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On triangular norms and uninorms definable in ŁΠ½

On triangular norms and uninorms definable in ŁΠ½
On triangular norms and uninorms definable in ŁΠ½
In this paper, we investigate the definability of classes of t-norms and uninorms in the logic ŁΠ½. In particular we provide a complete characterization of definable continuous t-norms, weak nilpotent minimum t-norms, conjunctive uninorms continuous on [0, 1), and idempotent conjunctive uninorms, and give both positive and negative results concerning definability of left-continuous t-norms (and uninorms). We show that the class of definable uninorms is closed under construction methods as annihilation, rotation and rotation-annihilation. Moreover, we prove that every logic based on a definable uninorm is in PSPACE, and that any finitely axiomatizable logic based on a class of definable uninorms is decidable. Finally we show that the Uninorm Mingle Logic (UML) and the Basic Uninorm Logic (BUL) are finitely strongly standard complete w.r.t. the related class of definable left-continuous conjunctive uninorms.
Complexity, Decidability, Fuzzy logics, Left-continuous t-norms, Left-continuous uninorms, Semialgebraic sets
0888-613X
179-201
Marchioni, Enrico
729c9984-5949-438e-8de7-0e079bdb9f96
Montagna, Franco
79e82ba3-7d33-40ba-ab38-7732c22ee736
Marchioni, Enrico
729c9984-5949-438e-8de7-0e079bdb9f96
Montagna, Franco
79e82ba3-7d33-40ba-ab38-7732c22ee736

Marchioni, Enrico and Montagna, Franco (2008) On triangular norms and uninorms definable in ŁΠ½. International Journal of Approximate Reasoning, 47 (2), 179-201. (doi:10.1016/j.ijar.2007.04.003).

Record type: Article

Abstract

In this paper, we investigate the definability of classes of t-norms and uninorms in the logic ŁΠ½. In particular we provide a complete characterization of definable continuous t-norms, weak nilpotent minimum t-norms, conjunctive uninorms continuous on [0, 1), and idempotent conjunctive uninorms, and give both positive and negative results concerning definability of left-continuous t-norms (and uninorms). We show that the class of definable uninorms is closed under construction methods as annihilation, rotation and rotation-annihilation. Moreover, we prove that every logic based on a definable uninorm is in PSPACE, and that any finitely axiomatizable logic based on a class of definable uninorms is decidable. Finally we show that the Uninorm Mingle Logic (UML) and the Basic Uninorm Logic (BUL) are finitely strongly standard complete w.r.t. the related class of definable left-continuous conjunctive uninorms.

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More information

Accepted/In Press date: 20 April 2007
e-pub ahead of print date: 30 April 2007
Published date: February 2008
Keywords: Complexity, Decidability, Fuzzy logics, Left-continuous t-norms, Left-continuous uninorms, Semialgebraic sets

Identifiers

Local EPrints ID: 417388
URI: http://eprints.soton.ac.uk/id/eprint/417388
ISSN: 0888-613X
PURE UUID: 645248c8-7ce3-4fbe-866d-6ab3a3dcc38e

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Date deposited: 30 Jan 2018 17:31
Last modified: 03 Jan 2019 10:30

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Contributors

Author: Enrico Marchioni
Author: Franco Montagna

University divisions

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