# The P-table and its application to second and third-order simplification of Boolean functions

TL;DR: In this paper, a technique for obtaining the product of sum expression from the sum of product expression of a Boolean function is presented, where a tabular representation is made with the product terms and the variables present in the function specified in the sum-of-product form and appropriate rows of the table are combined to give different sum terms.

Abstract: A technique has been developed in this article for obtaining the ‘ product of sum ’ expression from the ‘ sum of product ’ expression of a Boolean function. In this technique, first a tabular representation is made with the product terms and the variables present in the function specified in the ‘ sum of product ’ form and then appropriate rows of the table are combined to give different sum terms. The idea of the technique has also been extended for obtaining the third-order minimal expression of a Boolean function.

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TL;DR: The basic concepts are formulated, the general theorems are given, and all the distinct minimal ``sum of products of sums'' expressions are obtained in case the cell complex of f consists of two isolated points.

Abstract: The problem of economical synthesis of circuits for digital computers leads to the problem of finding Boolean expressions of minimal length equivalent to a Boolean expression f. Previous authors restricted themselves to ``sum of products'' expressions; dualizing this gives ``products of sums.'' The next more efficient step is to find minimal ``sums of products of sums'' expressions. In this paper, the basic concepts are formulated in Part I, and general theorems are given in Part II. In Part III, all the distinct minimal ``sum of products of sums'' expressions are obtained in case the cell complex of f consists of two isolated points. For the case of three isolated points partial results have been obtained which will be published in a later communication.

12 citations

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TL;DR: In this paper, a method of directly testing whether the AND-OR or OR-AND form of a switching function is more economic for some simple cases is presented, and simplified expressions leading to economic three-level synthesis have also been derived.

Abstract: A method of directly testing whether the AND-OR or the OR-AND form of a switching function is more economic for some simple cases is presented. Simplified expressions leading to economic three-level synthesis have also been derived.

3 citations

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TL;DR: In this paper, a simple and straightforward procedure for finding absolute minimal third-order expressions (in the sum-ofproduct-of-sum) of a special class of Boolean functions called unate functions is suggested.

Abstract: A simple and straightforward procedure for finding absolute minimal third-order expressions (in the ‘ sum-of-product-of-sum’ forms) of a special class of Boolean functions called unate functions is suggested in the paper. The central idea developed through the procedure involves a decomposition of the assigned Boolean function first into a group of sub-functions called maximal uniliteral sub-functions (MTJL's) each of which is realizable in a minimal second-order ‘ product-of-sum ’ form and then a selection of an appropriate sub-set of these maximal uniliteral sub-functions or MUL's (or their sub-functions) in order to cover all the prime implicants of the function minimally.

2 citations