filtration methods

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 paper resolves the decidability problem for quasi-dense modal logic. Prior work suffers from a fundamental flaw in canonical model construction and inaccurate complexity analysis. To address this, we introduce a novel path-based filtration method that carefully captures the semantic behavior of paths in the canonical model, enabling effective model reduction. Unlike traditional filtrations—which improperly truncate infinite branching—our approach preserves essential path structure, thereby significantly simplifying the decidability proof. Crucially, we correct and improve the complexity upper bound: whereas prior (erroneous) claims asserted EXPSPACE, we establish a tight NEXPTIME upper bound. Our results not only confirm decidability but also provide the first compact, rigorous, and constructive complexity characterization for this logic. This fills a key theoretical gap at the intersection of modal semantics and computational complexity.

This study addresses the computational complexity of the constructive modal logics CK* and WK*. By introducing their semantic characterizations and establishing mutual interpretability with fragments of propositional dynamic logic (PDL), the authors combine modal semantics with complexity-theoretic techniques to analyze these systems. They prove for the first time that both CK* and WK* are EXPTIME-complete and enjoy the exponential finite model property. Furthermore, the work confirms a conjecture by Afshari et al. regarding the EXPTIME-completeness of the diamond-free fragments of these logics and extends the result to show that the validity problems for CS4 and WS4 also reside in EXPTIME.

This paper addresses the cut-elimination problem for non-wellfounded proof systems in frugal logic, specifically under an interpretation of the exponential modality “!” as a finite datastream constructor—where global consistency and convergence must be ensured. We propose a progressing-criterion-based non-wellfounded cut-elimination method, yielding the first infinitary cut-elimination procedure in frugal logic that simultaneously preserves progressiveness and higher-order regularity. Using finite approximation techniques, we rigorously establish the convergence of this procedure to well-defined non-wellfounded proofs. Additionally, we develop a relational model semantics that provides a sound denotational foundation for the system. Our main contribution is the first structural proof-theoretic framework for frugal logic that jointly satisfies structural conservation (i.e., admissibility of cut), limit convergence of reduction sequences, and semantic soundness with respect to the relational model.

This paper addresses the fine-grained modeling of uncertainty and epistemic confusion (e.g., “being uncertain whether it is Monday or Tuesday”) by introducing KG_inv, a novel modal logic system. Methodologically, it pioneers the integration of involutive negation into Gödel modal logic, enabling the representation of belief tendencies and multi-alternative uncertainty. Semantically, KG_inv is grounded in [0,1]-valued Kripke models, for which a new finite-model semantics is established and proven equivalent to the standard semantics. Furthermore, a constraint tableaux calculus is devised—equipped with countermodel extraction—and shown to be sound and complete; logical validity is proven PSPACE-complete. The contributions thus include: (i) a semantically transparent, compact formal framework for uncertain belief reasoning; (ii) the first Gödel-based modal logic supporting involutive negation; (iii) a decidable, complexity-optimal proof system with effective countermodel generation.

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 finite-state reduction in finitely-valued Heyting modal logics that preserves the exact truth values of formulas. Building on relational bi-topological duality, the work proposes a minimality-preserving reduction method by constructing an observational quotient structure via evaluation maps induced by modal subalgebras, ensuring that all formulas retain their precise truth values in the reduced model. The main contributions include proving that this observational quotient is isomorphic to a finite image within its bi-topological dual; constructing tree-shaped certificates of exact truth values for any formula and state, whose depth is bounded by modal depth and whose branching depends on the height of the truth-value algebra and the number of boxed subformulas; and, for the first time, providing bounded counterexample certificates that preserve exact falsity values in cases of truth-value failure.

This work addresses the lack of the finite model property in traditional Gödel modal logics under standard Kripke semantics, which stems from their reliance on limit behaviors that yield non-constructive interpretations. To overcome this limitation, the paper introduces GW logic, equipped with a novel witnessed Kripke semantics that requires the truth of every modal formula to be explicitly witnessed by some accessible world, thereby eliminating non-constructive limit cases. Building on this semantics, the authors establish the first Gödel modal logic framework enjoying the finite model property and develop a corresponding refutation calculus together with a terminating backward proof-search algorithm. The calculus is proven sound and complete, enabling automated reasoning and countermodel generation, and substantially enhancing the constructivity and computability of the logical system.

This work establishes, for the first time, a Lindström-style maximality theorem for Fitting’s modal logic based on finite Heyting algebras and crisp Kripke frames, without assuming linearity or co-atomicity. By introducing precise truth-value tests, Booleanization techniques, and derived existential modalities, and by leveraging compactness, the Tarski union property, and strong bisimulation invariance, the paper develops an abstract model-theoretic framework tailored to non-classical settings. The main contribution is the proof that any abstract logic extending Fitting’s system and satisfying these three properties is 1-expressively equivalent to Maruyama’s version of Fitting logic, thereby establishing its maximal expressive power and yielding a definability characterization for each precise truth-value fiber.

This study addresses the challenge of uniformly formalizing complex statements that intertwine probability, action, and knowledge within fuzzy modal logic—such as “after performing action a, agent A knows that proposition p holds with probability 0.25.” To this end, the paper introduces a novel fuzzy modal logic equipped with a formal semantics based on Kripke frames augmented with probability measures. The primary contribution lies in the first unified integration of probability, action, and knowledge into a single fuzzy modal logical framework. Furthermore, the work identifies several logically distinct fragments of varying expressiveness, each admitting a satisfiability problem decidable in polynomial time, thereby establishing an upper bound on the satisfiability complexity of the logic over finitely branching models.