Abstract
AbstractThe orthogonalization of Boolean functions in disjunctive form, that means a Boolean function formed by sum of products, is a classical problem in the Boolean algebra. In this work, the novel methodologyORTH[ⴱ] of orthogonalization which is an universally valid formula based on the combination technique »orthogonalizing difference-building ⴱ« is presented. Therefore, the technique ⴱ is used to transform Sum of Products into disjoint Sum of Products. The scope of orthogonalization will be solved by a novel formula in a mathematically easier way. By a further procedure step of sorting product terms, a minimized disjoint Sum of Products can be reached. Compared to other methods or heuristicsORTH[ⴱ] provides a faster computation time.
Highlights
Introduction and PreliminariesA Boolean function of n variables is defined as the mapping f (x) : {0, 1}n → {0, 1}
The orthogonalization of Boolean functions in disjunctive form, that means a Boolean function formed by sum of products, is a classical problem in the Boolean algebra
Four normal forms of Boolean functions exist, the disjunctive normal form (DNF), conjunctive normal form (CNF), antivalence normal form (ANF) and equivalence normal form (ENF), which consist of either product terms pk(x)
Summary
A Boolean function of n variables is defined as the mapping f (x) : {0, 1}n → {0, 1}. An orthogonal representation of a SOP, that means dSOP, is characterized by product terms which are disjoint to one another in pairs [3, 4]. The dSOP is equivalent to dESOP consisting of the same product terms and differ only in the logical connectivity between the product terms This relationship can be explained well with the following definition out of [6], if both product terms pi(x) and pj(x) are disjoint to each other. That both products terms are disjoint, building their conjunction results to 0. North as the number of product terms in the orthogonal result corresponds to the number of literals presented in the subtrahend ps(x) and are not presented in the minuend pm(x) at the same time.
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