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Modal Equivalence and Bisimilarity in Many-valued Modal Logics with Many-valued Accessibility Relations

Diaconescu Denisa

Fundamenta Informaticae

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2020

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Vol. 173, nr 2-3

177--189

EN

In this paper we investigate the Hennessy-Milner property for models of many-valued modal logics defined based on complete MTL-chains having many-valued accessibility relations. Our main result gives a necessary and sufficient algebraic condition for the class of image-finite models for such modal logics to admit the Hennessy-Milner property.

2

Finite-valued Logics for Information Processing

Avron A., Konikowska B.

Fundamenta Informaticae

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2012

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Vol. 114, nr 1

1-30

EN

We examine the issue of collecting and processing information from various sources, which involves handling incomplete and inconsistent information. Inspired by the framework first proposed by Belnap, we consider structures consisting of information sources which provide information about the values of formulas of classical propositional logic, and a processor which collects that information and extends it by deriving conclusions following from it according to the truth tables of classical logic, applied forward and backward. Our model extends Belnap’s in allowing the sources to provide information also about complex formulas. As that framework cannot be captured using finite ordinary logical matrices, if we want to represent each of the relevant logics with a single matrix, we employ Nmatrices for that purpose. In opposition to the approach proposed in our earlier work, we assume that the information sources are reasonable, i.e. that they provide information consistent with certain coherence rules. We provide sound and complete sequent calculi admitting strong cut elimination for the logic of a single information source, and (several variants of) the logic generated by the source and processor structures described above. In doing this, we also provide new characterizations for some known logics. We prove that, in opposition to the variantwith unconstrained information sources considered earlier, the latter logic cannot be generated by structures with any bounded number of sources.

3

Similarity MV-algebras

Gerla B., Leustean I.

Fundamenta Informaticae

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2006

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Vol. 69, nr 3

287-300

EN

Similarities are an extension of equivalence relations to a fuzzy context. In this paper we introduce the class of similarity MV-algebras obtained as a generalization of the variety of MV-algebras by adding a binary operator playing the role of similarity. We further introduce the similarity ukasiewicz logic and we prove a completeness theorem.

4

Weaker Axioms, More Ranges

Koutras C.D., Peppas P.

Fundamenta Informaticae

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2002

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Vol. 51, nr 3

297-310

EN

In the family of many-valued modal languages proposed by M. Fitting in 1992, every modal language is based on an underlying Heyting algebra which provides the space of truth values. The lattice of truth values is explicitly represented in the language by a set of special constants and this allows for forming weak, generalized, many-valued analogs of all classical modal axioms. Weak axioms of this kind have been recently investigated from the canonicity, completeness and correspondence perspective. In this paper, we provide some results on the effect of adopting weak versions of the axioms D, T, 4, 5 and w5 in the family of many-valued modal non-monotonic logics, a` la McDermott and Doyle. For many-valued modal languages built on finite chains, we extend the results by proving two quite general range theorems. We then hint on the relation between the modal non-monotonic logics obtained: we prove that there exist ranges which selectively pick out some of the expansions produced by the many-valued autoepistemic logics, actually the ones with a confidence-bounded set of beliefs. However, an exact characterization of the relation between the various ranges created by the weak many-valued modal axioms still remains to be explored.

5

Many-valued modal non-monotonic reasoning: sequential stable sets and logics with linear truth spaces

Koutras C.D., Koletsos G., Zachos S.

Fundamenta Informaticae

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1999

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Vol. 38, Nr 3

281-324

EN

A family of many-valued modal logics which correspond to possible-worlds models with many-valued accessibility relations, has been recently proposed by M. Fitting. Non-monotonic extensions of these logics are introduced with a fixpoint construction a la McDermott & Doyle and employ sequential belief sets as epistemic states. In this paper we take a logical investigation of many-valued modal non-monotonic reasoning in Fitting's formal framework. We examine the notion of MV-stable sets which emerges as a sequential many-valued analog of Stalnaker-Moore stable sets and prove that several attractive epistemic properties are essentially retained in the many-valued setting, esp. when focusing on a syntactically simple epistemic fragment of MV-stable sets. We show that MV-stable sets are always closed under S4 consequence and identify three sufficient conditions for capturing axioms of negative introspection. Also, the relation of MV-stable sets to many-valued analogs of classical S5 models and to many-valued extensions of universal models is discussed. Finally, we pay special attention to the subclass of logics built on linear Heyting algebras and show that inside this subclass, the situation is very similar - in many respects - to the machinery devised by W. Marek, G. Schwarz and M. Truszczyński. In particular, the normal fragments of the two important classical ranges of modal non-monotonic logics remain intact: many-valued autoepistemic logic is captured by any non-monotonic logic in K5-KD45 and many-valued reflexive autoepistemic logic corresponds to KTw5-Sw5.