Formalization of the Resolution Calculus for First-Order Logic

Anders Schlichtkrull*

*Corresponding author for this work

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Abstract

I present a formalization in Isabelle/HOL of the resolution calculus for first-order logic with formal soundness and completeness proofs. To prove the calculus sound, I use the substitution lemma, and to prove it complete, I use Herbrand interpretations and semantic trees. The correspondence between unsatisfiable sets of clauses and finite semantic trees is formalized in Herbrand’s theorem. I discuss the difficulties that I had formalizing proofs of the lifting lemma found in the literature, and I formalize a correct proof. The completeness proof is by induction on the size of a finite semantic tree. Throughout the paper I emphasize details that are often glossed over in paper proofs. I give a thorough overview of formalizations of first-order logic found in the literature. The formalization of resolution is part of the IsaFoL project, which is an effort to formalize logics in Isabelle/HOL.

Original languageEnglish
JournalJournal of Automated Reasoning
Volume61
Issue number1–4
Pages (from-to)455–484
ISSN0168-7433
DOIs
Publication statusPublished - 20 Jan 2018

Keywords

  • Completeness
  • First-order logic
  • Herbrand’s theorem
  • Isabelle/HOL
  • Resolution
  • Semantic trees
  • Soundness

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