Game semantics for lattice-based modal {\mu}-calculus

EPiC series in computing, 2018

2006

Journal of Logic and Computation, 2009

This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones only evaluated in the idempotent elements and the ones evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truth-constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.

Information Sciences, 2011

An algebraic model of a kind of modal extension of de Morgan logic is described under the name MDS5 algebra. The main properties of this algebra can be summarized as follows: (1) it is based on a de Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N , K, T , and 5 (and as a consequence also B and 4) of modal logic is introduced; it allows one to introduce a modal possibility by the usual combination of necessity operation and de Morgan negation; (3) the necessity operator satisfies a distributivity principle over joins. The latter property cannot be meaningfully added to the standard Boolean algebraic models of S5 modal logic, since in this Boolean context both modalities collapse in the identity mapping. The consistency of this algebraic model is proved, showing that usual fuzzy set theory on a universe U can be equipped with a MDS5 structure that satisfies all the above points (1)-(3), without the trivialization of the modalities to the identity mapping. Further, the relationship between this new algebra and Heyting-Wajsberg algebras is investigated. Finally, the question of the role of these deviant modalities, as opposed to the usual nondistributive ones, in the scope of knowledge representation and approximation spaces is discussed.

Miskolc Mathematical Notes, 2008

In this work we give some results on the representation by means of relational structures of distributive lattices endowed with two modal operators and Þ, and two weak forms of negation and r.

International Journal of Theoretical Physics

In a recent work Foulis and Pulmannová [9] studied the logical connectives in lattice effect algebras. In this paper we extend their study and investigate further the logical calculus for which the lattice effect algebras can serve as semantic models. We shall first focus on some properties of lattice effect algebras and will then give a complete axiomatisation of this logic.

A non-empty universe with a collection of functions and predicates of finite arity, which is simply called a structure, induces a collection of corresponding accessibility relations constituting a generalized Kripke frame, so that a multimodal logic is introduced by a structure via the induced generalized Kripke frame. In this paper the authors discuss the problem of axiomatizing the multimodal logics thus introduced by structures in such a general setting as to cover many important classes of structures including, for instance, all expansions of implicative lattices, those of groups, and so on. First the least multimodal logic of the class of all structures with an arbitrarily fixed type is axiomatized featuring the “difference” modal operator, which is adjoined to those induced by the functions and predicates of given type. Then, on the basis of the axiomatization of the least multimodal logic, an axiomatization is given for the logic determined by a universal class of structures ...

PHD THESIS, INSTITUT FÜR INFORMATIK UND …, 2002

Advances in Modal Logic, 2024

Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal logics via algebraic representation theorems. We begin with complete lattices L equipped with an antitone operation ¬ sending 1 to 0, a completely multiplicative operation □, and a completely additive operation ◊. Such lattice expansions can be represented by means of a set X together with binary relations ◁, R, and Q, satisfying some first-order conditions, used to represent (L, ¬), □, and ◊, respectively. Indeed, any lattice L equipped with such a ¬, a multiplicative □, and an additive ◊ embeds into the lattice of propositions of a frame (X, ◁, R, Q). Building on our recent study of fundamental logic, we focus on the case where ¬ is dually self-adjoint (a ≤ ¬b implies b ≤ ¬a) and ◊¬a ≤ ¬□a. In this case, the representations can be constrained so that R = Q, i.e., we need only add a single relation to (X, ◁) to represent both □ and ◊. Using these results, we prove that a system of fundamental modal logic is sound and complete with respect to an elementary class of bi-relational structures (X, ◁, R).

arXiv (Cornell University), 2020

In this paper, we study logics of bounded distributive residuated lattices with modal operators considering ✷ and ✸ in a noncommutative setting. We introduce relational semantics for such substructural modal logics. We prove that any canonical logic is Kripke complete via discrete duality and canonical extensions. That is, we show that a modal extension of the distributive full Lambek calculus is the logic of its frames if its variety is closed under canonical extensions. After that, we establish a Priestleystyle duality between residuated distributive modal algebras and topological Kripke structures based on Priestley spaces.

A structure here is a non-empty universe together with a collection of functions and predicates. Such a structure is considered as a generalized Kripke frame whose set of possible worlds is endowed with a specific algebraic structure. Thus, a class of similar structures induces a certain multimodal logic. The authors axiomatize the basic modal logic of the class of algebras of arbitrary signature and give universal schemes for axiomatization of modal logic for universal and for Π 2 0 -classes of structures. They also discuss the connections of their approach with modal languages for complex algebras and promise to discuss a number of logical and algebraical consequences in a forthcoming full paper.

LICS 2013, 2013

We provide a characterization theorem, in the style of van Benthem and Janin-Walukiewicz, for the alternation-free fragment of the modal mu-calculus. For this purpose we introduce a variant of standard monadic second-order logic (MSO), which we call well-founded monadic second-order logic (WFMSO). When interpreted in a tree model, the second-order quantifiers of WFMSO range over subsets of well-founded subtrees. The first main result of the paper states that the expressive power of WFMSO over trees exactly corresponds to that of weak MSO-automata. Using this automata-theoretic characterization, we then show that, over the class of all transition structures, the bisimulation-invariant fragment of WFMSO is the alternation-free fragment of the modal mu-calculus. As a corollary, we find that the logics WFMSO and WMSO (weak monadic second-order logic, where second-order quantification concerns finite subsets), are incomparable in expressive power.

Metascience, 2014

The volume under review contains work dedicated to the memory of Leo Esakia, who died in 2010, after having worked for over 40 years towards developing duality theory for modal and intuitionistic logics. The collection comprises ten technical contributions that follow the first chapter, in which the reader can find information on Esakia's studies and career, as well as a complete list of his research publications. In the sequel, we will refer briefly to each of these ten chapters, following the order in the list of contents. B. Jónsson and A. Tarski, in two papers they published in the early 1950s in the American Journal of Mathematics, initiated the study of duality for Boolean algebras with additional operations, via the theory of canonical extensions. Esakia was among the first researchers who studied duality for lattices with additional operations [Topological Kripke models. Soviet Math. Dokl. 15 (1974), 147-151], in particular for Heyting algebras and S4 modal algebras. M. Gehrke, author of the second chapter, shows how distributive lattices, Heyting algebras and S4 modal algebras can be viewed as certain maps between distributive lattices and Boolean algebras. Furthermore, he shows how Stone duality follows from the canonical extension results and how both Priestley and Esakia duality can be derived from Stone duality. In the third chapter, N. Bezhanishvili, S. Ghilardi and M. Jibladze discuss the step-by-step method, i.e. how duality theory can be used to arrive at descriptions of finitely generated free algebras, thus shedding light on issues concerning modal propositional logics. The authors begin by recalling how this method works for free rank one modal logics and then, exploiting the method developed by D. Coumans and S. Van Gool [On generalizing free algebras for a functor. J. Logic Comput. 23 (2012), 645-672], show how it can be extended to work for logics of rank greater than one, such as T, K4 and S4. The paper ends with

1995

We reexamine the modal-calculus in the light of some classical theory of Boolean algebras and recent results on duality theory for a modal logic with xed points. We propose interpreting formulas into a eld of subsets of states instead of the full power set lattice used by Kozen. Under this interpretation we relate image compact modal frames with Scott continuity of the box modality, m-saturated transition systems and descriptive modal frames. Also, it is shown that the class of image compact modal frames satis es the Hennessy-Milner property. We conclude by showing that for descriptive modal-frames the standard interpretation coincides with the one we proposed. Re-interpreting the modal-calculus / 3

imm.dtu.dk

We study the general problem of axiomatizing s tructures in the framework of modal logic and present a u niform method for complete axiomatization of the modal l ogics determined by a large family of classes o f s tructures of any signature.