Boolean Methods in Operations Research and Related Areas by Professor Dr. Peter L. Hammer, Professor Dr. Sergiu Rudeanu

By Professor Dr. Peter L. Hammer, Professor Dr. Sergiu Rudeanu (auth.)

In classical research, there's a mammoth distinction among the category of difficulties which may be dealt with via the equipment of calculus and the category of difficulties requiring combinatorial ideas. With the arrival of the electronic desktop, the excellence starts off to blur, and with the expanding emphasis on difficulties related to optimization over buildings, tIlE' contrast vanishes. what's invaluable for the analytic and computational remedy of vital questions coming up in sleek keep an eye on concept, mathematical economics, scheduling conception, operations study, bioengineering, etc is a brand new and extra versatile mathematical idea which subsumes either the cla8sical non-stop and discrete t 19orithms. The paintings via HAMMER (IVANESCU) and RUDEANU on Boolean tools represents a tremendous step during this dnectlOn, and it really is therefore a very good excitement to welcome it into print. it's going to definitely stimulate loads of extra study in either conception and alertness. RICHARD BELLMAN college of Southern California FOf(,WOl'

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N times which can be expressed in the disjunctive canonical form (3) U I(XI""'X n )= a~, I(IXI, ... ,IXn)x11 ... x~n, , an or, equivalently, in the conjunctive canonical form (4) t(XI,""X n )= II [/(&I""'&n)vxl1v"'vx~n], where the vector (lXI, ... , IXn) runs over all 2" possible systems of 0 - 1 values and x' is defined by (5) XO = X, Xl = X. It is easy to prove - and, in fact, it is well-known - that the above definition is equivalent to the following one: Definition I'. By a Boolean function defined on a Boolean algebra B, we mean a function of the type (2) which is generated by a Bolean expression E (Definition 1.

34. i-I)]. Example 3. 3) bve~x~iivbve, a ve v b x ~ y~ (a b ve) x v Ii c x, a v b v e(x vii) v ex y ~ z ~ [(b ve a) x v Ii c x] [(a v be) y va c y]. Notice that equatIOn (30) is equivalent to the system of recurrent inequalities (35), whatever the Boolean algebra B may be (not only for the Boolean algebra B 2• considered in Example 2). The method of successive eliminations enables us to express also the solutions in a parametric form; in other words, it enables us to find the general solution, in the sense of the following Definition 4.

The Single. Equation Form of a System of Boolean Equations 25 these three basic operations. For the equivalence of Definitions 1 and 1', as well as for further properties of Boolean functions, see, for instance, S. RUDEANU [2, 4]. Definition 2. ,x n } [respectively, of the form t (Xl' ... , Xn) ~ g (Xl, ... , xn)], g are Boolean functions. Definition 3. Two (systems of) Boolean equations (inequalities) are called equivalent if they have the same solutions. Lemma 1. A Boolean equation 1 = g is equivalent to the Boolean equation 1{j u 1g = 0, (and also to 1g u 1{j = I} while a Boolean in· equality 1 ~ g is equivalent to the Boolean equation 1{j = 0 (and also tolug=l).

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