# Simplification of Switching Functions Involving a very Large Number of ‘Don't Care’ States†

TL;DR: In this paper, a method of simplification of switching functions involving a very large number of "don't care" states is suggested. But it is not shown how the knowledge of the sets of prime implicants thus obtained can be used for finding minimal or other irredundant sums of switching function.

Abstract: A method of simplification of switching functions involving a very large number of ‘ don't care’ states is suggested in the present paper. First a tabular technique is suggested which generates all the prime implicants starting from the maxterm type expressions of switching functions, avoiding generation of the prime implicants formed of ‘don't care’ states only. The technique presented is simple and iterative. Next it is suggested how the knowledge of the sets of prime implicants thus obtained can be utilized for finding minimal or other irredundant sums of switching functions.

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TL;DR: This note describes an iterative procedure for generating the prime implicants of switching functions by utilizing a new tabular mode of functional representation called clause-column table, which can be applied equally well to functions given in the sum-of-products or in the product- of-sums froms.

Abstract: This note describes an iterative procedure for generating the prime implicants of switching functions by utilizing a new tabular mode of functional representation called clause-column table. The procedure generates all the prime implicants and can be applied equally well to functions given in the sum-of-products or in the product-of-sums froms, both canonical and noncanonical. The procedure can also be readily adapted to determine the prime implicants of functions having a large number of unspecified or DON'T CARE terms.

23 citations

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TL;DR: The survey focuses on minimization of boolean functions in the class of disjunctive normal forms (d.n.f.s) and covers the publications from 1953 to 1986, and presents a classification of minimization algorithms.

Abstract: The survey focuses on minimization of boolean functions in the class of disjunctive normal forms (d.n.f.s) and covers the publications from 1953 to 1986. The main emphasis is on the mathematical direction of research in boolean function minimization: bounds of parameters of boolean functions and algorithmic difficulties of minimal d.n.f. synthesis). The survey also presents a classification of minimization algorithms and gives some examples of minimization heuristics with their efficiency bounds.

11 citations

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TL;DR: In this paper, an optimal controller which does not require prediction is derived for a discrete, single variable time delay system and compared with others which handle time delay problems and its performance tested on a simulation example of practical en...

Abstract: The fact that the most popular techniques available for handling time delay problems in process control involve prediction, coupled with the very notion of a time delay, and its unique effect on process dynamics, have led to the following general consensus: that any systematic treatment of the controller design problem for time delay systems must result in controllers that require prediction. The traditional problem faced by these controllers in practice however, has been that of obtaining acceptable predictions using unavoidably imperfect mathematical models of the plant's dvnamic behavior. While this may not necessarily constitute an insurmountable barrier, a means by which the prediction problem may be totally circumvented is presented in this paper. An optimal controller which does not require prediction is derived for a discrete, single variable time delay system. The controller is compared with others which handle time delay problems and its performance tested on a simulation example of practical en...

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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

##### References

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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 basic part of the general synthesis problem is the design of a two-terminal network with given operating characteristics, and this work shall consider some aspects of this problem.

Abstract: THE theory of switching circuits may be divided into two major divisions, analysis and synthesis. The problem of analysis, determining the manner of operation of a given switching circuit, is comparatively simple. The inverse problem of finding a circuit satisfying certain given operating conditions, and in particular the best circuit is, in general, more difficult and more important from the practical standpoint. A basic part of the general synthesis problem is the design of a two-terminal network with given operating characteristics, and we shall consider some aspects of this problem.

774 citations

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TL;DR: This paper develops a method for both disjunctive and conjunctive normal truth functions which is in some respects similar to Quine's but which does not involve prior expansion of a formula into developed normal form.

Abstract: In [1] Quine has presented a method for finding the simplest disjunctive normal forms of truth functions. Like the tabular methods of [2] and [3], Quine's method requires expansion of a formula into developed normal form as a preliminary step. This aspect of his method to a certain extent defeats one of the purposes of a mechanical method, which is to secure simplest forms in complicated cases (perhaps by using a digital computer) [4]. In the present paper we develop a method for both disjunctive and conjunctive normal truth functions which is in some respects similar to Quine's but which does not involve prior expansion of a formula into developed normal form. Familiarity with [1] is presupposed. We use the notations and conventions of [1] with the following exceptions and additions. ‘Φ’ names any formula, ‘Ψ’ any conjunction of literals, and ‘χ’ any disjunction of literals. Any disjunction of conjunctions of literals is a disjunctive normal formula and is designated by ‘ψ’; any conjunction of disjunctions of literals is a conjunctive normal formula and is designated by ‘X’. Note that we do not make use of Quine's notion of fundamental formulas. A formula Ψ occurring in a disjunctive normal formula ψ, provided it is a disjunct of ψ, is a clause ; similarly for χ. We use ‘≠” for logical equivalence of formulas and ‘=’ for identity of formulas to within the order of literals in clauses and the order of clauses in normal formulas.

96 citations

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TL;DR: In this paper, an iterative technique for simplifying Boolean functions is presented, which enables the user to obtain prime implicants by simple operations on a set of decimal numbers which describe the function.

Abstract: This article presents an iterative technique for simplifying Boolean functions. The method enables the user to obtain prime implicants by simple operations on a set of decimal numbers which describe the function. This technique may be used for functions of any number of variables.

33 citations