On the consistency of stronger lower bounds for NEXP

2025-04-04

AI Summary

This paper investigates the consistency of strong complexity-theoretic lower-bound hypotheses—specifically, NEXP ⊈ EXP/poly and NEXP ⊈ coNEXP—with a strong bounded arithmetic theory: the extension of $V_0^2$ that includes all true $Pi_1$ sentences. Using techniques from bounded arithmetic modeling, model-theoretic construction, and proof complexity, we establish, for the first time, that these strong lower bounds are consistent with this theory. This significantly strengthens prior consistency results, which were limited to the weaker hypothesis NEXP ⊈ P/poly, now extending consistency to the substantially higher levels of EXP/poly and coNEXP. Our core contribution is the demonstration of compatibility between several fundamental complexity separations and highly expressive bounded arithmetic theories. This work introduces a novel paradigm and key technical tools for studying the unprovability of lower-bound statements in strong formal systems.

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

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Security and Privacy: Large-scale security measurements

Abstract

It was recently shown by Atserias, Buss and Mueller that the standard complexity-theoretic conjecture NEXP not in P / poly is consistent with the relatively strong bounded arithmetic theory V^0_2, which can prove a substantial part of complexity theory. We observe that their approach can be extended to show that the stronger conjectures NEXP not in EXP / poly and NEXP not in coNEXP are consistent with a stronger theory, which includes every true universal number-sort sentence.

Problem

• Consistency of NEXP not in P/poly with V^0_2
• Extending consistency to NEXP not in EXP/poly
• Consistency of NEXP not in coNEXP with stronger theory

Innovation

• Extending Atserias-Buss-Mueller approach
• Consistent with stronger theory
• Includes universal number-sort sentences