analyze frame conditions

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 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.

Verifying low-level system software (e.g., OS kernels, device drivers) requires unified modeling of hardware-awareness, concurrency, and memory safety—properties that are notoriously difficult to capture coherently within a single formal framework. Method: This paper proposes a resource-contextual modal logic verification paradigm. It systematically identifies and formalizes modal verification design patterns, embedding resource constraints—such as memory regions, lock states, and device register configurations—directly into the semantics of modal operators. This enables structured, compositional specification and verification of system-level behaviors. Contribution/Results: We establish the first reusable modal modeling paradigm tailored to systems software, bridging the gap between modal logic theory and systems engineering practice. Evaluated on multiple real-world case studies, our approach demonstrates strong effectiveness and scalability, significantly enhancing both specification expressivity and verification automation.

This paper addresses the lack of a unified formal framework for modeling operational semantics of programming languages and verifying program correctness. We propose a novel unifying framework based on multi-sorted hybrid modal logic—the first application of such a logic to operational semantics modeling—significantly reducing representational distance in semantic encoding. Compared with dynamic logic, our approach more naturally captures program execution dynamics; relative to traditional weakest precondition calculi, it offers superior expressiveness and semantic clarity. The framework uniformly supports semantic definition, property specification, and formal verification. Crucially, we establish key completeness results, thereby laying a theoretically rigorous foundation that retains practical expressivity for formal program verification.

Craig interpolation fails for most normal modal logics extending K4.3 (excluding S5) and for Priorean temporal logics over ℤ, ℚ, ℝ, and finite linear orders—posing a fundamental challenge to structural proof theory and model-theoretic analysis. Method: We systematically investigate the decidability of *uniform* interpolation (i.e., existence of interpolants independent of proof details) and establish its first bisimulation-based semantic characterization. Using descriptive frames, canonical models, and model-theoretic techniques, we analyze interpolation across diverse temporal flows. Contribution/Results: We prove that uniform interpolation existence is coNP-complete for every finitely axiomatizable modal logic containing K4.3—matching the complexity of logical consequence. This result uniformly covers standard temporal logics over ℤ, ℚ, ℝ, and finite linear orders. Our work establishes both decidability and precise computational complexity, surpassing prior work in both accuracy and generality for uniform interpolation in modal and temporal logics.

This study addresses the structural relationship between relevant logic and normal modal logic by systematically developing, for the first time, a formal translation mechanism from relevant logic into normal modal logic. By integrating semantic analyses from both logical frameworks, the work rigorously establishes a precise correspondence between their structures. The resulting framework not only uncovers deep connections between the two systems and yields several significant corollaries, but also opens new avenues for comparative research across logical systems. Furthermore, it identifies promising directions for future investigation that warrant deeper exploration.

This study addresses the challenge of effectively transferring soundness and completeness from highly expressive modal logics to weaker linguistic systems. To this end, it proposes a novel strategy that leverages semantic insensitivity to carry over soundness and employs faithful translations to embed the canonical model of the target logic into the framework of normal modal logic, thereby inheriting completeness. The approach unifies various notions of operator definability and allows the standard relational semantics to be inherited without explicitly specifying an accessibility relation. Consequently, it offers a general and streamlined method for constructing semantics for weak modal languages, substantially reducing the complexity of their metatheoretic analysis.

This work investigates the syntactic embedding of the classical modal logic S4 into Boolean Bunched Implication logic (BBI), preserving validity under hypothetical reasoning and axiomatic extensions. Inspired by Gödel’s embedding of intuitionistic logic into S4, the authors present, for the first time, a complete Hilbert-style deductive theorem proof for BBI. Building on this result, they construct robust syntactic embeddings of S4 into BBI and several of its extensions—including hybrid BBI and classical BI. The proposed method uniformly applies to arbitrary axiomatic extensions of both S4 and BBI, thereby uncovering deep semantic connections between modal logics and resource-sensitive logics.

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 study addresses the lack of a unified treatment for four classes of modal connectives in intuitionistic modal logic by systematically integrating the Prenosil-style and Wijesekera-style pairs of connectives, thereby constructing a coherent syntactic and semantic framework. Employing model-theoretic methods, frame semantics, and axiomatic system construction techniques, the work clarifies the semantic relationships among these four connective classes and establishes a unified analytical framework. The main contributions include proving that the minimal intuitionistic modal logics determined by the corresponding frame classes are all decidable and providing effective axiomatizations for several important frame classes, thus resolving their semantic definability and strong completeness problems.