0.2, 0.44, 0.38, 1.02) 0s infinite normal none running icon-shimmer; background-repeat: no-repeat; mask: linear-gradient(80deg, currentcolor 30%, rgba(0, 0, 0, 0.333), currentcolor 70%) right center / 300% 100%; }
@keyframes icon-shimmer {
0% { mask-position: right center; }
20% { mask-position: left center; }
100% { mask-position: left center; }