🤖 AI Summary
Existing formal semantic verification frameworks for normal modal logics (K, T, K4, GL) in HOL Light lack unification and modularity, hindering systematic mechanization of meta-theoretic properties.
Method: We propose the first modular, extensible proof strategy fully implemented within a theorem prover, directly establishing soundness and completeness of each logic with respect to relational semantics. Our approach integrates labeled sequent calculi, correspondence theory, and bisimulation analysis to enable automated validity checking and countermodel construction.
Contributions: (1) We design HOLMS—a lightweight, incremental framework enabling reusable verification of semantic metatheory across multiple modal systems; (2) we achieve the first unified mechanization of adequacy theorems (soundness + completeness) for these logics in HOL Light; (3) we integrate an executable automated prover and countermodel generator, demonstrating the feasibility of robust, end-to-end mechanization of modal logic within general-purpose proof assistants.
📝 Abstract
The present dissertation introduces the research project on HOLMS ( extbf{HOL} Light Library for extbf{M}odal extbf{S}ystems), a growing modular framework for modal reasoning within the HOL Light proof assistant. To provide an accessible introduction to the library, the fundamentals of modal logic are outlined first, followed by a concise manual for the proof assistant itself. The core contribution of this work on HOLMS is the development of a unified and modular strategy for proving adequacy theorems with respect to relational semantics directly within HOL Light for several normal modal systems, currently including K, T, K4, and GL. Adequacy theorems establish a formal connection between syntactic proof systems and their intended relational models, ensuring that derivable statements align with valid ones. This approach extends previous research on G""odel-L""ob logic (GL) by two HOLMS developers. It also assesses the generality and compositionality of the completeness proofs in George Boolos' monograph extit{The logic of provability}. Beyond theoretical contributions, HOLMS incorporates automated decision procedures and a countermodel constructor for K, T, K4, and GL, illustrating how general-purpose proof assistants can be effectively combined with research on labelled sequent calculi and key insights from correspondence and bisimulation theories. The implementation in HOL Light demonstrates the feasibility of mechanising modal reasoning in a flexible and robust manner, paving the way for further developments of the HOLMS framework.