Maxterm Type Expressions of Switching Functions and Their Prime Implicants
TL;DR: Algorithms have been formulated for this purpose which first generate all possible prime implicants corresponding to a specified switching function and then select minimal subsets of these primeimplicants for use in the formation of the minimal networks.
Abstract: One of the basic problems of combinational switching circuit theory is that of designing circuits with a minimum number of AND-gates or prime implicants. Algorithms have been formulated for this purpose which first generate all possible prime implicants corresponding to a specified switching function and then select minimal subsets of these prime implicants for use in the formation of the minimal networks [1]-[6]. In practically all the currently available methods of simplification of switching functions, use is made of the minterm type expression specified in the algebraic or its equivalent binary or decimal form. Operations with binary or decimal numbers have become very popular because of their inherent advantages.
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20 Nov 2014
TL;DR: An algorithm for reduction from multi input into two input variables, if it involves more than six variables for the simplification of Boolean functions, is proposed.
Abstract: Modern society developed on the base of technology improvement. Progress in technologies based on computer, communication and automatic controls engineering all are depends on Integrated Circuit(IC). In the first half of 1990s, the electronic industry experienced an explosion in the demand for personal computers, cellular telephones, and high speed data communication devices. Vying for market share, vendors built products increasingly greater functionality, higher performance; lower cost, lower power consumption, and smaller in dimensions. To do these, vendors created highly integrated, complex systems with fewer IC devices and less printed-circuit-board (PCB) area. Minimizations of combinational logic functions before implementation in hardware have many advantages. A reduced number of gates considerably decreases the cost of the hardware, reduces the heat generated by the chip and most importantly increases the speed. Boolean functions may be simplified by algebraic means by applying the basic laws and theorems. However, this procedure is awkward because it lacks specific rules to predict each succeeding step in the minimization process for multi-input variables. The simplification by K- mapping has many disadvantages of redundant grouping and non-unique solution. But no method is effective for the simplification of Boolean functions, if it involves more than six variables. A few algorithm and modified technique are found in literature. This paper proposes an algorithm for reduction from multi input into two input variables.
7 citations
Cites methods from "Maxterm Type Expressions of Switchi..."
...This basic idea of Nelson was subsequently utilized by Das and Choudhury in developing a tabular method for a more efficient generation of all the prime implicants of a Boolean function starting from the maxterm type expression represented in decimal mode [5]....
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TL;DR: This paper describes a technique for extending the clause table approach to the minimization of multi-terminal networks and extends the algorithm of Slagle et al. to generate the important implicants of a set of switching functions and to determine a minimum realization.
Abstract: There are a number of methods which use a clause table to generate the prime implicants of a switching function. This paper describes a technique for extending the clause table approach to the minimization of multi-terminal networks. The specific extension described here extends the algorithm of Slagle et al. to generate the important implicants of a set of switching functions and to determine a minimum realization.
2 citations
Cites background from "Maxterm Type Expressions of Switchi..."
...111 112 S121 s131 (1) * (1) * (1) * (1) * S2 S S4 S5 S s2 3 s4 5 s6...
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...x3 x3 3/ \X~~~~~~3 S211 S212 (O) (0)* (O) (O) (1) 3) (1) x3...
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...This step is recursive and continued until 1) the node represents an empty set, which is defined as a success (1) or 2) the node represents a set containimng an empty clause, which is defined as a failure (0)....
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...x(3x4 ( 3 (°) (O) (1) (1) K(1x) (0) (0) (0)...
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23 May 2016
TL;DR: An assessment of the performance of this iterative procedure for the generation of the prime implicants of switching functions by utilizing a new tabular mode of functional representation called clause-column table indicates that the algorithm performs quite elegantly with respect to the CPU time and memory usage.
Abstract: Of late, an iterative procedure for the generation of the prime implicants of switching functions by utilizing a new tabular mode of functional representation called clause-column table was developed by Das and Khabra. This approach of Das and Khabra generates all the prime implicants of the functions given either in the sum-of-products or in the product-of-sums forms, both canonical and noncanonical, and is computationally very efficient. The method can be applied for finding the prime implicants of functions having a large number of unspecified or don't care terms as well. In a clause-column table representation of a function, the set of literals or clauses defining a product or sum term in the sum-of-products or product-of-sums form, respectively, appears in a column, the total number of columns in the table being equal to the number of such product or sum terms. Although the basic principle utilized in the procedure is that of successive expansion around the literals as employed in some of the other existing methods, the manner the expansion is carried through is entirely different and greatly reduces the generation of the redundant terms. For a switching function specified in the product-of-sums form, the method produces all the prime implicants, while for a switching function given in the sum-of-products form, the procedure yields all the prime implicates first and is next applied to this set of prime implicates to obtain the prime implicants of the function. In the paper, an assessment of the performance of this prime implicant generation algorithm has been made by simulating it on a computer under conditions of different numbers of variables and different numbers of terms in the product-of-sums (sum-of-products) representations of the functions. The experimental outcome indicates that the algorithm performs quite elegantly with respect to the CPU time and memory usage.
2 citations
Cites methods from "Maxterm Type Expressions of Switchi..."
...Das and Choudhury also extended their method for generating the prime implicants of functions having a large number of unspecified or don’t care terms....
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...Several other wellknown methods of prime implicant determination include those of Scheinman [6], Hall [7], Das and Choudhury [8] and Slagle et al. [9]....
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...This basic idea of Nelson was later utilized by Das and Choudhury in developing a tabular method for a more efficient generation of all the prime implicants of switching functions starting from the maxterm-type expression given in the decimal form....
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...Several other wellknown methods of prime implicant determination include those of Scheinman [6], Hall [7], Das and Choudhury [8] and Slagle et al....
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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
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
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
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
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