Weak interpolation in extensions of the logics S 4 and K 4

Karpenko, A. V.

doi:10.1007/s10469-008-9035-8

Cited by 8 publications

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“…Lemma 9 [5,Lemma 8]. Suppose that the class of finitely generated simple algebras of a variety M of modal algebras is amalgamable and contains V 0 n for some n ≥ 3.…”

Section: Proof Suppose That the Class Of Finitely Generated Simple Amentioning

confidence: 99%

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Interpolation properties in the extensions of the logic of inequality

Karpenko

2010

Sib Math J

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We consider the modal logics wK4 and DL as well as the corresponding weakly transitive modal algebras and DL-algebras. We prove that there exist precisely 16 amalgamable varieties of DL-algebras. We find a criterion for the weak amalgamation property of varieties of weakly transitive modal algebras, solve the deductive interpolation problem for extensions of the logic of inequality DL, and obtain a weak interpolation criterion over wK4.The interpolation theorem, proved by Craig in 1957 for the classical predicate logic, initiated the study of interpolation in various formal theories. In the classical predicate logic the interpolation theorem of Craig has several equivalent statements, all becoming inequivalent for modal logics. In this regard a few cases of interpolation properties were formulated: the Craig interpolation property (CIP), the deductive interpolation property (IPD), the restricted interpolation property (IPR), and finally the weak interpolation property (WIP) [1].It is known that CIP and IPD are decidable [2] over the logic S4 but undecidable [3, 4] over K4, while WIP is decidable [5] over K4. The proofs of these facts base on the existence of a duality isomorphism between the class of extensions of a logic and the class of subvarieties of the corresponding variety of algebras (see [6] for instance). Therefore, the study of interpolation reduces to studying the amalgamation property of varieties of modal algebras.Maksimova studied various cases of the amalgamation property of varieties of modal algebras. In particular, she showed [2] that precisely 50 varieties of topoboolean algebras are amalgamable, among which 38 enjoy the strong amalgamation property. Maksimova found [7] the necessary and sufficient conditions for the amalgamation property, which enables us to reduce the question of the presence of the properties for this variety of modal algebras to the consideration of a subclass of finitely generated and finitely indecomposable algebras.The joint article of the author and Maksimova [8] deals mainly with weakly transitive modal algebras and DL-algebras. These classes of modal algebras became known in connection with the study of the modal logics wK4 and DL [9-11] presenting their natural algebraic semantics. In [8] we showed that the class of weakly transitive modal algebras coincides with the class of simple DL-algebras and found a full description for finitely generated simple DL-algebras and their embeddings. We proved that the variety of DL-algebras is not weakly amalgamable and the corresponding modal logics wK4 and DL lack all interpolation properties. In this article we study the amalgamation property of varieties of DL-algebras in more detail.In Section 2 we prove that the family of varieties of DL-algebras has the cardinality of the continuum. We establish the existence of a bijective correspondence between the varieties of DL-algebras and the downward closed subclasses of the class K (DL) = {V m n | n + m > 0}. In Section 3 we obtain necessary conditions for the amalgamation propert...

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“…Lemma 9 [5,Lemma 8]. Suppose that the class of finitely generated simple algebras of a variety M of modal algebras is amalgamable and contains V 0 n for some n ≥ 3.…”

Section: Proof Suppose That the Class Of Finitely Generated Simple Amentioning

confidence: 99%

“…The classes K (L) = V 0 n | n > 0 and K (L) = V 0 n | n > 0 ∪ V 1 0 are amalgamable as well (see [5]). Moreover, the presence of V 1 0 in a class of algebras does not affect the amalgamation property; thus, the number of possible amalgamable subclasses of K (DL) is even.…”

Section: Amalgamable Varieties Of Dl-algebrasmentioning

confidence: 99%

Interpolation properties in the extensions of the logic of inequality

Karpenko

2010

Sib Math J

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“…there are at least two argument places having truth-values which do not coincide with 1 2 . Therefore, the right part of the equation is equal to 0, and its left part must, according to definition of the function f m (x 1 , ..., x m ), be equal to 1 which is impossible.…”

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confidence: 98%

“…It is worth mentioning that the formula ∼⊢ x∩ ∼⊢∼ x represents the Rosser-Turquette operator (J-operator) for the value 1 2 in B 3 1 . So, the formula, expressing functions f i (x 1 , ..., x i ), may be simplified with the use of J-operator for the value 1 2 just as follows:…”

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confidence: 99%

Cardinality of sets of closed functional classes in weak 3-valued logics

Преловский¹

2013

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“…A description of simple weakly transitive modal algebras found in [12] made it possible to fully resolve the weak interpolation problem in extensions of K4 and the weak amalgamation problem in varieties of transitive algebras.…”

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confidence: 99%

Simple weakly transitive modal algebras

Karpenko

Maksimova

2010

Algebra Logic

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Weakly transitive modal algebras are studied. It is proved that the class of simple weakly transitive algebras coincides with the class of simple DL-algebras. A full description is given for finitely generated simple DL-algebras together with their embeddings. As a consequence, it is shown that the varieties of weakly transitive algebras and of DLalgebras are not weakly amalgamable, and that modal logics wK4 and DL do not possess the weak interpolation property.We consider weakly transitive modal algebras. These algebras provide a natural algebraic interpretation for a modal logic wK4, which is a bit weaker than a well-known modal system K4. The logic wK4 and its extensions have been widely studied in connection with topological semantics for modal logic [1]. Relations between extensions of the wK4 logic and extensions of the S4 logic were treated in [2]. A most interesting extension of wK4 is the difference logic DL (see [1,3,4]). Previously, it was proved that adding a modal difference operator to topological modal logics expands the expressive capacity of languages essentially [1,[5][6][7].Many known systems of modal logics, in particular, the minimal logic K, the transitive logic K4, the logics S4 and S5, the Grzegorczyk logic and the Gödel-Löb logic, possess the Craig interpolation property (see, e.g., [8]). The study of the interpolation problem in modal logics clearly recognized several principal versions of the interpolation property and revealed their connections with natural versions of the amalgamation property in varieties of modal algebras [9]. In examining a weak interpolation property (weakest among the versions considered) for modal logics, it was *

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Weak interpolation in extensions of the logics S 4 and K 4

Cited by 8 publications

References 8 publications

Interpolation properties in the extensions of the logic of inequality

Interpolation properties in the extensions of the logic of inequality

Cardinality of sets of closed functional classes in weak 3-valued logics

Simple weakly transitive modal algebras

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