States in Łukasiewicz logic correspond to probabilities of rational polyhedra
type: Item
instance of: scholarly article
title: States in Łukasiewicz logic correspond to probabilities of rational polyhedra
zbMATH Open document ID: 1260.03050
author: Tomáš Kroupa
published in: International Journal of Approximate Reasoning
publication date: 2012-09-11T00:00:00Z
reviewed by: Brunella Gerla
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 06D35
zbMATH DE Number: 6080889
zbMATH Keywords: Łukasiewicz logic
zbMATH Keywords: McNaughton functions
zbMATH Keywords: many-valued probability
zbMATH Keywords: rational polyhedron
MaRDI profile type: Publication
full work available at URL: j.ijar.2011.10.007
OpenAlex ID: W2026981903
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DOI: 10.1016/J.IJAR.2011.10.007
review text: A state on an MV-algebra can be seen as a probability of many-valued events (described by formulas of Łukasiewicz logic). The free MV-algebra over \(n\) generators is an algebra of \([0,1]\)-valued functions, called McNaughton functions. States on the free MV-algebra over \(n\) generators coincide with integrals of the McNaughton functions with \(n\) variables (this result applies to the more general class of semisimple MV-algebras; see [the author, Fuzzy Sets Syst. 157, No. 20, 2771--2782 (2006; Zbl 1107.06007); \textit{G. Panti}, Commun. Algebra 36, No. 8, 2849--2861 (2008; Zbl 1154.06008)].NEWLINENEWLINEIn this paper this result is strengthened, and it is shown that states over free MV-algebras over \(n\) generators coincide with measures of rational polyhedra in \([0,1]^n\). Indeed, a state of a McNaughton function is completely determined by the measure of its one-set. The proof is new and elementary and does not rely on the previous results of integral representation: this makes the paper both deep in its results and pleasant (and self-contained) to read.
description: scientific article; zbMATH DE number 6080889