axiomatization and decidability

This work addresses the absence of a unified proof-theoretic framework for mainstream modal logics by developing a hypersequent calculus that uniformly encompasses K and its standard extensions—including T, D, 4, B, and 5. The paper systematically elucidates the construction mechanisms underlying sequent calculi for various modal logics and provides syntactic cut-elimination proofs for all major systems except KB, KDB, and KTB. This framework not only clarifies the origins of both sequent and hypersequent formulations for logics such as S5 but also lays the groundwork for further extensions to quantified modal logics. By doing so, it significantly advances the systematicity and unification of proof theory in modal logic.

This study addresses the long-standing challenge of proving the finite model property (FMP) for a broad class of modal logics and rule-based systems. We introduce a novel method based on subpartition construction, integrating the framework of stable canonical rules, finite-height modal algebras, and modal space techniques—marking the first application of subpartitions to generate finite countermodels and establishing a synergistic analytical pathway linking algebraic and Kripke semantics. Our main contributions are: (1) a proof that all systems axiomatized by stable canonical formulas and rules over finite-height modal algebras possess the FMP; and (2) a characterization of a class of systems whose corresponding lattices admit splitting joins, revealing that their Kripke incompleteness degree is exactly 1. These results uniformly extend the scope of known FMP results and provide a new paradigm for investigating metalogical properties of modal systems.

This work addresses the long-standing lack of a systematic semantic characterization and effective proof system for intuitionistic monotonic modal logic (IM). It introduces, for the first time, a constructive neighborhood semantics that provides exact models for IM and its natural extensions. Building upon the structured sequent calculus for classical monotonic modal logic M, the paper develops a tailored structured sequent calculus for IM. Through proof-theoretic analysis, the calculus is shown to be sound, complete, and decidable. Furthermore, the study uncovers a deep analogy between IM and intuitionistic modal logic K, thereby establishing IM as the faithful intuitionistic counterpart of M.

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.

This paper addresses the conceptual fragmentation in sequent-style proof systems across modal, temporal, intuitionistic, conditional, and substructural logics, as well as the ill-defined distinction between “internal” and “external” calculi. We propose a unified classification framework grounded in the fundamental data structure of sequents. By systematically analyzing extant sequent calculi, we develop a hierarchical taxonomy and conduct meta-theoretic analysis of upward and downward logical translations. Crucially, we formally define—and thereby eliminate—the spurious internal/external distinction for the first time. We prove that upward translation preserves structural integrity, whereas downward translation is generally infeasible. Our results establish a principled comparability benchmark and formal assessment toolkit for cross-logical proof systems, advancing the unified modeling of structural proof theory and enabling robust automation support.

This study investigates the axiomatizability, finite model property, and decidability of products and semi-products of modal logics L and S5 under locally bounded depth. By integrating bisimulation games with algebraic semantics and model-theoretic techniques, the authors establish minimal axiomatizations for several product and semi-product logics and prove that these logics enjoy the product (semi-product) finite model property. They also construct explicit counterexamples demonstrating that certain such logics are not minimally axiomatizable. Furthermore, the paper establishes the local tabularity of these logics, from which it derives the decidability of first-order modal logic QL and its one-variable fragment extended with the Barcan formula.

This study addresses the problem of systematically constructing logical systems corresponding to program abstractions to support formal reasoning. By associating logical systems with finite abstract domains, the work proposes a general method: for a given abstract lattice, it constructs a logic whose Lindenbaum–Tarski algebra is isomorphic to the abstraction, and derives corresponding axioms and inference rules. This approach establishes, for the first time, a systematic connection between abstract interpretation and proof theory as well as algebraic logic, enabling logical modeling of non-Cartesian abstractions such as octagons. The resulting logical connectives and inference systems preserve the concretization map and, under suitable conditions, satisfy soundness and completeness. The framework naturally extends to Cartesian products, multi-variable settings, and non-Cartesian abstract domains.

This work proposes a novel semantic and proof-theoretic framework for the μ-calculus over the constructive modal logic CK. We first introduce a game semantics and establish its equivalence with the standard two-sorted Kripke semantics. Building on this game-theoretic foundation, we develop the first non-wellfounded labeled proof system tailored to the constructive μ-calculus. By integrating game semantics with non-wellfounded proof methods—a combination previously unexplored in this setting—our approach yields a sound and complete deductive system for the logic. This contribution not only provides a robust reasoning framework for constructive μ-calculi but also opens new avenues for proof-theoretic investigations of other non-classical modal logics.

This work addresses the need for reasoning in intuitionistic logic under temporal evolution and information updates by introducing modal and fixed-point operators to construct five intuitionistic dynamic logics. Building on bi-relational Kripke semantics, it develops corresponding Hilbert-style axiomatizations and circular sequent calculi, and employs model checking and compositional analysis to systematically investigate expressiveness, the finite model property, decidability, and computational complexity. The main contributions include the first analytic circular sequent calculus for intuitionistic epistemic and public announcement logics, a proof of the finite model property and decidability for bi-intuitionistic modal logic, and a resolution of the long-standing open problem of finite axiomatizability for intuitionistic linear temporal logic.

This study addresses the long-standing open problem of proof complexity for the intuitionistic modal logic FIK, which lies between CCDL and IK but whose computational complexity has remained unknown. We introduce, for the first time, a shallow sequent calculus for FIK featuring at most one level of nesting, and establish its syntactic completeness by proving the admissibility of the cut rule. Building upon this calculus, we combine cut elimination with careful complexity analysis to show that the decision problem for FIK lies within EXPSPACE. This result substantially improves upon the previously conjectured non-elementary upper bound for IK and represents a significant breakthrough in overcoming the high-complexity barrier commonly associated with such logics.