kripke model construction

This paper addresses the lack of higher-order semantic frameworks for intuitionistic and non-classical modal logics. We introduce *higher-order Kripke models*, defining standard Kripke models as 0th-order, and recursively constructing an *n*-th-order model whose domain comprises all (*n*−1)-th-order models; modal operators are interpreted via accessibility relations between lower-order models. This recursive hierarchy enables the first higher-order abstraction of “possible worlds”, endowing them with the intuitive interpretation of “alternative timelines”. We construct first-order models for intuitionistic modal logic *IK* and a novel logic *MK*, verifying the modularity and extensibility of semantic clauses. The framework uniformly supports modal extensions of diverse non-classical logics—including intuitionistic, paraconsistent, and many-valued systems—and yields several open conjectures concerning expressive power and strong completeness.

Establishing foundational metatheoretic properties—particularly decidability, finite model property, and interpolation—for multi-agent S5 modal logic with agent alternation. Method: We introduce a novel proof system that integrates hypersequential and nested sequential structures, uniformly capturing both S5 modal reasoning and T-type agent alternation. The system is cut-free, sound, complete, and terminating. Contribution/Results: We rigorously establish decidability and the finite model property for the logic. This work delivers the first proof of Lyndon interpolation for multi-agent S5—including agent symbols—and achieves joint interpolation over both agent atoms and propositional atoms in the common language; Craig interpolation follows as a corollary. Furthermore, we identify and analyze fundamental obstructions to extending these results to distributed knowledge logic and to realizing deductive interpolation.

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 paper addresses the fundamental semantic divergence between CK and IK—two prominent intuitionistic modal logics—regarding the treatment of the possibility operator (◇), resolving their inconsistency over the ◇-free fragment and clarifying their conservativity over pure necessity (□) axiom systems. Methodologically, it extends CK’s Kripke semantics into a unified semantic framework, enabling the first precise characterization of frame conditions for IK and several classical axioms (e.g., T, 4, B). Building on this, the paper establishes definitive (non-)conservativity results for over a dozen intuitionistic modal logics with respect to ◇-free intuitionistic modal logic. All results are supported by machine-checked formal proofs in Coq, thereby settling multiple long-standing open problems on conservativity. The work provides a rigorous semantic foundation and principled guidance for axiomatization in intuitionistic modal logic.

This work addresses the limitation of existing distributed epistemic logics, which are typically grounded in simplicial complexes and thus struggle to express belief. For the first time, it formalizes belief within the simplicial complex semantics by introducing a multi-colored simplicial complex structure. By incorporating plausibility relations among states, the framework naturally accommodates multiple notions of belief. This approach significantly extends the expressive power of traditional simplicial models, offering a novel semantic foundation and modeling tool for reasoning about uncertain epistemic attitudes—such as belief—in multi-agent systems.

Existing simplicial complex belief models struggle to simultaneously satisfy the KD45 axioms, the knowledge-to-belief implication (Kφ → Bφ), and the standard coloring constraint—requiring each simplex to contain exactly one vertex of each color—without collapsing belief into knowledge, and typically rely on the fragile “properness” technical assumption, which is easily violated. This paper introduces the first nontrivial simplicial-set-based belief semantics for epistemic logic. Methodologically, it abandons simplicial complexes entirely in favor of simplicial sets, thereby eliminating the coloring constraint and dispensing with properness. The resulting model rigorously separates belief from knowledge—demonstrating that belief can be strictly weaker than knowledge—while fully validating KD45, Kφ → Bφ, and the semantic coloring condition. It is both logically sound and expressively adequate, establishing a novel mathematical foundation for distributed cognition and multi-agent belief reasoning.

This study investigates the semantic foundations of constructive modal logic CK, aiming to establish a systematic connection between its algebraic semantics and bi-relational semantics and to resolve the frame definability problem. To this end, the paper constructs, for the first time, a categorical duality framework for CK that unifies these two semantic perspectives. Building on this duality, the classical Sahlqvist correspondence theory and the Goldblatt–Thomason definability theorem from modal logic are extended to the constructive setting. The authors successfully prove Sahlqvist-type correspondence and strong completeness results for CK and provide a Goldblatt–Thomason-style characterization of the classes of frames definable in this logic.

Existing simplicial models support only knowledge representation and lack the capacity to formalize agent beliefs—particularly introspection and multi-agent belief consistency. Method: We introduce a novel doxastic logic semantics based on directed hypergraphs—the first such application in belief logic—combining the expressive power of simplicial structures with rigorous belief modeling. We develop a sound and complete axiomatization, construct canonical hypergraph models, and establish semantic completeness. Additionally, we design bidirectional translation algorithms between Kripke models and hypergraph models. Contribution/Results: The framework unifies knowledge and belief reasoning within a single semantic structure, overcoming fundamental limitations of traditional models in capturing belief semantics. It enables principled modeling of introspective beliefs and inter-agent belief alignment, while preserving logical tractability and compositional expressivity. This advances the theoretical foundations of epistemic and doxastic logic, particularly for distributed and self-aware multi-agent systems.

This work establishes a duality between relations among computational systems—such as bisimulation—and relations among logical predicates, thereby enabling cross-system logical reasoning. By extending Tarski duality and Thomason duality to the relational level for the first time, and integrating tools from category theory, Kripke semantics, and infinitary modal logic, the authors construct a dual framework that systematically links system relations with predicate relations. Building on this foundation, they develop a novel proof system capable of formally relating formulas across distinct systems. The resulting framework provides a robust theoretical basis for program logics and verification of concurrent systems, while significantly broadening the scope of classical duality theory within relational semantics.

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.