Showing papers on "Boolean function published in 1972"
TL;DR: The investigation of the formal systems obtained by extending well-known connectives to continuous arguments, called ''soft algebras,'' are investigated and it is shown that every soft algebra is a bounded, distributive and symmetric lattice.
46 citations
17 citations
01 Jan 1972
TL;DR: It is shown that, starting from the extended truth vector, it is possible to compute the Taylor expansions of a given function and of its derivatives with a (3/4)" saving of complexity with respect to the known algorithms.
Abstract: The paper studies the 'usefulness of the representation of a Boolean function by means of an easily constructed binary vector having 3" components and called extended truth vector. It is shown that, starting from the extended truth vector, it is possible (a) to compute the Taylor expansions of a given function and of its derivatives with a (3/4)" saving of complexity with respect to the known algorithms; (b) to compute the weightsof these expansions, and accordingly to choose an optimal expansion point.
16 citations
TL;DR: Optimal networks consisting of NOR-OR gates (each gate produces the NOR and/or the OR of its inputs) are tabulated for all Boolean functions of three variables.
Abstract: Optimal networks consisting of NOR-OR gates (each gate produces the NOR and/or the OR of its inputs) are tabulated for all Boolean functions of three variables. Optimality is defined as minimizing first the number of gates and then the number of interconnections. The optimal networks were synthesized for each Boolean function by using an integer programming synthesis technique.
16 citations
TL;DR: By considering universal logic modules whose terminals may be interconnected, new designs are presented for six, nine, and ten arguments, whose numbers of terminals represent substantial improvements over previously known designs.
Abstract: By considering universal logic modules whose terminals may be interconnected, new designs are presented for six, nine, and ten arguments, whose numbers of terminals represent substantial improvements over previously known designs.
12 citations
01 Jan 1972
TL;DR: It is shown how an algebraization of the Ashenurst's theory directly leads to a decomposition condition much simpler than that of Akers, which allows the building of a simple scheme for disjunctive and nondisjunctive decompositions of Boolean functions.
Abstract: A fast algorithm for the computation of the set of zl-operators of a given Boolean function is presented This algorithm together with some new theorems on functional decomposition derived in this paper allow the building of a simple scheme for disjunctive and nondisjunctive decompositions of Boolean functions 1 Introduetion The general problem of functional decomposition of Boolean functions has proven to be of a high computational complexity One approach to this problem is the decomposition chart of Ashenurst 1)Ashenurst's theory has been used by Curtis 2)as a basis for a more general theory in order to develop a systematic method of deriving economical multiple-stage switching circuits Unfortunately, the procedure proposed by Ashenurst and Curtis requires the testing of2n-2-n decomposition charts, where n is the number of input variables Another approach has been proposed by Akers 3) The algebraic decomposition condition obtained by Akers however requires the computation of zl-zl-operators: this leads to algebraic expressions 'of overwhelming size In this paper, we will show how an algebraization of the Ashenurst's theory directly leads to a decomposition condition much simpler than that of Akers The algebraization is performed through a systematic use of the Boolean differential operators defined in ref 4 In particular, we present a fast algorithm for the simple, disjunctive or not, decompositions of Boolean functions when the Jatter are represented by the sets of all their prime implicants and of all their prime implicates In secs 2 and 3 the problem is stated and the main results by Ashenurst and Curtis are, briefly recalled The theoretical results of this paper relative to decomposition conditions and to algebraic expressions of the zl-operator are gathered in secs 4 and 5 respectively Finally an algorithm for simple decompositions is presented in sec 6 The notations 'used in this paper are those of ref 4
11 citations
IBM1
TL;DR: A theorem is presented that simplifies the computations necessary for complementing a Boolean function.
Abstract: A theorem is presented that simplifies the computations necessary for complementing a Boolean function.
11 citations
10 citations
TL;DR: The authors show how mutual 2-asummobility can be utilized for the testing of isobaricity of threshold functions containing no more than seven variables.
Abstract: A simple algorithm for testing the 2-summability of Boolean functions is presented in this note. The concept of asummobility has been extended and the idea of "mutual 2-asummability" has been introduced. The authors show how mutual 2-asummobility can be utilized for the testing of isobaricity of threshold functions containing no more than seven variables.
9 citations
5 citations
01 Aug 1972
TL;DR: In this article, the Boolean difference technique for fault test derivation in logic networks is described through the use of simple examples and it is shown that the technique makes use of familiar Boolean algebra manipulations and eliminates the need for "chasing" 1's and 0's through the networks.
Abstract: The Boolean difference technique for fault test derivation in logic networks is described through the use of simple examples. It is shown that the technique makes use of familiar Boolean algebra manipulations and eliminates the need for "chasing" 1's and 0's through the networks. A hint of the additional complexity of sequential networks is shown through the introduction of state variables and the derivation of test sequences.
TL;DR: In this paper, the use of Boolean matrices for the manipulation of Boolean functions is described, and also a method whereby any function may be re-expressed in terms of any logical combination of its defining variables.
Abstract: The Boolean matrices employed here were first described by J.O. Campeau, and were used to analyse swtiching and counting circuits. This letter describes their use in the manipulation of Boolean functions, and also a method whereby any function may be re-expressed in terms of any logical combination of its defining variables. Such manipulations, it is shown, may be applied to a number of functions simultaneously, and leads to valuable techniques in logic-network synthesis.
01 Jun 1972
TL;DR: Properties of the cellular n-cube representation are used to advantage in developing a fast algorithm for finding the Prime Implicants, Essential Primeimplicants and Non-essential PrimeImplicants of a Boolean function.
Abstract: : Properties of the cellular n-cube representation are used to advantage in developing a fast algorithm for finding the Prime Implicants, Essential Prime Implicants and Non-essential Prime Implicants of a Boolean function. The algorithm is discussed and several examples are included showing computer solutions to selected Boolean function minimization problems. The complete FORTRAN source program listing for the automated algorithm is included. (Author)
TL;DR: The process of mapping Boolean functions is generalized from the Karnaugh map, and it is shown that there are eight useful maps corresponding to any function, from which eight two-layer networks may be derived for implementation of the function.
Abstract: The process of mapping Boolean functions is generalized from the Karnaugh map, and it is shown that there are eight useful maps corresponding to any function, from which eight two-layer networks may be derived for implementation of the function. Applications to e.c.l. and m.o.s. logic are mentioned.
TL;DR: The author shows that this new algorithm does not require the Boolean function to be in conjunctive normal form and is complete in the sense that it generates all the prime implicants, and can be used to efficiently solve the selection problem.
TL;DR: The system described in this paper represents a set of several hundreds of programs expressed in the LYaPAS algoritmical language and realizable with the help of an automatic programming system having the same name.
TL;DR: An algorithm is described which generates the set of representative functions of the negation and/or permutation of variables and negation of the function (NPN) equivalence classes of unate Boolean functions.
Abstract: An algorithm is described which generates the set of representative functions of the negation and/or permutation of variables and negation of the function (NPN) equivalence classes of unate Boolean functions. The algorithm is based upon integer programming techniques. The set of representative functions for the NPN equivalence classes of unate functions of six or fewer variables was obtained using this algorithm. The set of pseudothreshold functions of six or fewer variables was also tabulated by using this algorithm as a basis.
TL;DR: In this paper, a simple tabular method has been presented for the near minimal realization of arbitrary Boolean function by networks composed of And and Exclusive-Or circuits, which leads to significant circuit reduction.
Abstract: The use of Exclusive-Or circuits often lead to significant circuit reduction. However, the minimization of circuits containing Exclusive-Or gates is quite different from the minimization of conventional And-Or circuits. In this paper, a simple tabular method has been presented for the near minimal realization of arbitrary Boolean function by networks composed of And and Exclusive-Or circuits.
TL;DR: A method is presented which enables any Boolean function, defined on n variables, to be partitioned, and its exhaustive application enables a fast implementation of the Quine-McCluskey1-3 minimisation algorithm.
Abstract: A method is presented which enables any Boolean function(s), defined on n variables, to be partitioned. Each partition so extracted is independent of a selected number of the defining variables, and hence each member of such a partition may be defined on the remaining variables. The method is applicable to logic-network synthesis, and its exhaustive application enables a fast implementation of the Quine-McCluskey1–3 minimisation algorithm.
TL;DR: A lower bound on the complexity is derived in terms of the number of such switching function realizers required to realize all Boolean functions of n variables.
Abstract: In this note, arbitrary switching function realizers and their complexity are defined. Then, a lower bound on the complexity is derived in terms of the number of such switching function realizers required to realize all Boolean functions of n variables.