Combinatorial Proofs and Decomposition Theorems for First-order Logic
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.
Cite
@article{arxiv.2104.13124,
title = {Combinatorial Proofs and Decomposition Theorems for First-order Logic},
author = {Dominic Hughes and Lutz Straßburger and Jui-Hsuan Wu},
journal= {arXiv preprint arXiv:2104.13124},
year = {2021}
}
Comments
To be published in LICS 2021. This is the author version of the paper with full proofs in the appendix