Representation and extension of states on MV-algebras

Kroupa, Tomáš

doi:10.1007/s00153-005-0286-y

Cited by 37 publications

(21 citation statements)

References 18 publications

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“…Clearly, the modeling of this kind of knowledge cannot be done using the classical approach to probability since, given the un-sharp nature of events like chaotic traffic, the structure of such fuzzy events cannot be considered to be a Boolean algebra any longer. The study of finitely additive measures in the context of MV-algebras, structures more general than Boolean algebras, was started by Mundici in [20] and further developed by Mundici and Riečan in [21], as well as by Kroupa [17].…”

Section: (Fp1) P (¬ ∨ ) → (P ( ) → P ( )) (Fp2) P (¬ ) ≡ ¬P ( ) (Fpmentioning

confidence: 99%

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A logic for reasoning about the probability of fuzzy events

Flaminio

Godo

2007

Fuzzy Sets and Systems

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In this paper we present the logic F P (Ł n , Ł) which allows to reason about the probability of fuzzy events formalized by means of the notion of state in a MV-algebra. This logic is defined starting from a basic idea exposed by Hájek [Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998]. Two kinds of semantics have been introduced, namely the class of weak and strong probabilistic models. The main result of this paper is a completeness theorem for the logic F P (Ł n , Ł) w.r.t. both weak and strong models. We also present two extensions of F P (Ł n , Ł): the first one is the logic F P (Ł n , RP L), obtained by expanding the F P (Ł n , Ł)-language with truth-constants for the rationals in [0, 1], while the second extension is the logic F CP (Ł n , Ł 1 2 ) allowing to reason about conditional states.

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Section: (Fp1) P (¬ ∨ ) → (P ( ) → P ( )) (Fp2) P (¬ ) ≡ ¬P ( ) (Fpmentioning

confidence: 99%

“…• A weak probabilistic model for Ł∀ evaluates a modal formula P ( ) by means of a finitely additive measure (or state) defined over the MV-algebra of provably equivalent Łukasiewicz formulas (see [16][17][18][19][20][21] for a detailed definition of state over an MV-algebra).…”

Section: (Fp1) P (¬ ∨ ) → (P ( ) → P ( )) (Fp2) P (¬ ) ≡ ¬P ( ) (Fpmentioning

confidence: 99%

A logic for reasoning about the probability of fuzzy events

Flaminio

Godo

2007

Fuzzy Sets and Systems

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“…Kroupa 2005) we have Corollary 7 Semi-divisible residuated lattices admit Riečan states. Now consider Bosbach states.…”

Section: Theorem 6 Riečan States On a Semi-divisible Residuated Lattimentioning

confidence: 99%

“…Kroupa (2005) proved that a state s on a M V -subalgebra of an M V -algebra L can be extended on the whole L, however, this extension is not unique. Thus, in particular, we have Remark 10 In any semi-divisible residuated lattice L, a state s on a Boolean subalgebra B ⊂ M V (L) ⊂ L can be extended on the whole L. This extension is not unique, however, it is a Riečan state.…”

Section: We Concludementioning

confidence: 99%

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States on semi-divisible residuated lattices

Turunen

Mertanen

2007

Soft Comput

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Given a residuated lattice L, we prove that the subset M V (L) of complement elements x * of L generates an M V -algebra if, and only if L is semi-divisible. Riečan states on a semi-divisible residuated lattice L, and Riečan states on M V (L) are essentially the very same thing. The same holds for Bosbach states as far as L is divisible. There are semi-divisible residuated lattices that do not have Bosbach states.

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Towards a Standard Completeness for a Probabilistic Logic on Infinite-Valued Events

Flaminio

2019

Lecture Notes in Computer Science

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MV-algebras with internal states, or SMV-algebras for short, are the equivalent algebraic semantics of the logic SFP( L, L) which allows to represent and reason about the probability of infinite-valued events. In this paper we will make the first steps towards establishing completeness for SFP( L, L) with respect to the class of standard SMV-algebras, a problem which has been left open since the first paper on SMV-algebras was published. More precisely we will prove that, if we restrict our attention to a particular, yet expressive, subclass of formulas, then theorems of SFP( L, L) are the same as tautologies of a class of SMV-algebras that can be reasonably called "standard".

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Representation and extension of states on MV-algebras

Cited by 37 publications

References 18 publications

A logic for reasoning about the probability of fuzzy events

A logic for reasoning about the probability of fuzzy events

States on semi-divisible residuated lattices

Towards a Standard Completeness for a Probabilistic Logic on Infinite-Valued Events

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