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Litan Kumar Das

Publications and source records attributed to Litan Kumar Das.

7 recordsLinked to original sources
arXiv2026

A Lindstr\"om Theorem for Fitting's Modal Logic over a Finite Heyting Algebra

We establish a Lindstr\"om-style maximality theorem for Maruyama's exact-truth-test presentation of Fitting's modal logic over a fixed finite Heyting algebra and crisp Kripke frames. Unlike the existing characterization over finite MTL-chains, no linearity or distinguished coatom is assumed. Exact truth tests yield Boolean tests for designated and non-designated values and a derived existential modality sufficient for the saturation argument. We prove that every abstract extension which is compact, has the Tarski Union Property, and is strongly invariant under bisimulation is $1$-expressively equivalent to Maruyama's version of Fitting's Heyting-valued modal logic. As a consequence, every exact-value fibre of an extension formula is definable in Maruyama's exact-truth-test modal language.

cs.LO↗
arXiv2026

A Bitopological Approach to Finite Reduction and Bounded Exact-Value Certificates for Fitting's Finite Heyting-valued Modal Logic

Fitting's finite Heyting-valued modal logic interprets modal formulas over a finite Heyting algebra. We use a relational bitopological representation to obtain a finite-state reduction. For a finite model and a finite vocabulary, the modal subalgebra generated by the atomic valuations determines a state-evaluation map. We prove that the observational quotient is isomorphic to its finite image in the bitopological dual and that the quotient relation is the restriction of the canonical dual relation. Hence every formula over the vocabulary preserves its exact truth value, and the quotient is minimal among surjective reductions through which all generated observations factor. In addition, for any formula and state, we construct a finite tree-like exact-value certificate whose depth is bounded by modal depth and whose branching depends only on the height of the truth-value algebra and the number of boxed subformulas. Failed formulas therefore admit bounded reduced counterexamples preserving their precise failure values.

cs.LO↗
arXiv2026

Positive Instantial Neighbourhood logic: Typed Completeness and Admissible-Open Representation

Instantial neighbourhood logic is a modal language for neighbourhood frames in which formulas can express information about the kinds of worlds occurring inside a neighbourhood of a given world. In this paper, we study a positive, negation- and implication-free version of instantial neighbourhood logic with two primitive instantial modalities, one of \(\Box\)-type and one of \(\Diamond\)-type. Since classical negation is not available, the two modalities are treated independently. We introduce the language and proof system of positive instantial neighbourhood logic (PINL) and interpret it over persistent two-sided neighbourhood models. We then define a typed persistent neighbourhood semantics, used as an auxiliary canonical semantics to control witness and co-witness conditions. This yields a truth lemma and a typed completeness theorem for PINL. On the algebraic side, we introduce \(2\)-$\mathrm{DLIO}$s, bounded distributive lattices equipped with two families of instantial operations, as the algebraic semantics of PINL. We prove algebraic soundness and completeness via the Lindenbaum \(2\)-$\mathrm{DLIO}$. Finally, we construct the canonical bitopological PINL-space and show that the algebra of its admissible positive opens is isomorphic to the Lindenbaum \(2\)-$\mathrm{DLIO}$. Thus the paper establishes a canonical admissible-open representation of positive instantial neighbourhood logic, providing a first step toward a future duality theory.

cs.LO↗
arXiv2023

Duality for Fitting's Multi-valued Modal logic via bitopology and biVietoris coalgebra

Fitting's Heyting-valued logic and Heyting-valued modal logic have already been studied from an algebraic viewpoint. In addition to algebraic axiomatizations with the completeness of Fitting's Heyting-valued logic and Heyting-valued modal logic, both topological and coalgebraic dualities have also been developed for algebras of Fitting's Heyting-valued modal logic. Bitopological methods have recently been employed to investigate duality for Fitting's Heyting-valued logic. However, the concepts of bitopology and bi-Vietoris coalgebras are conspicuously absent from the development of dualities for Fitting's many-valued modal logic. With this study, we try to bridge that gap. The main results are bitopological and coalgebraic duality for Fitting's many-valued modal logic. We develop a bitopological duality for algebras of Fitting's Heyting-valued modal logic by extending known bitopological duality for Fitting's non-modal logic. To develop coalgebraic duality, we adapt Lauridsen's bi-Vietoris construction from the category of pairwise Stone spaces to the category $PBS_{\mathcal{L}}$ of $\mathcal{L}$-valued (with $\mathcal{L}$ a bounded finite distributive lattice, i.e., a Heyting algebra) pairwise Boolean spaces by incorporating a structure map, and from this obtain the $\mathcal{L}$-biVietoris functor. Finally, we establish dual equivalence between coalgebras for the $\mathcal{L}$-biVietoris functor and algebras of Fitting's $\mathcal{L}$-valued modal logic. As a result, we conclude that Fitting's Heyting-valued modal logic is sound and complete with respect to the coalgebras of the $\mathcal{L}$-biVietoris functor. We also apply this coalgebraic approach to the bitopological duality to show the existence of cofree and final coalgebras and to establish a Hennessy-Milner property.

cs.LO↗
arXiv2022

Coalgebraic Fuzzy geometric logic

The paper aims to develop a framework for coalgebraic fuzzy geometric logic by adding modalities to the language of fuzzy geometric logic. Using the methods of coalgebra, the modal operators are introduced in the language of fuzzy geometric logic. To define the modal operators, we introduce a notion of fuzzy-open predicate lifting. Based on coalgebras for an endofunctor $T$ on the category $\textbf{Fuzzy-Top}$ of fuzzy topological spaces and fuzzy continuous maps, we build models for the coalgebraic fuzzy geometric logic. Bisimulations for the defined models are discussed in this work.

cs.LO↗
arXiv2018

Categorical study for Algebras of lattice-valued logic and lattice-valued modal logic

The paper explores categorical interconnections between lattice-valued Relational systems and algebras of Fitting's lattice-valued modal logic. We define lattice-valued boolean systems, and then we study co-adjointness, adjointness of functors. As a result, we get a duality for algebras of lattice-valued logic. Following this duality results, we establish a duality for algebras of lattice-valued modal logic

math.CT↗