[Paper Review] A Finite-Model-Theoretic View on Propositional Proof Complexity

Erich Grädel, Martin Grohe+2 more|arXiv (Cornell University)|Jan 1, 2019

Logic, Reasoning, and Knowledge5 citations

TL;DR

This paper establishes precise logical characterizations of propositional proof systems—Horn resolution, bounded-width resolution, and polynomial calculus with bounded degree—using fixed-point logics from finite model theory. It shows that Horn resolution captures least fixed-point logic, bounded-width resolution captures existential least fixed-point logic, and polynomial calculus over rationals captures fixed-point logic with counting, enabling new finite-model-theoretic tools for proving proof complexity lower bounds.

ABSTRACT

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. By exploring these connections further, we establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and over finite fields.

Motivation & Objective

• To bridge propositional proof complexity and finite model theory by identifying logical characterizations of key proof systems.
• To understand the expressive power of proof systems like Horn resolution, bounded-width resolution, and polynomial calculus through the lens of fixed-point logics.
• To develop finite-model-theoretic tools for proving lower bounds in proof complexity, particularly for polynomial calculus over rationals and finite fields.

Proposed method

• Using logical interpretations to map proof systems to fragments of fixed-point logic, such as least fixed-point logic and its extensions.
• Establishing equivalence between proof-theoretic proof length and definability in fixed-point logic with and without counting.
• Applying model-theoretic techniques, including Ehrenfeucht-Fraïssé games and pebble games, to analyze proof complexity lower bounds.
• Leveraging the finite model theory framework to analyze the expressive power of polynomial calculus with bounded degree over rationals.
• Proving that the solvability of propositional formulas in these systems corresponds exactly to definability in specific fixed-point logic variants.
• Using logical invariants and interpretations to transfer results from model theory to proof complexity.

Experimental results

Research questions

• RQ1 How can propositional proof systems like Horn resolution be logically characterized in terms of fixed-point logics?
• RQ2 What is the exact logical fragment captured by bounded-width resolution, and how does it relate to existential quantification in fixed-point logic?
• RQ3 To what extent does polynomial calculus with bounded degree over the rationals correspond to fixed-point logic with counting?
• RQ4 Can finite-model-theoretic methods be used to derive lower bounds for polynomial calculus over finite fields and rationals?
• RQ5 What logical invariants underlie the proof complexity of bounded-degree polynomial calculus?

Key findings

• Horn resolution has the same expressive power as least fixed-point logic, meaning it can define exactly the same classes of formulas.
• Bounded-width resolution captures existential least fixed-point logic, providing a logical characterization of its proof-theoretic limitations.
• Polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting.
• The logical characterizations enable new finite-model-theoretic tools for proving lower bounds in proof complexity.
• The results establish tight connections between syntactic proof systems and semantic logical definability in finite model theory.
• These connections allow for the transfer of model-theoretic techniques, such as pebble games, to the analysis of proof complexity.