A Formally Verified Checker for First-Order Proofs

The Verified TESC Verifier (VTV) is a formally verified checker for the new Theory-Extensible Sequent Calculus (TESC) proof format for first-order ATPs. VTV accepts a TPTP problem and a TESC proof as input, and uses the latter to verify the unsatisfiability of the former. VTV is written in Agda, and the soundness of its proof-checking kernel is verified in respect to a first-order semantics formalized in Agda. VTV shows robust performance in a comprehensive test using all eligible problems from the TPTP problem library, successfully verifying all but the largest 5 of 12296 proofs, with >97% of the proofs verified in less than 1 second

A Proof-Theoretic Approach to Certifying Skolemization

International audienceWhen presented with a formula to prove, most theorem provers for classical first-order logic process that formula following several steps, one of which is commonly called skolemization. That process eliminates quantifier alternation within formulas by extending the language of the underlying logic with new Skolem functions and by instantiating certain quantifiers with terms built using Skolem functions. In this paper, we address the problem of checking (i.e., certifying) proof evidence that involves Skolem terms. Our goal is to do such certification without using the mathematical concepts of model-theoretic semantics (i.e., preservation of satisfiability) and choice principles (i.e., epsilon terms). Instead , our proof checking kernel is an implementation of Gentzen's sequent calculus, which directly supports quanti-fier alternation by using eigenvariables. We shall describe deskolemization as a mapping from client-side terms, used in proofs generated by theorem provers, into kernel-side terms, used within our proof checking kernel. This mapping which associates skolemized terms to eigenvariables relies on using outer skolemization. We also point out that the removal of Skolem terms from a proof is influenced by the polarities given to propositional connectives

A verified prover based on ordered resolution

International audienceThe superposition calculus, which underlies first-order theorem provers such as E, SPASS, and Vampire, combines ordered resolution and equality reasoning. As a step towards verifying modern provers, we specify, using Isabelle/HOL, a purely functional first-order ordered resolution prover and establish its soundness and refutational completeness. Methodologically, we apply stepwise refinement to obtain, from an abstract nondeterministic specification, a verified de-terministic program, written in a subset of Isabelle/HOL from which we extract purely functional Standard ML code that constitutes a semidecision procedure for first-order logic

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book