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The categorical models of the differential lambda-calculus are additive categories because of the Leibniz rule which requires the summation of two expressions. This means that, as far as the differential lambda-calculus and differential…
In this paper, we study logics of bounded distributive residuated lattices with modal operators considering $\Box$ and $\Diamond$ in a noncommutative setting. We introduce relational semantics for such substructural modal logics. We prove…
Propositional term modal logic is interpreted over Kripke structures with unboundedly many accessibility relations and hence the syntax admits variables indexing modalities and quantification over them. This logic is undecidable, and we…
We define a framework for incorporating alternation-free fixpoint logics into the dual-adjunction setup for coalgebraic modal logics. We achieve this by using order-enriched categories. We give a least-solution semantics as well as an…
We investigate the expressive power of the two main kinds of program logics for complex, non-regular program properties found in the literature: those extending propositional dynamic logic (PDL), and those extending the modal mu-calculus.…
The paper is dedicated to the problem of adding a modality to the \Lukasiewicz many-valued logics in the purpose of obtaining completeness results for Kripke semantics. We define a class of modal many-valued logics and their corresponding…
First-order game logic GL and the first-order modal mu-calculus Lmu are proved to be equiexpressive and equivalent, thereby fully aligning their expressive and deductive power. That is, there is a semantics-preserving translation from GL to…
We prove a conjecture stated in a previous paper by the author about the existence of canonical filtrations for a family of vertex operator algebras in rational levels.
In this paper, we investigate the many-valued version of coalgebraic modal logic through predicate lifting approach. Coalgebras, understood as generic transition systems, can serve as semantic structures for various kinds of modal logics. A…
The paper explores properties of the {\L}ukasiewicz {\mu}-calculus, or {\L}{\mu} for short, an extension of {\L}ukasiewicz logic with scalar multiplication and least and greatest fixed-point operators (for monotone formulas). We observe…
Modal separability for modal fixpoint formulae is the problem to decide for two given modal fixpoint formulae $\varphi,\varphi'$ whether there is a modal formula $\psi$ that separates them, in the sense that $\varphi\models\psi$ and…
We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies…
This paper proposes a basic proof theoretic framework for major modal logics: {\sf S5} and some of its subsystems. The framework is based on a version of hypersequent calculus, and the basic modal systems we handle here are the system {\sf…
In this paper, we generalize modal $\mu$-calculus to the non-distributive (lattice-based) modal $\mu$-calculus and formalize some scenarios regarding categorization using it. We also provide a game semantics for the developed logic. The…
This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e. the original problem is first reformulated as a nonconvex optimization problem, its well-posedness…
We study the completeness problem for propositionally quantified modal logics on quantifiable general frames, where the admissible sets are the propositions the quantifiers can range over and expressible sets of worlds are admissible, and…
We investigate quantitative extensions of modal logic and the modal mu-calculus, and study the question whether the tight connection between logic and games can be lifted from the qualitative logics to their quantitative counterparts. It…
We explore the theory of illfounded and cyclic proofs for the propositional modal $\mu$-calculus. A fine analysis of provability for classical and intuitionistic modal logic provides a novel bridge between finitary, cyclic and illfounded…
Modal formulae express monadic second-order properties on Kripke frames, but in many important cases these have first-order equivalents. Computing such equivalents is important for both logical and computational reasons. On the other hand,…
The paper explores properties of {\L}ukasiewicz mu-calculus, a version of the quantitative/probabilistic modal mu-calculus containing both weak and strong conjunctions and disjunctions from {\L}ukasiewicz (fuzzy) logic. We show that this…