Combinatorial Proofs and Decomposition Theorems for First-order Logic

TL;DR

This paper explores the relationship between combinatorial and syntactic proofs in first-order logic, introducing a new normal form and proving soundness, completeness, and full completeness of combinatorial proofs.

Contribution

It establishes a deep connection between combinatorial and syntactic proofs, providing a new normal form and proving key completeness theorems.

Findings

Deep inference decomposition theorem for proofs

Normal form for syntactic proofs

Full completeness of combinatorial proofs

Abstract

We uncover a close relationship between combinatorial and syntactic proofs for first-order logic (without equality). Whereas syntactic proofs are formalized in a deductive proof system based on inference rules, a combinatorial proof is a syntax-free presentation of a proof that is independent from any set of inference rules. We show that the two proof representations are related via a deep inference decomposition theorem that establishes a new kind of normal form for syntactic proofs. This yields (a) a simple proof of soundness and completeness for first-order combinatorial proofs, and (b) a full completeness theorem: every combinatorial proof is the image of a syntactic proof.