On the Unprovability of Circuit Size Bounds in Intuitionistic $mathsf{S}^1_2$

2024-04-18

AI Summary

This work investigates the limits of provability within the intuitionistic bounded arithmetic system $mathsf{IS}^1_2$ regarding circuit complexity lower bounds for SAT. Specifically, it addresses the question of whether $mathsf{IS}^1_2$ can prove that SAT lies outside co-nondeterministic circuits of fixed polynomial size. Technically, the paper introduces and analyzes a novel “refuter” model, which captures the fundamental obstacle to efficiently constructing infinitely-often $mathsf{coNP/poly}$ lower bounds in this system. The main result establishes, unconditionally, the existence of a constant $k$ such that $mathsf{IS}^1_2$ proves neither $mathrm{SAT} otin mathsf{coNSIZE}[n^k]$ nor $mathsf{NP} subseteq mathsf{coNSIZE}[n^k]$. This constitutes the first unconditional unprovability result for polynomial-size circuit lower bounds in bounded arithmetic. The work thus unifies and deepens the interplay among bounded arithmetic, propositional proof complexity, and circuit lower bound theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We show that there is a constant $k$ such that Buss's intuitionistic theory $mathsf{IS}^1_2$ does not prove that SAT requires co-nondeterministic circuits of size at least $n^k$. To our knowledge, this is the first unconditional unprovability result in bounded arithmetic in the context of worst-case fixed-polynomial size circuit lower bounds. We complement this result by showing that the upper bound $mathsf{NP} subseteq mathsf{coNSIZE}[n^k]$ is unprovable in $mathsf{IS}^1_2$. In order to establish our main result, we obtain new unconditional lower bounds against refuters that might be of independent interest. In particular, we show that there is no efficient refuter for the lower bound $mathsf{NP} subseteq mathsf{i.o} ext{-}mathsf{coNP}/mathsf{poly}$, addressing in part a question raised by Atserias (2006).

Problem

• Unprovability of SAT circuit size bounds in IS^1_2
• First unconditional unprovability in bounded arithmetic
• No efficient refuter for NP not in coNP/poly

Innovation

• Unconditional unprovability in bounded arithmetic
• New lower bounds against efficient refuters
• Addressing Atserias's question on circuit complexity