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On computational complexity of semilinear varieties

On computational complexity of semilinear varieties
On computational complexity of semilinear varieties
We propose a method for characterizing the complexity of satisfiability and tautologicity of equational theories of varieties of algebras by relying on their representability in the theory of the ordered additive group of reals R with rational constants. We call semilinear those varieties which are generated by a subclass of algebras in which the operations are representable as semilinear functions with rational coefficients. Those functions are definable in the theory of R which admits quantifier elimination and whose existential theory is NP-complete. We prove that there is a polynomial time translation of the equational theories of semilinear varieties into the existential theory of R. Then, if the variety is generated (up to isomorphism) by one semilinear algebra, the satisfiability problem is in NP, while the tautologicity problem is in co-NP. We apply this method in order to provide a comprehensive study of complexity of several varieties related to logics based on left-continuous conjunctive uninorms and left-continuous t-norms. © The Author, 2008. Published by Oxford University Press. All rights reserved.
Computational complexity, Ordered divisible Abelian groups, Semilinear functions, Triangular norms, Uninorms, Varieties
941-958
Marchioni, Enrico
729c9984-5949-438e-8de7-0e079bdb9f96
Marchioni, Enrico
729c9984-5949-438e-8de7-0e079bdb9f96

Marchioni, Enrico (2008) On computational complexity of semilinear varieties. Journal of Logic and Computation, 18 (6), 941-958. (doi:10.1093/logcom/exn017).

Record type: Article

Abstract

We propose a method for characterizing the complexity of satisfiability and tautologicity of equational theories of varieties of algebras by relying on their representability in the theory of the ordered additive group of reals R with rational constants. We call semilinear those varieties which are generated by a subclass of algebras in which the operations are representable as semilinear functions with rational coefficients. Those functions are definable in the theory of R which admits quantifier elimination and whose existential theory is NP-complete. We prove that there is a polynomial time translation of the equational theories of semilinear varieties into the existential theory of R. Then, if the variety is generated (up to isomorphism) by one semilinear algebra, the satisfiability problem is in NP, while the tautologicity problem is in co-NP. We apply this method in order to provide a comprehensive study of complexity of several varieties related to logics based on left-continuous conjunctive uninorms and left-continuous t-norms. © The Author, 2008. Published by Oxford University Press. All rights reserved.

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e-pub ahead of print date: 27 June 2008
Published date: December 2008
Keywords: Computational complexity, Ordered divisible Abelian groups, Semilinear functions, Triangular norms, Uninorms, Varieties

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Local EPrints ID: 416722
URI: http://eprints.soton.ac.uk/id/eprint/416722
PURE UUID: adde7678-67f5-495e-ae61-8670d51a3a72

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Date deposited: 05 Jan 2018 17:30
Last modified: 13 Mar 2019 19:04

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