Expanding basic fuzzy logic with truth constants for component delimiters
type: Item
instance of: scholarly article
title: Expanding basic fuzzy logic with truth constants for component delimiters
zbMATH Open document ID: 1262.03041
author: Zuzana Haniková
published in: Fuzzy Sets and Systems
publication date: 2012-11-21T00:00:00Z
review text: The paper introduces and studies expansions/extensions of Hájek's basic fuzzy logic BL based on the ordinal sum decomposition of BL-chains. Each saturated BL-chain is decomposable into an ordinal sum whose non-singleton components are isomorphic to either the standard (Łukasiewicz) MV-chain or the standard Gödel chain or the standard product chain. The author introduces first the logic \(\mathrm{BL}_{\mathrm{EP}(*)}\), whose language is BL expanded with a new constant for each idempotent element of a continuous t-norm \(*\) that is the end-point of a component in the aforementioned ordinal-sum decomposition. Book-keeping axioms express idempotency and relative order of these elements. Finite strong standard completeness is proved for each \(\mathrm{BL}_{\mathrm{EP}(*)}\) with respect to its natural algebraic semantics. The logic \(\mathrm{BL}_{\mathrm{EP}(*)}\) only captures the external structure of the ordinal sum expressing the t-norm \(*\). To be able to deal with the isomorphism types of each component, the paper introduces first the logic \(\mathrm{BL}_{\mathrm{COMP}(*)}\), adding axioms characterising each component of \(*\). A canonical \(\mathrm{BL}_{\mathrm{COMP}(*)}\)-algebra is a standard \(\mathrm{BL}_{\mathrm{COMP}(*)}\)-algebra such that each one of the new constants is interpreted by its corresponding idempotent end-point element. It is proved that \(\mathrm{BL}_{\mathrm{COMP}(*)}\) is complete with respect to its canonical algebra for each continuous t-norm \(*\) which is the ordinal sum of finitely many components. A further extension of \(\mathrm{BL}_{\mathrm{COMP}(*)}\), with axioms characterising the intervals of the ordinal-sum decomposition of \(*\), gives the logic \(\mathrm{BL}_{\mathrm{COMP}(*)}\) that is proved to enjoy completeness with respect to its canonical algebra for each continuous t-norm \(*\). The paper ends with some results of computational complexity. In particular, it is shown that the t-norm \(*\) can be chosen in such a way…
reviewed by: Stefano Aguzzoli
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 03B52
Mathematics Subject Classification ID: 03G25
zbMATH DE Number: 6109486
zbMATH Keywords: basic fuzzy logic BL
zbMATH Keywords: propositional constants
zbMATH Keywords: propositional logic of standard algebras
zbMATH Keywords: BL-chain
MaRDI profile type: Publication
full work available at URL: j.fss.2011.11.006
OpenAlex ID: W2065116368
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DOI: 10.1016/J.FSS.2011.11.006
description: scientific article; zbMATH DE number 6109486