# 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.

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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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Topics: Decimal (53%), Binary number (50%)

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TL;DR: It is shown that all presented algorithms are polynomial in the number of minterms occurring in the canonical disjunctive normal form representation of a Boolean function.

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Abstract: A notion of a neighborhood cube of a term of a Boolean function represented in the canonical disjunctive normal form is introduced. A relation between neighborhood cubes and prime implicants of a Boolean function is established. Various aspects of the problem of prime implicants generation are identified and neighborhood cube-based algorithms for their solution are developed. The correctness of algorithms is proven and their time complexity is analyzed. It is shown that all presented algorithms are polynomial in the number of minterms occurring in the canonical disjunctive normal form representation of a Boolean function. A summary of the known approaches to the solution of the problem of the generation of prime implicants is also included.

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52 citations

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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.

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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.

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22 citations

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TL;DR: It is shown that factoring accounts for a dramatic increase in efficiency over Nelson's algorithm, and the increased efficiency is illustrated with results obtained from several examples that were implemented for both algorithms using the symbolic manipulation systems SETS.

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Abstract: An algorithm for finding the prime implicants of a Boolean function is given. The algorithm is similar to Nelson's algorithm since both involve the operations of complementing, expanding, and simplifying, but the new algorithm includes the additional operation of factoring. The algorithm with factoring is proved, and it is shown that factoring accounts for a dramatic increase in efficiency over Nelson's algorithm. The increased efficiency is illustrated with timing results obtained from several examples that were implemented for both algorithms using the symbolic manipulation systems SETS.

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21 citations

### Cites methods from "Maxterm Type Expressions of Switchi..."

...For example, Scheinman [8], Das and Choudhury [1], and Slagle, Chang, and Lee [9] have developed similar [2] tabular methods of constructing trees, in which literals are assigned to either nodes or branches and in which certain paths correspond to prime implicants....

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TL;DR: A new algorithm for the generation of all the prime implicants of a Boolean function is described, which is different from those previously given in the literature and works equally well with either the conjunctive or the disjunctive form of the function.

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Abstract: One of the major areas in switching theory research has been concerned with obtaining suitable algorithrns for the minimization of Boolean functions in connection with the general problem of their economic realization. A solution of the minimization problem, in general, involves consideration of two distinct phases. In the first phase all the prime implicants of the function are found, while in the second phase, from this set of all the prime implicants, a minimal subset (according to some criterion of minimality) of prime implicants is selected such that their disjunction is equivalent to the function and from which none of the prime implicants can be dropped without sacrificing equivalence. Many different algorithms exist for solving both the first and the second phase of this minimization problem. In a recent paper,' Slagle et al. describe a new algorithm for the generation of all the prime implicants of a Boolean function. As claimed by the authors, this algorithm is different from those previously given in the literature. The algorithm is efficient, does not generate the same prime implicant more than once (though the algorithm sometimes generates some non-prime implicants), and does not need large capacity of memory for implementation on a digital computer. The algorithm works equally well with either the conjunctive or the disjunctive (both canonical and noncanonical) form of the function.

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9 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.

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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.

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9 citations

##### 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.

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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

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1,063 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.

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

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821 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.

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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.

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737 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.

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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.

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96 citations

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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.

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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.

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59 citations