A Finite-Model-Theoretic View on Propositional Proof Complexity

Abstract

We establish new, and surprisingly tight, connections between propositionalproof complexity and finite model theory. Specifically, we show that the powerof several propositional proof systems, such as Horn resolution, bounded-widthresolution, and the monomial calculus of bounded degree, can be characterisedin a precise sense by variants of fixed-point logics that are of fundamentalimportance in descriptive complexity theory. Our main results are that Hornresolution has the same expressive power as least fixed-point logic, thatbounded-width resolution captures existential least fixed-point logic, and thatthe polynomial calculus with bounded degree over the rationals solves preciselythe problems definable in fixed-point logic with counting. We also study thebounded-degree polynomial calculus. Over the rationals, it captures fixed-pointlogic with counting if we restrict the bit-complexity of the coefficients. Forunrestricted coefficients, we can only say that the bounded-degree polynomialcalculus is at most as powerful as bounded variable infinitary counting logic,but a precise logical characterisation of its power remains an open problem.These connections between logics and proof systems allow us to establishfinite-model-theoretic tools for proving lower bounds for the polynomialcalculus over the rationals and also over finite fields. This is a corrected version of the paper (arXiv:1802.09377) publishedoriginally on January 23, 2019