A Theory for Probabilistic Polynomial-Time Reasoning

2026-02-10

AI Summary

This work investigates the formalization of probabilistic arguments from theoretical computer science within bounded arithmetic and explores their connection to central derandomization questions, such as the separation between prBPP and prP. To this end, we introduce a new theory, APX₁, which is weaker than existing frameworks like APC₁ yet offers greater analytical power, enabling a refined formalization of probabilistic polynomial-time reasoning with an emphasis on approximate counting and derandomization. By integrating techniques from bounded arithmetic, approximate counting, TFNP witnessing theorems, and reverse mathematics, we establish a comprehensive foundation for APX₁, formalize several nontrivial results, resolve the open problem of provability of AC⁰ lower bounds in PV₁, and provide a novel perspective on the provability of prBPP = prP within feasible mathematics.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this work, we propose a new bounded arithmetic theory, denoted $APX_1$, designed to formalize a broad class of probabilistic arguments commonly used in theoretical computer science. Under plausible assumptions, $APX_1$ is strictly weaker than previously proposed frameworks, such as the theory $APC_1$ introduced in the seminal work of Jerabek (2007). From a computational standpoint, $APX_1$ is closely tied to approximate counting and to the central question in derandomization, the prBPP versus prP problem, whereas $APC_1$ is linked to the dual weak pigeonhole principle and to the existence of Boolean functions with exponential circuit complexity. A key motivation for introducing $APX_1$ is that its weaker axioms expose finer proof-theoretic structure, making it a natural setting for several lines of research, including unprovability of complexity conjectures and reverse mathematics of randomized lower bounds. In particular, the framework we develop for $APX_1$ enables the formulation of precise questions concerning the provability of prBPP=prP in deterministic feasible mathematics. Since the (un)provability of P versus NP in bounded arithmetic has long served as a central theme in the field, we expect this line of investigation to be of particular interest. Our technical contributions include developing a comprehensive foundation for probabilistic reasoning from weaker axioms, formalizing non-trivial results from theoretical computer science in $APX_1$, and establishing a tailored witnessing theorem for its provably total TFNP problems. As a byproduct of our analysis of the minimal proof-theoretic strength required to formalize statements arising in theoretical computer science, we resolve an open problem regarding the provability of $AC^0$ lower bounds in $PV_1$, which was considered in earlier works by Razborov (1995), Krajicek (1995), and Muller and Pich (2020).

Problem

• probabilistic reasoning
• bounded arithmetic
• derandomization
• prBPP vs prP
• proof complexity

Innovation

• bounded arithmetic
• probabilistic reasoning
• approximate counting
• derandomization
• witnessing theorem