# Coincidence between Boolean products and its application to third-order simplification†

TL;DR: The second-order expressions of Boolean functions can have either sum-ofproduct or product-of-sum forms, and the concept of coincidence between the p terms of the function is introduced in this article.

Abstract: The second-order expressions of Boolean functions can have either sum-of-product or product-of-sum forms For a Boolean function specified in the irredundant sum-of-product form as the disjunction of a number of prime implicants or p terms, groups of these p terms can sometimes be more economically realized in the minimal product-of-sum forms than in the sum-of-product forms To know whether a group of p terms in the irredundant sum-of-product form of the function has a more economic realization in the product-of-sum form, the concept of coincidence between the p terms of the function is introduced in the paper and a number of interesting properties of the function in relation to coincidence are established The coincidence between a pair of p terms in a function is defined as the number of literals occurring as mutually common in their algebraic representations It is next shown that the study of the properties of Boolean functions in relation to coincidence also aids in readily obtaining the economic th

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TL;DR: A systematic procedure is presented for writing a Boolean function as a minimum sum of products and specific attention is given to terms which can be included in the function solely for the designer's convenience.

Abstract: A systematic procedure is presented for writing a Boolean function as a minimum sum of products This procedure is a simplification and extension of the method presented by W V Quine Specific attention is given to terms which can be included in the function solely for the designer's convenience

1,103 citations

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TL;DR: The Problem of Simplifying Truth Functions is concerned with the problem of reducing the number of operations on a graph to a simple number.

Abstract: (1952). The Problem of Simplifying Truth Functions. The American Mathematical Monthly: Vol. 59, No. 8, pp. 521-531.

885 citations

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TL;DR: A thorough algebraic method is described for the determination of the complete set of irredundant normal and conjunctive forms of a Boolean function that is mechanical and therefore highly programmable on a computer.

Abstract: A thorough algebraic method is described for the determination of the complete set of irredundant normal and conjunctive forms of a Boolean function. The method is mechanical and therefore highly programmable on a computer.

63 citations

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General Mills

^{1}TL;DR: A numerical easily programmed procedure is given with which it is possible to treat problems with a greater number of variables than has heretofore been practical.

Abstract: The topology of the n-dimensional cube is used to reduce the problem of determining the minimal forms of a Boolean function of n variables to that of finding the minimal coverings of the essential vertices of the basic cell system associated with the given function. The proof of this statement is contained in the central Theorem 4. A numerical easily programmed procedure is given with which it is possible to treat problems with a greater number of variables than has heretofore been practical. The procedure by-passes the determination of the basic cells (the prime implicants of W. V. Quine) and locates the essential vertices, from which in turn the irredundant and minimal forms are obtained.

59 citations

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TL;DR: By applying repeatedly the rule of consensus to the prime implicants, it is possible to derive alist of implication relations that express the necessary and sufficient conditions of eliminability of the primeimplicants in terms of which the irredundant normal forms can be computed.

Abstract: This paper describes a new algebraic way of determining irredundant forms from the prime implicants. The method does not require using the developed normal form, and it makes novel application of Quine's technique of iterative consensus-taking. Thus, by applying repeatedly the rule of consensus to the prime implicants, it is possible to derive alist of implication relations that express the necessary and sufficient conditions of eliminability of the prime implicants in terms of which the irredundant normal forms can be computed. The extension of Quine's technique to this phase of simplification serves to shorten considerably the logical machinery needed for complete solution of the simplification problem. By the same token, it renders the method suitable for use with a digital computer.

42 citations