By Janusz Kacprzyk, Dimitar Filev, Gleb Beliakov
This publication deals a entire document at the state-of-the artwork within the broadly-intended box of “intelligent systems”. After introducing key theoretical concerns, it describes a few promising types for facts and process research, determination making, and keep watch over. It discusses vital theories, together with danger conception, the Dempster-Shafer conception, the speculation of approximate reasoning, in addition to computing with phrases, including novel purposes in quite a few parts, resembling info aggregation and fusion, linguistic info summarization, participatory studying, structures modeling, etc. through proposing the equipment of their program contexts, the booklet indicates how granular computing, delicate computing and fuzzy common sense recommendations delivers novel, effective recommendations to real-world difficulties. it really is devoted to Professor Ronald R. Yager for his nice clinical and scholarly achievements, and for his long-lasting carrier to the bushy good judgment, and the substitute and computational intelligence groups. it's been encouraged by means of the authors’ appreciation of his unique pondering and groundbreaking principles, with a different inspiration to his worthy study at the automated implementation of varied features of human cognition for decision-making and problem-solving.
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Extra info for Granular, Soft and Fuzzy Approaches for Intelligent Systems: Dedicated to Professor Ronald R. Yager
The constraints satisﬁed by the cube of Fig. 2 can be generalized in the following way : (i) (ii) (iii) (iv) (v) (vi) (vii) ???? = n(o), ???? = n(????) and ???? ′ = n(o′ ) and ????′ = n(????′ ); ???? ≤ ????, ???? ≤ o and ???? ′ ≤ ????′ , ????′ ≤ o′ ; ???? ∗ ???? = 0 and ???? ′ ∗ ????′ = 0; n(????) ∗ n(o) = 0 and n(????′ ) ∗ n(o′ ) = 0; ???? ≤ ????′ , ???? ′ ≤ ???? and ????′ ≤ o, ???? ≤ o′ ; ???? ′ ∗ ???? = 0, ???? ∗ ????′ = 0; n(????′ ) ∗ n(o) = 0, n(????) ∗ n(o′ ) = 0. In the paper, we restrict to the numerical setting and let n(a) = 1 − a. It leads to deﬁne ∗ = max(0, ⋅ + ⋅ − 1) (the Łukasiewicz conjunction).
6. Indeed, if m(∅) = 0, we have Belm (A) ≤ Plm (A) ⇔ Belm (A) + Belm (A) ≤ 1 ⇔ Plm (A) + Plm (A) ≥ 1, which gives birth to the square of Q Organizing Families of Aggregation Operators into a Cube of Opposition 37 Fig. 6 Cube of opposition of evidence theory ∑ opposition ????????????????. We can check as well that Belm (A) = E⊆A m(E) ≤ m (A) = ∑ 1 − A⊆E m(E). Similar inequalities ensure that Qm (A) ≤ Plm (A) = 1 − Belm (A), or Q Belm (A) + Qm (A) ≤ 1, for instance, which ensures that the constraints of the cube hold.
Fuzzy sets are deﬁnable in universes whatsoever. Provided the universe X is structured as a Boolean algebra, and two diﬀerent fuzzy sets in X were deﬁned by ????(x) = prob1 (x), and ????(x) = prob2 (x), there is no possibility of having ???? ≤ ????, since prob1 (x) ≤ prob2 (x) ⇒ prob2 (x′ ) ≤ prob1 (x′ ), for all x′ in X, or ???? ≤ ????, and then ???? = ????. That is, two of these fuzzy sets can only be identical, or not comparable under the usual pointwise ordering of fuzzy sets; this ordering is not suitable for probabilities.