By Rafik A Aliev, Oleg H Huseynov, Rashad R Aliyev, Akif A Alizadeh

Real-world details is imperfect and is mostly defined in ordinary language (NL). in addition, this knowledge is frequently in part trustworthy and a level of reliability is additionally expressed in NL. In view of this, the idea that of a Z-number is a extra enough thought for the outline of real-world details. the most serious challenge that evidently arises in processing Z-numbers-based info is the computation with Z-numbers. these days, there is not any mathematics of Z-numbers advised in present literature. This ebook is the 1st to give a accomplished and self-contained concept of Z-arithmetic and its purposes. the various options and methods defined within the booklet, with rigorously worked-out examples, are unique and seem within the literature for the 1st time. The booklet could be useful for pros, lecturers, managers and graduate scholars in fuzzy common sense, choice sciences, man made intelligence, mathematical economics, and computational economics.

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9575-The Arithmetic of Z-numbers 18 The Arithmetic of Z-numbers. Theory and Application 6. Equate precisiated p to CF*(p). 7. CF*(p) defines the possibility distribution of ED given p, Poss(ED|p). 8. CF*(p) defines the truth distribution of the truth value or p given ED, Tr(p|ED). 9. Poss(ED|p) = Tr(p|ED). 10. Define the precisiated (computational) meaning of p as the possibility distribution of ED given p, Poss(ED|p). More informatively, the precisiated (computational) meaning of p is the possibility distribution, Poss(ED|p), together with the procedure which computes Poss(ED|p).

R Additional restrictions on p X and pY are: ∫p X (u )du = 1, Y (u ) du = 1, R ∫p R ∫ up ∫ uµ X (u )du = R AX (u )du R ∫µ AX (u )du (compatibility), R ∫ up Y R ∫ uµ (u )du = AY (u ) du R ∫µ R AY (u )du (compatibility). 9575-The Arithmetic of Z-numbers 38 The Arithmetic of Z-numbers. 6) . 7) R ∫ up Y R ∫ uµ (u )du = AY (u ) du R ∫µ AY (u )du R In this case, the combined restriction on the arguments is expressed as a conjunction of their restrictions, with ^ interpreted as min. In effect, application of the extension principle reduces computation of pZ to a problem in functional optimization.

Informally, the internal numerical truth value is defined as the degree of agreement of p with an instantiation of ED. Informally, an external numerical truth value of p is defined as the degree of agreement of p with factual information, F. 2. Int(ntp)=tr(ED). 9575-The Arithmetic of Z-numbers The General Concept of a Restriction and Z-numbers 19 In this equation, ED is an instantiation of the explanatory database, Int(ntp) is the internal numerical truth value of p, and tr is the truth function which was defined earlier.