Some characterization and preservation theorems in modal logic

Perkov, Tin; Vuković, Mladen

doi:10.1016/j.apal.2012.07.001

Cited by 4 publications

(5 citation statements)

References 8 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: 71%

“…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: 71%

Section: Introductionmentioning

confidence: 99%

Existential definability of modal frame classes

Perkov

Mikec

2020

Mathematical Logic Qtrly

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“…Theorem 70. With respect to model and frame definability, we have the following hierarchies: We can now extend the characterisations of model and frame definability (i.e., Theorems 8,16,28,32,33,and 36) to cover also team-based logics.…”

Section: Connecting Team Semantics and Universal Modalitymentioning

confidence: 99%

“…Also restricted versions of frame definability, such as definability within the class of finite transitive frames [3,10], have been considered. 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].…”

Section: Introductionmentioning

confidence: 99%

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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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A Generalization of Modal Frame Definability

Perkov

2014

Pristine Perspectives on Logic, Language, and Computation

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Some characterization and preservation theorems in modal logic

Cited by 4 publications

References 8 publications

Existential definability of modal frame classes

Existential definability of modal frame classes

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

A Generalization of Modal Frame Definability

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