Combinatorial Proofs and Decomposition Theorems for First-order Logic

Lutz Strassburger, Research Director at Inria Research Centre Saclay – Île-de-France and Laboratoire d’Informatique (LIX).

Abstract:

In this talk I will discuss the problem of proof identity, and then give a gentle introduction to combinatorial proofs for first-order logic. I will also investigate the relationship between combinatorial and syntactic proofs. 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 will see 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.