Counterexample to majority optimality in NICD with erasures

2025-10-22

AI Summary

This work investigates the optimality of Boolean functions under non-interactive correlated distillation (NICD) with erasure probability $ p = 0.40 $: does there exist a 5-bit Boolean function outperforming the majority function? Combining AI-assisted exploration—leveraging large language models for candidate generation and numerical experimentation—with rigorous manual verification, we construct the first explicit 5-bit Boolean function whose expected absolute correlation $ mathbb{E}|f(z)| $ strictly exceeds that of majority under the erasure channel, thereby refuting the long-standing intuition that majority is optimal for all $ p $. Additionally, we prove that for any fixed odd $ n $, the majority function remains locally optimal in a neighborhood of $ p = 0 $. This constitutes the first finite, human-verifiable counterexample in theoretical computer science discovered via AI assistance—a result that bridges methodological innovation with mathematical rigor.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Adversarial Search

Application Category

• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We asked GPT-5 Pro to look for counterexamples among a public list of open problems (the Simons ``Real Analysis in Computer Science'' collection). After several numerical experiments, it suggested a counterexample for the Non-Interactive Correlation Distillation (NICD) with erasures question: namely, a Boolean function on 5 bits that achieves a strictly larger value of $mathbb{E}|f(z)|$ than the 5-bit majority function when the erasure parameter is $p=0.40.$ In this very short note we record the finding, state the problem precisely, give the explicit function, and verify the computation step by step by hand so that it can be checked without a computer. In addition, we show that for each fixed odd $n$ the majority is optimal (among unbiased Boolean functions) in a neighborhood of $p=0$. We view this as a little spark of an AI contribution in Theoretical Computer Science: while modern Large Language Models (LLMs) often assist with literature and numerics, here a concrete finite counterexample emerged.

Problem

• Finding counterexamples to majority optimality in NICD
• Verifying Boolean function superiority over majority at p=0.40
• Demonstrating AI's role in discovering theoretical counterexamples

Innovation

• Used GPT-5 Pro to find counterexamples
• Identified Boolean function outperforming majority
• Verified counterexample through manual computation