# Minimal Third-order Expressions of Boolean Unate Functions

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.

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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: 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: In this article, a method of direct determination of all the minimal prime implicant covers of switching functions has been presented, where the columns of the cover table cannot be arranged in a single connected cover term matrix or in a number of connected cover terms with mutually disjoint sots of the implicants.

Abstract: A method of direct determination of all the minimal prime implicant covers of switching functions has been presented in the paper. It has been shown that some of the difficulties encountered in finding directly all the minimal prime implicant covers of the function for which the columns of the cover table cannot be arranged in a single connected cover term matrix or in a number of connected cover term matrices with mutually disjoint sots of prime implicants can be overcome by first dividing the cover table into a number of sub-tables such that the columns of one of the sub-tables can be arranged to form a connected cover term matrix by ignoring the presence of some of the prime implicants from some of its columns. Next by associating the different irredundant covers of the other sub-tables with this connected cover term matrix, all the minimal prime implicant covers of the function can be found out.

11 citations

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TL;DR: A method of detection of component functions of fewer number of variables, disjunct of which forms a given switching function, which yields bettor economy in several cases with regard to number of logical circuit elements.

Abstract: This paper describes a method of detection of component functions of fewer number of variables, disjunct of which forms a given switching function. The multi-level electronic circuit synthesized from the component functions yields bettor economy in several cases with regard to number of logical circuit elements compared with the case when the same function is synthesized from the ‘ minimal’ two-stage forms.

6 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