A Generalization of Modal Frame Definability

Perkov, Tin

doi:10.1007/978-3-662-44116-9_10

Cited by 7 publications

(9 citation statements)

References 11 publications

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“…This paper resumes previous work in which analogues of the Goldblatt-Thomason theorem for existential definability and a generalized notion which combines standard and existential definability have been obtained [10,11]. We can think of these notions of modal definability as being about certain fragments of modal logic enriched with the universal modality, for which a characterization theorem was proved by Goranko and Passy in full generality [6].…”

Section: Introductionsupporting

confidence: 69%

“…Nevertheless, characterization theorems have intuitively clearer closure conditions under the assumption of the first-order definability. Such results are given in [10]. In the present paper, we give general analogue for existential definability, without the assumption that the class of frames is elementary.…”

Section: Introductionmentioning

confidence: 92%

“…In [10], the following characterizations with more natural constructions, but under the assumption of first-order definability, are proved using model-theoretic techniques (in particular, saturation). By closure under generated interframes we mean that for all F 1 , F 2 ∈ K and for any frame F we have: if F 1 is a generated subframe of F and F is a generated subframe of F 2 , then F ∈ K. This is a natural condition for ∀∃-definability, since both validity and ∃-validity is preserved under it.…”

Section: Definition 46mentioning

confidence: 99%

“…While the paper [10] deals with frames as usual, the paper [11] provides analogous characterizations for existentially definable and generalized definable classes of modal models, following the characterization for standard modal definability of model classes given by de Rijke and Sturm [2]. Modally definable model classes are always elementary (by standard translation), but this is not so in the case of frame classes.…”

Section: Introductionmentioning

confidence: 99%

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Existential definability of modal frame classes

Perkov

Mikec

2020

Mathematical Logic Qtrly

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We prove an existential analogue of the Goldblatt‐Thomason Theorem which characterizes modal definability of elementary classes of Kripke frames using closure under model theoretic constructions. The less known version of the Goldblatt‐Thomason Theorem gives general conditions, without the assumption of first‐order definability, but uses non‐standard constructions and algebraic semantics. We present a non‐algebraic proof of this result and we prove an analogous characterization for an alternative notion of modal definability, in which a class is defined by formulas which are satisfiable under any valuation (the so‐called existential validity). Continuing previous work in which model theoretic characterization for this type of definability of elementary classes was proved, we give an analogous general result without the assumption of the first‐order definability. Furthermore, we outline relationships between sets of existentially valid formulas corresponding to several well‐known modal logics.

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Section: Introductionsupporting

confidence: 69%

Section: Introductionmentioning

confidence: 92%

Section: Definition 46mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Existential definability of modal frame classes

Perkov

Mikec

2020

Mathematical Logic Qtrly

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“…Also model theoretic characterisations of definable model classes have been given, e.g., for ML [6] and ML( u ) [28]. For related work, see also [27]. This paper initiates the study of model and frame definability in the framework of team semantics.…”

Section: Introductionmentioning

confidence: 99%

Characterising modal definability of team-based logics via the universal modality

Sano

Virtema

2019

Annals of Pure and Applied Logic

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We study model and frame definability of various modal logics. Let ML( u + ) denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We show that a class of Kripke models is definable in ML( u + ) if and only if the class is elementary and closed under disjoint unions and surjective bisimulations. We also characterise the definability of ML( u + ) in the spirit of the well-known Goldblatt-Thomason theorem. We show that an elementary class F of Kripke frames is definable in ML( u + ) if and only if F is closed under taking generated subframes and bounded morphic images, and reflects ultrafilter extensions and finitely generated subframes. In addition we study frame definability relative to finite transitive frames and give an analogous characterisation of ML( u + )-definability relative to finite transitive frames. Finally, we initiate the study of model and frame definability in team-based logics. We study (extended) modal dependence logic, (extended) modal inclusion logic, and modal team logic. We establish strict linear hierarchies with respect to model definability and frame definability, respectively. We show that, with respect to model and frame definability, the before mentioned team-based logics, except modal dependence logic, either coincide with ML( u + ) or plain modal logic ML. Thus as a corollary we obtain model theoretic characterisation of model and frame definability for the team-based logics.This article subsumes and extends the conference articles [30] and [31].

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From Simplified Kripke-Style Semantics to Simplified Analytic Tableaux for Some Normal Modal Logics

Petrukhin

Zawidzki

2019

Lecture Notes in Computer Science

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A Generalization of Modal Frame Definability

Cited by 7 publications

References 11 publications

Existential definability of modal frame classes

Existential definability of modal frame classes

Characterising modal definability of team-based logics via the universal modality

From Simplified Kripke-Style Semantics to Simplified Analytic Tableaux for Some Normal Modal Logics

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