Download e-book for iPad: Binary Functions and their Applications by Horand Störmer

By Horand Störmer

ISBN-10: 3540528121

ISBN-13: 9783540528128

ISBN-10: 3642615198

ISBN-13: 9783642615191

In this ebook binary services and their illustration by means of implicants or implicates are defined. particularly minimum representations via leading implicants or top implicates are given. Such representations generalize the minimum representations of the standard Boolean capabilities. it's proven that implicants (implicates) of discrete features could be developed with the aid of implicants (implicates) of binary capabilities. One immense software is the outline of the reliability constitution of technical platforms, one other is using binary respectively discrete services to categorise items that are defined by way of the grades of convinced attributes. ultimately a category of Boolean algebras of functional significance (set algebras, indicator algebras, algebras of sessions of propositions) are thought of. the weather of such algebras have representations that are strongly hooked up with the representations of binary functions.

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1 we obtained representations of a discrete function binary functions (indicators) IU=y'} and lU~Yi},i E {1, ... ,k}. 13) f by the Now we may represent these indicators by their implicants (prime implicants) or by their implicates (prime implicates). So we obtain f as an expression in such special indicator implicants or implicates. Consider for instance minimal representations by prime implicants. li max(Cil, ... ) = ~ax(Yi max(Cil, ... EN. i=l I, = max (yoCo 0) iEN. jEN"i • where the functions YiCij satisfy the condition Thus we may regard them as special" implicants" of f, because Cij is an implicant of lU=Yi} and 80 Cij = I implies Ci/j = 0 for all C"j with if :1= i.

5). 2) we see that anyanticube indicator F is a prime implicate of 1U>y} if and only if 1 - F is prime implicant of l{f~y}. 12) holds. 14). In the same way we may prove the next result. 15 Theorem Let f be antitone, U ~ y}. 16) Y E {O, ... ) : Z E G~y}. The only representation of l{f>y} by prime implicates is given by the minimum (product) of all prime implicates: 1{I >y} -- min FE:F". ({I>}) p y F -- min 1-(z,a·) - zEG< _11 II zEG~1I 1-(z,a·)" By the last theorems we obtained unique representations of the indicators 1U~Y}' 1U>y} by prime implicants respectively prime implicates, where f was assumed to 51 be monotone.

B) For each maximplicate F(Ci) of 1r, there is at least one Fp E {Fl , ... , Fr } with Fp ~ F(Ci). 10) Every indicator has a representation by all its implicates (prime implicates). 11) A prime implicate F of Ir is essential if and only if there exists a maximplicate F(Ci) of Ir such that F is the only prime implicate of Ir with F ~ F(Ci). The following statements are the complete implicate analogues to the results concerning the representations of Ir by implicants given in chapter 2. 13 Notation Let again F(r) be the set of all implicates of Ir, further Fp(r) all prime implicates of Ir.

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Binary Functions and their Applications by Horand Störmer

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