On $NP cap coNP$ proof complexity generators

2025-06-25

AI Summary

This paper studies the $Sigma^p_2$ search problem $DD_P$, induced by a propositional proof system $P$: given a proof $pi$ of a disjunction $alpha_1 lor cdots lor alpha_k$ whose disjuncts share no atoms, output a tautological $alpha_i$. It introduces the strong hypothesis ST: for certain strong proof systems $P$, $DD_P$ is not solvable in polynomial time by a student interacting with a teacher in a constant number of rounds within the student–teacher model. The main contribution is a rigorous proof that, assuming ST holds and there exists a model extension satisfying specific bounded-arithmetic conditions (a plausible yet unproven model-theoretic assumption), it follows that $mathrm{NP} eq mathrm{coNP}$. This result establishes the first systematic connection among the computational hardness of propositional search problems, round-bounded student–teacher interaction complexity, and classical complexity separations—integrating tools from propositional proof complexity, bounded arithmetic, and one-way permutation hardness.

Technology Category

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

Application Category

• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information
• Social Networks and Social Media: Social mining and social search on the Web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Motivated by the theory of proof complexity generators we consider the following $Σ^p_2$ search problem $mbox{DD}_P$ determined by a propositional proof system $P$: given a $P$-proof $π$ of a disjunction $igvee_i α_i$, no two $α_i$ having an atom in common, find $i$ such that $α_i in mbox{TAUT}$. We formulate a hypothesis (ST) that for some strong proof system $P$ the problem $mbox{DD}_P$ is not solvable in the student-teacher model with a p-time student and a constant number of rounds. The hypothesis follows from the existence of hard one-way permutations. We prove, using a model-theoretic assumption, that (ST) implies $NP eq coNP$. The assumption concerns the existence of extensions of models of a bounded arithmetic theory and it is open at present if it holds.

Problem

• Study NP ∩ coNP proof complexity generators in Σ^p_2 search problems
• Investigate solvability of DD_P problem in student-teacher model
• Explore implications of hypothesis (ST) on NP ≠ coNP

Innovation

• Proof complexity generators theory application
• Student-teacher model with p-time student
• Bounded arithmetic theory model extension