Frame definability in finitely-valued modal logics (original) (raw)

First-Order Modal Logic: Frame Definability and a Lindström Theorem

Studia Logica, 2017

We generalize two well-known model-theoretic characterization theorems from propositional modal logic to first-order modal logic (FML, for short). We first study FMLdefinable frames and give a version of the Goldblatt-Thomason theorem for this logic. The advantage of this result, compared with the original Goldblatt-Thomason theorem, is that it does not need the condition of ultrafilter reflection and uses only closure under bounded morphic images, generated subframes and disjoint unions. We then investigate Lindström type theorems for first-order modal logic. We show that FML has the maximal expressive power among the logics extending FML which satisfy compactness, bisimulation invariance and the Tarski union property.

Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages

Journal of Logic and Computation, 2020

We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, varPi0_1\varPi ^0_1varPi0_1-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics textbfGL\textbf{GL}textbfGL, textbfGrz\textbf{Grz}textbfGrz and textbfKTB\textbf{KTB}textbfKTB—among them, textbfK\textbf{K}textbfK, textbfT\textbf{T}textbfT, textbfD\textbf{D}textbfD, textbfKB\textbf{KB}textbfKB, textbfK4\textbf{K4}textbfK4 and textbfS4\textbf{S4}textbfS4.

Recursive enumerability and elementary frame definability in predicate modal logic

Journal of Logic and Computation, 2019

We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On one hand, it is well known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e. a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on ‘natural’ propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a...

Canonicity and Completeness Results for Many-Valued Modal Logics

Journal of Applied Non-classical Logics, 2002

We prove frame determination results for the family of many-valued modal logics introduced by M. Fitting in the early '90s. Each modal language of this family is based on a Heyting algebra, which serves as the space of truth values, and is interpreted on an interesting version of possible-worlds semantics: the modal frames are directed graphs whose edges are labelled with an element of the underlying Heyting algebra. We introduce interesting generalized forms of the classical axioms D, T, B, 4, and 5 and prove that they are canonical for certain algebraic frame properties, which generalize seriality, reflexivity, symmetry, transitivity and euclideanness. Our results are quite general as they hold for any modal language built on a complete Heyting algebra.

Frame constructions, truth invariance and validity preservation in many-valued modal logic

Journal of Applied Non-classical Logics, 2005

In this paper we define and examine frame constructions for the family of many-valued modal logics introduced by M. Fitting in the '90s. Every language of this family is built on an underlying space of truth values, a Heyting algebra H. We generalize Fitting's original work by considering complete Heyting algebras as truth spaces and proceed to define a suitable notion of H-indexed families of generated subframes, disjoint unions and bounded morphisms. Then, we provide an algebraic generalization of the canonical extension of a frame and model, and prove a preservation result inspired from Fitting's canonical model argument in . The analog of a complex algebra and of a principal ultrafilter is defined and the embedding of a frame into its canonical extension is presented. s s s J J J J J J J J J

Algebras and relational frames for G\"{o}del modal logic and some of its extensions

arXiv (Cornell University), 2021

Gödel modal logics can be seen as extenions of intutionistic modal logics with the prelinearity axiom. In this paper we focus on the algebraic and relational semantics for Gödel modal logics that leverages on the duality between finite Gödel algebras and finite forests, i.e. finite posets whose principal downsets are totally ordered. We consider different subvarieties of the basic variety GAO of Gödel algebras with two modal operators (GAOs for short) and their corresponding classes of forest frames, either with one or two accessibility relations. These relational structures can be considered as prelinear versions of the usual relational semantics of intuitionistic modal logic. More precisely we consider two main extensions of finite Gödel algebras with operators: the one obtained by adding Dunn axioms, typically studied in the fragment of positive classical (and intuitionistic) logic, and the one determined by adding Fischer Servi axioms. We present Jónsson-Tarski like representation theorems for the different types of finite GAOs considered in the paper.

Undecidable problems for modal definability: Table 1

Journal of Logic and Computation, 2016

The core of our article is the computability of the problem of deciding the modal definability of first-order sentences with respect to classes of frames. It gives a new proof of Chagrova's Theorem telling that, with respect to the class of all frames, the problem of deciding the modal definability of first-order sentences is undecidable. It also gives the proofs of new variants of Chagrova's Theorem.

Complexity of finite-variable fragments of propositional modal logics of symmetric frames

Logic Journal of the IGPL, 2018

While finite-variable fragments of the propositional modal logic S5-complete with respect to reflexive, symmetric, and transitive frames-are polynomialtime decidable, the restriction to finite-variable formulas for logics of reflexive and transitive frames yields fragments that remain "intractable." The role of the symmetry condition in this context has not been investigated. We show that symmetry either by itself or in combination with reflexivity produces logics that behave just like logics of reflexive and transitive frames, i.e., their finite-variable fragments remain intractable, namely PSPACE-hard. This raises the question of where exactly the borderline lies between modal logics whose finite-variable fragments are tractable and the rest.