Provable Reductions in TFNP

2026-06-26

AI Summary

This work investigates how to characterize the strength of propositional proof systems via provable reductions to TFNP search problems. To this end, it introduces a novel class of implicit proof systems ⟨EF, R⟩, where R is a TFNP problem such that the task of finding falsifying assignments for unsatisfiable formulas reduces polynomially to R, and the correctness of this reduction is verifiable in Extended Frege (EF). The main contributions include establishing, for the first time, a polynomial equivalence between ⟨EF, Resolution⟩ and the classical sequent calculus G₁; proving that ⟨EF, Iter⟩ is likewise equivalent to both G₁ and ⟨EF, Resolution⟩; and demonstrating that EF-provably correct reductions are strictly stronger than FP-computability. Moreover, for any sufficiently strong proof system P, there exists a search problem Rₚ in FP such that ⟨EF, Rₚ⟩ is polynomially equivalent to P.

Abstract

We introduce a new family of propositional proof systems, denoted <EF, R>, for an arbitrary TFNP search problem $R$. Informally, a refutation of a CNF formula $F$ in <EF, R> is given by a polynomial-time reduction from the false-clause search problem $Search_F$ to $R$, combined with an Extended Frege proof that the reduction is correct. These are motivated in two ways: 1. They are the propositional translations of witnessing theorems in bounded arithmetic, by which proofs of $\forall Σ^b_1$ formulas $φ$ in a theory $T$ imply algorithms solving the search problem for $φ$ in a TFNP class corresponding to $T$. 2. They are a white-box analogue of the characterizations of proof systems using decision tree reductions to black-box TFNP problems. We consider the proof system <EF, Iter>, where Iter is a complete problem for PLS. We prove that <EF, Iter> is polynomially equivalent to the sequent calculus $G_1$, and also to the implicit Resolution proof system [EF, Resolution]. Hence $G_1$ and [EF, Resolution] are equivalent, which is the first characterization of an implicit proof system by a classical proof system beyond the work of Wang. We also consider <EF, R> for general TFNP relations $R$. We observe that if EF can prove that a search problem $R$ is in FP, then <EF, R> is polynomially equivalent to EF. This contrasts to our above result, which shows that Extended-Frege provable reductions to $Iter$, a problem widely believed not to be in FP, yields a proof system ($G_1$) that is believed to be stronger than Extended Frege. Finally, we show that for any proof system $P$ which is sufficiently strong, there is a polynomial-time computable search problem $R_P \in $ FP such that <EF, $R_P$> is polynomially equivalent to $P$. Letting $P =$ [EF, Resolution] and combining our two results shows that <EF, Iter> is polynomially equivalent to <EF, $R_{[EF, Resolution]}$>.

Problem

• TFNP
• propositional proof systems
• polynomial reductions
• Extended Frege
• search problems

Innovation

• propositional proof systems
• TFNP reductions
• Extended Frege
• implicit proof systems
• bounded arithmetic