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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.
• 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 language | English |
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Publisher | Technical University of Denmark |
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Number of pages | 175 |
Publication status | Published - 2023 |
Keywords
- Propositional logic
- Henkin-style completeness
- Isabelle/HOL
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Formally Correct Deduction Methods for Computational Logic
From, A. H. (PhD Student), Paulson, L. C. (Examiner), Villadsen, J. (Main Supervisor), Gierasimczuk, N. (Supervisor) & Braüner, T. (Examiner)
01/02/2020 → 14/06/2023
Project: PhD