By Célia da Costa Pereira, Andrea G. B. Tettamanzi (auth.), Francesco Masulli, Sushmita Mitra, Gabriella Pasi (eds.)
Read 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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Extra info for Applications of Fuzzy Sets Theory: 7th International Workshop on Fuzzy Logic and Applications, WILF 2007, Camogli, Italy, July 7-10, 2007. Proceedings
If xi x, by isotonicity f (xi ) S f (x). Then for x ∈ f (x) we can choose xi+1 ∈ f (xi ) such that xi+1 x and obviously xi xi+1 by inﬂation. For a limit ordinal α, as stated above, xα = supi<α xi ; now, by induction we have that xi x for every i < α, hence xα x. The transﬁnite chain (xi )i∈I constructed this way is increasing, therefore there is an ordinal α such that xα = xα+1 ∈ f (xα ), so xα is a ﬁxed point and xα x ✷ but by minimality of the ﬁxed point x, we have that x = xα . The usual way to approach the problem of reachability is to consider some kind of ‘continuity’ in our multi-valued functions, understanding continuity in the sense of preservation of suprema and inﬁma.
The new concept of P E−reductant is aiming at the reduction of the aforementioned negative eﬀects. Since P E−reductants are partially evaluated before being introduced in the target program, the computational eﬀort done (once) at generation time is saved (many times) at execution time. Intuitively, given a program P and a ground atomic goal A, a P E−reductant can be constructed following these steps: i) Construct an unfolding tree5 , τ , for P 4 5 That is, there exists a θi such that A = Ci θi .
We have applied these cost measures to compare the eﬃciency of two semantically equivalent notions of reductants. In the near future, we also plan to take advantage of these cost criteria to formally prove the eﬃciency of the fuzzy fold/unfold  and partial evaluation techniques  we are developing. References 1. : Fril-Fuzzy and Evidential Reasoning in Artiﬁcial Intelligence. John Wiley & Sons, Inc, West Sussex, England (1995) 2. : Fuzzy prolog: A new approach using soft constraints propagation.