Combinatorial Proofs and Decomposition Theorems for First-order Logic

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 a...

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Bibliographic Details
Published in:Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science pp. 1 - 13
Main Authors: Hughes, Dominic J. D., Strasburger, Lutz, Wu, Jui-Hsuan
Format: Conference Proceeding
Language:English
Published: IEEE 29.06.2021
Subjects:
Computer science
Syntactics
Online Access:Get full text
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Summary: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.
DOI:10.1109/LICS52264.2021.9470579