Showing papers on "Boolean function published in 1981"
28 Oct 1981
TL;DR: This work provides an intimate link between PDL as defined by the Segerberg axioms and the mu-calculi of de Bakker and Park and shows that its satisfiability problem is decidable in exponential time.
Abstract: We describe a mu-calculus which amounts to modal logic plus a minimization operator, and show that its satisfiability problem is decidable in exponential time. This result subsumes corresponding results for propositional dynamic logic with test and converse, thus supplying a better setting for those results. It also encompasses similar results for a logic of flowgraphs. This work provides an intimate link between PDL as defined by the Segerberg axioms and the mu-calculi of de Bakker and Park.
91 citations
TL;DR: In this article, the complexity of three types of programmable logic arrays (PLA's) is obtained by the theory of multiple-valued decomposition, and it is shown that a generalized Boolean function f(X1, X2,··, Xr) can be directly realized by a PLA with decoders or a three-level PLA.
Abstract: Generalized Boolean functions are shown to be useful for the design of programmable logic arrays (PLA's), and the complexity of three types of PLA's is obtained by the theory of multiple- valued decomposition. A two-level PLA consists of an AND array and an OR array, and they are cascaded to perform a two-level AND-OR circuit. A PLA with decoders consists of decoders, an AND array, and an OR array. A three-level PLA consists of a D array, an AND array, and an OR array, and they are cascaded to perform a three-level OR- AND-OR circuit. It is shown that a generalized Boolean function f(X1, X2,··, Xr):X Bni → B, where B = {0,1}, is represented by a generalized Boolean expression of 2ni-valued variables X i ; and f can be directly realized by a PLA with decoders or a three-level PLA. To realize a function of n-variables (n = 2r), the following sizes are shown to be sufficient: for a two-level PLA, (n + ½) 2n; for a PLA with two-bit decoders, 4(n + 4) 2n; for a three-level PLA, 2n+ (3n + l)√2n+ 2n2Especially in the case of PLA with two-bit decoders, the following sizes are shown to be necessary and sufficient: for an arbitrary symmetric function, 3/2(n + ½) √3n; and for a parity function, (n + ½)√ 2n.
84 citations
TL;DR: Through proper coding of the objects in a binary picture it is shown that the binary and Minkowski operators can be implemented in such a way as to decrease significantly computational complexity.
54 citations
28 Oct 1981
TL;DR: A projection of a Boolean function is a function obtained by substituting for each of its variables a variable, the negation of a variable or a constant as discussed by the authors, and it is shown that much of what is of everyday relevance in Turing machine-based complexity theory can be replicated easily and naturally in this elementary framework.
Abstract: A projection of a Boolean function is a function obtained by substituting for each of its variables a variable, the negation of a variable, or a constant. Reducibilities among computational problems under this relation of projection are considered. It is shown that much of what is of everyday relevance in Turing-machine-based complexity theory can be replicated easily and naturally in this elementary framework. Finer distinctions about the computational relationships among natural problems can be made than in previous formulations and some negative results are proved.
50 citations
TL;DR: A sequence of monotone Boolean functions hn:{0, 1}n→{0,1}n, such that the monotones complexity of hn is of order n2/log n is constructed, which includes the largest known lower bound of this kind.
29 citations
01 Jan 1981
TL;DR: It is proved that the search problem and the decision problem associated with each self-transformable problem are equally hard, which means that self- transformability bridges over the distinction between pure existential proofs and existential proofs constructing the object.
Abstract: A combinatorial problem is called self-transformable if each instance of the problem can be reduced in polynomial time to a set of smaller instances of the same problem. Most natural NP-complete problems are self-transformable as well as other important problems such as graph isomorphism and feasibility of linear inequalities. We prove that the search problem and the decision problem associated with each self-transformable problem are equally hard. This means that self-transformability bridges over the distinction between pure existential proofs and existential proofs constructing the object. This carries over to random algorithms. As a consequence every algorithm which efficiently detects non-isomorphic pairs of graphs, possibly by using random tests and which fails at most on a sparse set of non-isomorphic pairs of graphs, yields an efficient method for constructing isomorphisms between pairs of isomorphic graphs. In an independent section we show that the efficiency of man machine interaction in computing a Boolean function f is strongly limited by the network complexity of f. Any complex Boolean function f can be computed efficiently only if the human problem solver supplies as much information as is necessary to encode the minimal network for f.
24 citations
[...]
IBM1
TL;DR: A heuristic approach to the boolean equivalence problem is described which yields information important for understanding structural differences and an algorithm for approximating this difference is presented.
Abstract: This report deals with the problem of discovering the differences between two implementations of the same partially specified function. It describes a heuristic approach to the boolean equivalence problem which yields information important for understanding structural differences. It also contains a formalization for the notion of "structural difference" and presents an algorithm for approximating this difference.
18 citations
TL;DR: In this paper, a 15-term minimal form for the tree per both K&H and W&S is presented, identical to the list of p.i.'s.
Abstract: The verification by Worrell & Stack (WS this facilitates Boolean algebraic operations, so that both sets are described economically in their minimal forms, a subset of the prime implicants (p.i.'s). Quine's consensus operation is used to minimize and to find the p.i.'s. Corresponding to the MOCUS output for the inverse reported by KH 13 of these are a unique minimal form. Instead of 352 p.i.'s for the tree per both K&H and W&S, we have a 15-term minimal form, identical to the list of p.i.'s. The results are further analyzed as a contribution to the continuing discussion of the utility of minimal forms vis-a-vis the p.i.'s and of the consensus method.
15 citations
TL;DR: The laws of binary Boolean algebra are extended to encompass multistate variables, and simplification and consensus algorithms whereby prime implicants for non-coherent systems containing multistates variables can be obtained are developed.
Abstract: For s-coherent fault trees containing only AND/OR gates, many algorithms can be used to obtain the sum-of-products (s.o.p.), cut set, expressions for the top event. If the tree contains non-coherences such as XOR (exclusive OR) gates, these s.o.p. expressions can be reduced to irredundant prime implicant form by algorithms such as Kumamoto & Henley's or by applying simplification and consensus algorithms such as Nelson's or Quine's. If, however, the trees contain multistate variables, then the Boolean binary logic expressions on which present algorithms are based no longer apply. This paper extends the laws of binary Boolean algebra to encompass multistate variables, and develops simplification and consensus algorithms whereby prime implicants for non-coherent systems containing multistate variables can be obtained. A computer code based on this has been developed.
10 citations
DOI•
01 Nov 1981TL;DR: A new and efficient technique for the generation of near-minimal covers of Boolean functions is described, using a fast in-place transformation for the identification of all prime implicants covering a particular minterm.
Abstract: A new and efficient technique for the generation of near-minimal covers of Boolean functions is described. For the identification of all prime implicants covering a particular minterm, use is made of a fast in-place transformation. Crucial points in such covering procedures are the decision which of the minterms to process next and which of its prime implicants to use in the near-minimal cover. Both these problems are handled by a weighting transformation. A computer program QUAMIN, implementing the method, has been compared to other programs. It is shown that the new method performs well with respect to accuracy, computer time and storage requirements.
9 citations
TL;DR: An algorithm is presented for evaluating exact system reliability without using a Boolean polynomial equation and a Boolean expression is used to define logical system configuration.
Abstract: An algorithm is presented for evaluating exact system reliability without using a Boolean polynomial equation. A Boolean expression is used to define logical system configuration. The Boolean expression is converted to Polish postfix notation. System reliability is computed by converting the postfix notation into a sequence of arithmetic and stack operations. Errors are analyzed to identify a possible rounding or chopping error during floating point calculation for system reliability by digital computer.
TL;DR: Two algorithms have been described to realise single fault-tolerant sequential machines so that the resulting excitation functions can be implemented by special TLM and MTLM modules.
TL;DR: In this article, a Boolean function of n variables is completely determined by its 2 n Walsh coefficients, and if the function is given as monotone, it can be specified by a smaller number of its Walsh coefficients.
Abstract: A Boolean function of n variables is completely determined by its 2 n Walsh coefficients. However, if the function is given as monotone, it can be specified by a smaller number of its Walsh coefficients. A method to determine this subset of coefficients is described and illustrated. Tabulated results are presented for monotone Boolean functions of four variables or less; e.g., n ? 4.
TL;DR: This paper proves the existence of a Boolean function f with the following properties, and states the best possible hierarchy result on the depth of all nondegenerate Boolean functions.
Abstract: Circuit depth is an important complexity measure for a Boolean function. Let some Boolean function of n variables have depth k according to an arbitrary binary basis Ω. For each j where [log n]?j?k we prove the existence of a Boolean function f with the following properties. f depends essentially on n variables and the depth of f according to Ω is exactly j Thus we state the best possible hierarchy result on the depth of all nondegenerate Boolean functions.
TL;DR: A consistent theory for the minimization of fuzzy functions given in disjunctive normal form by studying the structure of such functions by introducing the notion of standard complex minterms and standard minterms is presented.
Abstract: In this paper, we present a consistent theory for the minimization of fuzzy functions given in disjunctive normal form by studying the structure of such functions. An O(N 2 ) time algorithm is given for minimization of a fuzzy function consisting entirely of elementary phrases. The minimal form in this case is shown to be unique (i.e., the minimal form consists of a unique set of phrases). By introducing the notion of standard complex minterms and standard minterms a tabular method is presented and illustrated for the minimization of a disjunctive normal form consisting of entirely complex phrases. The method resembles the well-known Quine-McCluskey tabular procedure for Boolean function minimization. The minimal form in this case may not be unique in general. Finally, the approach is extended for the case of minimization of an arbitrary disjunctive normal form consisting of both elementary as well as complex phrases.
TL;DR: In this paper, a method is proposed by which the prime implicants of a boolean function can be obtained without generating the lower-order terms, and an adjacency table is formed in which all the adjacent minterms of lower and higher weights are shown for every true (and every don't caro ", if any) minterm of the function.
Abstract: A method is proposed by which the prime implicants of a boolean function can be obtained without generating the lower order terms. An adjacency table is formed in which all the adjacent minterms of lower and higher weights are shown for every true (and every ’ don't caro ‘, if any) minterm of the function. Prime implicants are obtained by grouping these minterms according to certain observations.
01 Jan 1981
TL;DR: It is shown that a generalized Boolean function f(X1, X2,··, Xr):X Bni → B, where B = {0,1}, is represented by a generalized boolean expression of 2-valued variables Xi ; and f can be directly realized by a PLA with decoders or a three-level PLA.
Abstract: wforthedesign ofprogrammable logic arrays (PLA's), andthecomplexity ofthree types ofPLA'sisobtained bythetheory ofmultiple-xi valued decomposition.. Atwo-level PLAconsists ofanANDarray and anORarray, andtheyarecascaded toperform atwo-level AND-OR 2 circuit. APLAwith decoders consists ofdecoders, anANDarray, and ANDar2n anORarray. Athree-level PLAconsists ofaD array, anANDarray, andanORarray, andthey arecascaded toperform athree-level ORAND-ORcircuit. Itisshownthat ageneralized Boolean functionf(XI, xn X2,,Xr):X Bni B,whereB = 10, 11, isrepresented bya generalized Boolean expression of2ni-valued variables Xi;andf can 1 I
TL;DR: In this article, all the decompositions of a Boolean function of four variables can be determined directly from its prime implicants if it is decomposable with one free variable.
TL;DR: In this paper, a method is proposed to determine various symmetries of a completely or incompletly specified Boolean function, such as equivalence, nonequivalence, and single-variable symmetry.
TL;DR: In this paper, the probability of a minterm being a prime implicant or an essential prime implcant was calculated for n ≥ 10 variables in a computer program, assuming that all boolean functions of n variables are randomly generated.
Abstract: Some properties of boolean functions are presented. Assuming that all boolean functions of n variables are randomly generated, those properties are used to calculate the probabilities of a minterm being a prime implicant or an essential prime implicant or of a prime implicant being an essential primo implicant. Numerical results for n≤10 were obtained usign a computer program.
TL;DR: Theorems relating the most important concepts of switching theory to a new kind of difference operator are stated and this difference operator is used as a unifying concept to solve, by means of a new type of algorithm, several problems arising in logical design.
01 Sep 1981
TL;DR: An iterative algorithm for testing two-asummability of any given Boolean function is described in this letter, based on the decomposition of Boolean functions in terms of reduced functions, suitable for machine computation.
Abstract: An iterative algorithm for testing two-asummability of any given Boolean function is described in this letter. The method is based on the decomposition of Boolean functions in terms of reduced functions, and it is suitable for machine computation.
01 Jan 1981
TL;DR: This work describes the kinetic logic formalization of a model as a complementary tool to the classical continuous formalism which is more limited in the number of state variables it can handle and is very sensitive to the nonlinear character of the kinetic differential equations.
Abstract: The description of complex systems by means of logical variables is a method which allows arriving rapidly at a qualitative understanding of that system [1,2,3]. Such an approach offers a complementary tool to the classical continuous formalism which is more limited in the number of state variables it can handle and is very sensitive to the nonlinear character of the kinetic differential equations. To the reader who is not familiar with the kinetic logic formalization of a model, we give here a short introduction.
TL;DR: In this paper, the problem of finding the maximum upper zero of a monotonic function of finite-valued logic is solved in the Shannon formulation, and it is proved that the solution has a form similar to well-known solutions for Boolean functions and functions of k -valued logic.
Abstract: THE PROBLEM of finding the maximum upper zero of a monotonic function of finite-valued logic is solved in the Shannon formulation. It is proved that in the finite-valued case the solution has a form similar to well-known solutions for Boolean functions and functions of k -valued logic.