Formally Correct Deduction Methods for Computational Logic

Asta Halkjær From

Research output: Book/ReportPh.D. thesis

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Abstract

Some logics make it easier to say what we want than others and some methods of deduction are simpler to work with than others. One thing typically remains important: that we can deduce all valid things (completeness). In this thesis I tackle completeness of various deduction methods for various logics. The work is formalized in the proof assistant Isabelle/HOL which offers a common language for mathematics and computer science where proofs can be checked mechanically. This thesis discusses the following contributions of my PhD project in particular:
• A historical overview of formalized completeness proofs and a formalization that explains the essence of synthetic completeness proofs.
• A formalization of a concise completeness proof for first-order logic in Isabelle/HOL, including solutions to the issues of formalizing quantifiers and proving completeness for formulas with free variables.
• A verified prover for first-order logic with functions, with a formalized completeness proof that takes the search strategy of the prover into account.
• A synthetic completeness proof for a tableau system for basic hybrid logic which is both terminating and in the Seligman style where the proof rules reflect the local perspective that modal logic is based on.
• Formalized soundness and completeness results for epistemic and public announcement logic, instantiated to a range of concrete axiom systems.
• A best-first proof search tactic for the proof assistant Lean 4.
• An abstract framework for synthetic completeness proofs.
Original languageEnglish
PublisherTechnical University of Denmark
Number of pages175
Publication statusPublished - 2023

Keywords

  • Propositional logic
  • Henkin-style completeness
  • Isabelle/HOL

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