Application of super-sticking algebraic operation of variables for Boolean functions minimization by combinatorial method

Abstract

The simplification of the problem of Boolean function minimization by a combinatorial method is a new procedure for the algebra of logic – super-sticking of variables. This procedure is performed if there is a complete binary combinatorial system with repetition or an incomplete binary combinatorial system with repetition in the truth table structure. The procedure for reducing the total perfect disjunctive normal form (PDNF) of the logical function gives unity. And since the complete PDNF uniquely determines the complete binary combinatorial system with repetition and vice versa, this gives grounds to delete all the blocks of the complete binary combinatorial system from the truth table, whose structure allows to carry out the rules of super-sticking of variables. The efficiency of the algebraic operation of supers-sticking of variables greatly simplifies the algorithm for Boolean function minimization and allows manual minimization of functions with a number of variables up to 10. The complexity of the algorithm for finding the minimal function by a combinatorial method is O(n) and is linear for n<7. With an increase in the number of variables from n=6 to 8, the growth dynamics of the number of transformations is characterized by the law O(n 2 ), followed by the growth of O(f(n)) with the increase in the Boolean function capacity according to the polynomial law. The introduction of an algebraic operation of super-sticking of variables to the problem of Boolean function minimization is more advantageous in comparison with analogs in the following factors: – lower cost of development and implementation, since a significant proportion of functions are minimized by functions with a number of variables of no more than 16, and therefore, in general, the need for automation of the process of minimizing the function decreases; – increase in manual minimization of 4–10 bit functions, facilitates control and study of the algorithm for minimizing the logic function. The combinatorial method of Boolean functions minimization can find practical application in the design of electronic computer systems, because: – minimization of the DNF function is one of the multiextremal logic-combinatorial problems, the solution of which is, in particular, the combinatorial device of the block-diagram with repetition; – extends the capabilities of the algorithm for Boolean functions minimization for their application in information technology; – improves the algebraic method of Boolean function minimization due to the tabular organization of the method, the introduction of the shaped transformation apparatus and the rules of super-sticking of variables.