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Soft Computing
Published in: Soft Computing 11/2012

01-11-2012 | Focus

Geometrical aspects of possibility measures on finite domain MV-clans

Authors: Tommaso Flaminio, Lluís Godo, Enrico Marchioni

Published in: Soft Computing | Issue 11/2012

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Abstract

In this paper, we study generalized possibility and necessity measures on MV-algebras of [0, 1]-valued functions (MV-clans) in the framework of idempotent mathematics, where the usual field of reals \({\mathbb{R}}\) is replaced by the max-plus semiring \({\mathbb{R}}_{\rm max}.\) We prove results about extendability of partial assessments to possibility and necessity measures, and characterize the geometrical properties of the space of homogeneous possibility measures. The aim of the present paper is also to support the idea that idempotent mathematics is the natural framework to develop the theory of possibility and necessity measures, in the same way classical mathematics serves as a natural setting for probability theory.
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Footnotes
1
Papers on idempotent and tropical mathematics usually adopt the following notation: the idempotent operation is denoted by \(\oplus, \) while \(\odot\) denotes the usual sum. This notation is justified because the idempotent operation substitutes the sum, and the sum substitutes the product in the real field \({\mathbb{R}}.\) In this paper, conversely, we will not adopt this notation because it would be misleading with respect to those used in many-valued logic (see Sect. 2), where \(\oplus\) and \(\odot\) represent, respectively, a t-conorm and a t-norm. For this reason, we will keep the standard notation for max,  min,  + and ·.
 
2
It is worth noticing that, while Kroupa proved Theorem 1 in the case of semisimple MV-algebras, Panti showed that the hypothesis on the semisimplicity of the MV-algebra can be relaxed, since, for every MV-algebra \({\mathcal A}, \) there is a canonical bijection between the class \({\mathcal{S}}({\mathcal A})\) of all the states on \(A, \) and the class \({\mathcal{S}}({\mathcal A}/Rad({\mathcal A}))\) of all the states on its most general semisimple quotient \({\mathcal A}/Rad({\mathcal A}).\)
 
3
It is worth noticing that Kühr and Mundici [21, Corollary 4.3] extend de Finetti’s coherence criterion to any algebraizable (cf. Blok and Pigozzi 1989) logic \({\mathcal{L}}_\Upomega\) whose equivalent algebraic semantics is given by the algebraic variety generated by the algebra \(([0, 1],\Upomega), \) where \(\Upomega\) denotes a set of continuous operations on [0, 1]. Therefore, the following theorem can be reasonably seen as a particular case of [21, Corollary 4.3].
 
4
In (Flaminio et al. 2011a), we actually introduced the slightly more general notion of \(L\)-valued possibility (necessity) measure on an MV-algebra, where \(L\) is in any MV-chain. In this paper, we concentrate on [0, 1]-valued maps, and hence we will simply speak about “possibility” (resp. “necessity”) measures, without specifying that \([0, 1]_{\rm MV}\) serves as range for the measures.
 
5
A De Morgan triplet (see e.g., Garcia and Valverde (1989)) is a 3-tuple \((\mathop{\hat{\odot}}, {\hat{\oplus}}, \neg)\) where \(\mathop{\hat{\odot}}\) is a t-norm, \({\hat{\oplus}}\) a t-conorm, \(\neg\) a strong negation function such that \(x {\hat{\oplus}} y = \neg (\neg x \mathop{\hat{\odot}} \neg y)\) for all \(x, y \in [0, 1].\)
 
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Metadata
Title
Geometrical aspects of possibility measures on finite domain MV-clans
Authors
Tommaso Flaminio
Lluís Godo
Enrico Marchioni
Publication date
01-11-2012
Publisher
Springer-Verlag
Published in
Soft Computing / Issue 11/2012
Print ISSN: 1432-7643
Electronic ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-012-0838-0

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