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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Notes
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 ·.
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 \({\mathcal A}/Rad({\mathcal A}).\)
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].
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.
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 \(x, y \in [0, 1].\)
References
Blok WJ, Pigozzi D (1989) Algebraizable logics. Memoirs of the American Mathematical Society, 396, vol 77
Butnariu D, Klement EP (1993) Triangular norm based measures and games with fuzzy coalitions. Kluwer, Dordrecht
Calvo T, Mayor G, Mesiar R (eds) (2002) Aggregation operators—new trends and applications, volume 97 Studies in fuzziness and soft computing. Springer
Chang CC (1958) Algebraic analysis of many-valued logics. Trans Am Math Soc 88:467–490
Chang CC (1959) A new proof of the completeness of the Łukasiewicz axioms. Trans Am Math Soc 93:74–80
Cignoli R, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning, volume 7 of Trends in logic—Studia Logica Library. Kluwer Academic Publishers, Dordrecht
de Finetti B (1993) Sul significato soggettivo della probabilità, Fundamenta Mathematicae 17, 298–329, 1931 (Translated into English as “On the subjective meaning of probability”. In: Monari P, Cocchi (eds) Probabilità e Induzione, Clueb, Bologna, pp 291–321)
Develin M, Sturmfels B (2004) Tropical convexity. Doc Math 9:1–27
Dubois D, Prade H (1988) Possibility theory. An approach to computerized processing of uncertainty. Plenum Press, New York (with the collaboration of Farreny H, Martin-Clouaire R, Testemale C)
Flaminio T, Godo L, Marchioni E (2011a) On the logical formalization of possibilistic counterpart of states over n-valued Łukasiewicz events. J Logic Comput 21(3):447–464
Flaminio T, Godo L, Marchioni E (2011b) Reasoning about uncertainty of fuzzy events: an overview. In: Cintula P, Fermüller CG, Godo L, Hájek P (eds) Understanding vagueness—logical, philosophical, and linguistic perspectives. Studies in logic, vol 36. College Publications, London, pp 367–401
Garcia P, Valverde L (1989) Isomorphisms between De Morgan triplets. Fuzzy Sets Syst 30:27–36
Gaubert S, Katz RD (2007) The Minkowski theorem for max-plus convex sets. Linear Algebra Appl 421(2–3):356–369
Goodearl KR (1986) Partially ordered abelian groups with interpolation. AMS Math Surv Monogr 20
Grabisch M, Murofushi T, Sugeno M (1992) Fuzzy measure of fuzzy events defined by fuzzy integrals. Fuzzy Sets Syst 50:293–313
Hájek P (1998) Metamathematics of fuzzy logic, volume 4 of Trends in Logic—Studia Logica Library. Kluwer Academic Publishers, Dordrecht
Halpern JY (2003) Reasoning about uncertainty. MIT Press
Hopf E (1950) The partial differential equation \(u_{t}+uu_{x}=\mu u_{xx}.\) Commun Pure Appl Math 3:201–230
Kroupa T (2006) Every state on semisimple MV-algebra is integral. Fuzzy Sets Syst 157:2771–2782
Kühr J, Mundici D (2007) De Finetti theorem and Borel states in [0, 1]-valued algebraic logic. Int J Approx Reason 46(3):605–616
Litvinov GL (2007) Maslov dequantization, idempotent and tropical mathematics: a brief introduction. J Math Sci 140(3):426–444
Mundici D (1995) Averaging the truth-value in Łukasiewicz Logic. Studia Logica 55(1):113–127
Mundici D (2006) Bookmaking over infinite-valuede vents. Int J Approx Reason 43:223–240
Navara M (2005) Triangular norms and measures of fuzzy sets. In: Klement EP, Mesiar R (eds) Logical, algebraic, analytic, and probabilistic aspects of triangular norms. Elsevier, pp 345–390
Panti G (2008) Invariant measures on free MV-algebras. Commun Algebra 36(8):2849–2861
Richter-Gebert J, Sturmfels B, Theobald T (2005) First steps in tropical geometry. In: Litvinov GL, Maslov VP (eds) Idempotent mathematics and mathematical physics. Proceedings Vienna 2003. American Mathematical Society, Contemporary Mathematics 377, pp 289–317
Schrödinger E (1926) Quantization as an eigenvalue problem. Ann Phys 364:361–376
Sugeno M (1974) Theory of fuzzy integrals and its applications. PhD dissertation. Tokyo Institute of Technology, Tokyo
Weber S (1984) \(\bot\)-decomposable measures integrals for Archimedean t-conorms \(\bot.\) J Math Anal Appl 101:114–138
Zadeh LA (1978) Fuzzy sets as the basis for a theory of possibility. Fuzzy Sets Syst 1:3–28
Zimmermann U (1981) Linear and combinatorial optimization in ordered algebraic structures. Ann Discret Math 10:viii+380
Acknowledgments
The authors would like to thank the anonymous referees for their relevant suggestions and helpful remarks. They also acknowledge partial support from the Spanish projects TASSAT (TIN2010- 20967-C04-01), Agreement Technologies (CONSOLIDER CSD2007-0022, INGENIO 2010) and ARINF (TIN2009-14704-C03-03), as well as the ESF Eurocores-LogICCC/MICINN project (FFI2008-03126-E/FILO). Flaminio and Marchioni acknowledge partial support from the Juan de la Cierva Program of the Spanish MICINN.
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Flaminio, T., Godo, L. & Marchioni, E. Geometrical aspects of possibility measures on finite domain MV-clans. Soft Comput 16, 1863–1873 (2012). https://doi.org/10.1007/s00500-012-0838-0
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DOI: https://doi.org/10.1007/s00500-012-0838-0