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By Célia da Costa Pereira, Andrea G. B. Tettamanzi (auth.), Francesco Masulli, Sushmita Mitra, Gabriella Pasi (eds.)

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Read Online or Download Applications of Fuzzy Sets Theory: 7th International Workshop on Fuzzy Logic and Applications, WILF 2007, Camogli, Italy, July 7-10, 2007. Proceedings PDF

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Computing Systems, UCLM, 02071 Albacete, Spain 3 Dep. es Abstract. Multi-adjoint logic programming represents an extremely flexible attempt for introducing fuzzy logic into logic programming (LP). t. a given program is done in two separate phases. During the operational one, admissible steps are systematically applied in a similar way to classical resolution steps in pure LP, thus returning an expression where all atoms have been exploited. This last expression is then interpreted under a given lattice during the so called interpretive phase.

1 D. Bianucci, G. Cattaneo, and D. Ciucci Entropy and (Possible Negative) Co–entropy for Fuzzy Granulation In the case of a fuzzy granulation, for any of its fuzzy granule ωi it is possible to assign the non–negative number (and compare with (9)): p(ωi ) = 1 1 m(ωi ) = M (F ) M (F ) ωi (x) (12) x∈X whose collection p(F ) = (p(ω1 ), p(ω2 ), . . , p(ωN )) is a probability distribution N since trivially: (pd-1) for any i it is p(ωi ) ≥ 0 and (pd-2) i=1 p(ωi ) = 1. The entropy of the fuzzy granulation F is then the one generated by this probability distribution, which as usual is given by the real non–negative quantity N 0 ≤ H(F ) = − p(ωi ) log p(ωi ) ≤ log N (13) i=1 Trivially, by (12) and recalling that the total measure of the fuzzy covering F is M (F ) = |X|, one gets that H(F ) = − 1 M (F ) N m(ωi ) log i=1 m(ωi ) M (F ) (14) and so also in this case we can introduce the co–entropy of the fuzzy granulation F as the quantity E(F ) = 1 M (F ) N m(ωi ) log m(ωi ) (15) i=1 obtaining from (14) the following identity which is true whatever be the fuzzy granulation F : H(F ) + E(F ) = log M (F ) = log |X| (16) This identity is an extension to fuzzy granulations of the identity (5) previously seen in the case of partitions (crisp granulations).

We will consider the Smyth ordering among sets, and we will write X S Y if and only if for every y ∈ Y there exists x ∈ X such that x y. Definition 4. Let f : P −→ 2P be a multi-valued function on a poset P : – We say that f is isotone if and only if for all x, y ∈ P we have that x y implies f (x) S f (y). – We say that f is inflationary if and only if {x} S f (x) for every x ∈ P . As our intended application is focused on multilattice-based logic programs, we can assume the existence of minimal fixed points for a given multi-valued function on a multilattice (since in [7] the existence of minimal fixed points was proved for the TP operator).

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