Showing papers on "Boolean function published in 1982"
05 May 1982
TL;DR: A new method for proving lower bounds on the complexity of VLSI - computations and more generally distributed computations which only applies to deterministic computations.
Abstract: In this paper we describe a new method for proving lower bounds on the complexity of VLSI - computations and more generally distributed computations. Lipton and Sedgewick observed that the crossing sequence arguments used to prove lower bounds in VLSI (or TM or distributed computing) apply to (accepting) nondeterministic computations as well as to deterministic computations. Hence whenever a boolean function f is such that f and f (the complement of f, f = 1 - f) have efficient nondeterministic chips then the known techniques are of no help for proving lower bounds on the complexity of deterministic chips. In this paper we describe a lower bound technique (Thm 1) which only applies to deterministic computations
252 citations
TL;DR: The technique is based on analysing and modifying the Boolean expression prior to the conversion process, originally developed as an aid to fault-tree analysis but it applies to general problems of reliability assessment.
Abstract: A reliability block diagram for complex systems is often analyzed by applying the series/parallel product laws or, where this is not possible, by using a conditional probability result (Bayes theorem). In both cases, the analysis is conducted in the probabilistic domain and, for complex systems, is lengthy. An alternative method is to consider the component reliability parameters to be Boolean variables rather than probabilistic variables and to treat the whole problem as if it were Boolean. This has the advantage of allowing the use of powerful Boolean reduction theorems to contain the size of the problem. Unfortunately, much of this advantage is lost when conversion back into the probabilistic domain takes place. This paper presents a technique for overcoming this disadvantage; the technique is based on analysing and modifying the Boolean expression prior to the conversion process. The technique was originally developed as an aid to fault-tree analysis but it applies to general problems of reliability assessment. I claim no originality for the procedure. The motivation to write the paper is quite simple: the procedure is not as well-known as it should be either amongst practising reliability engineers or amongst those who teach the subject. The purpose of the paper is therefore tutorial.
57 citations
IBM1
TL;DR: New upper bounds on the (unbounded fan-in) circuit complexity of symmetric Boolean functions are proved and several reducibilities and equivalences among problems are given.
Abstract: A complexity theory for unbounded fan-in parallelism is developed where the complexity measure is the simultaneous measure (number of processors, parallel time). Two models of unbounded fan-in parallelism are (1) parallel random access machines that allow simultaneous reading from or writing to the same common memory location, and (2) circuits containing AND's, OR's and NOT's with no bound placed on the fan-in of gates. It is shown that these models can simulate one another with the number of processors preserved to within a polynomial and parallel time preserved to within a constant factor. Reducibilities that preserve the measure in this sense are defined and several reducibilities and equivalences among problems are given. New upper bounds on the (unbounded fan-in) circuit complexity of symmetric Boolean functions are proved.
43 citations
TL;DR: A property of Boolean functions of n variables is described and shown to imply lower bounds as large as $\Omega (n\log n)$ on the number of literals in any Boolean formula for any function with the property.
Abstract: A property of Boolean functions of n variables is described and shown to imply lower bounds as large as $\Omega (n\log n)$ on the number of literals in any Boolean formula for any function with the property. Formulas over the full basis of binary operations $( \wedge , \oplus ,{\text{ etc.}})$ are considered. The lower bounds apply to all but a vanishing fraction of symmetric functions, in particular, to all threshold functions with sufficiently large threshold and to the “congruent to zero modulo k” function for $k > 2$. In the case $k = 4$, the bound is optimal.
32 citations
TL;DR: In this article, it was shown that there is a direct relation between the Reed-Muller expansions and orthogonal expansions for Boolean functions which, from the mathematical point of view, rest upon completely different bases.
Abstract: In this note we will show that there is a direct relation between the Reed-Muller expansions and orthogonal expansions for Boolean functions which, from the mathematical point of view, rest upon completely different bases.
24 citations
TL;DR: A common measure of Boolean function complexity is transformed to the Rademacher/Walsh spectral domain and has an appealing visual interpretation not found in the functional domain.
Abstract: A common measure of Boolean function complexity is transformed to the Rademacher/Walsh spectral domain. The resulting spectral measure has an appealing visual interpretation not found in the functional domain. The relevance of this spectral measure to spectral translation and to the testability of certain classes of combinational networks is examined.
19 citations
01 Jan 1982
18 citations
01 Jan 1982
TL;DR: The conceptual frame work and algorithms for performing implication on MOS networks are presented, and extensions of the algorithm to test generation, and to a first order instead of zero order MOS network are discussed.
Abstract: In this paper we introduce the concept of implication for MOS switch level circuits. The implication performed on these circuits is an extension of the classical implication now applied to Boolean logic networks. Given the ability to perform implication on MOS circuits we can then; generate functional macromodels of MOS circuits, use these macromodels to verify the Boolean function realized by the MOS circuit extracted from the mask set, generate, directly from the MOS circuit sets of tests for nodes stuck-at-1 and stuck-at-0 as well as transistors stuck open and stuck short. We present the conceptual frame work and algorithms for performing implication on MOS networks. We present examples of MOS implication and discuss extensions of the algorithm to test generation, and to a first order instead of zero order MOS network.
15 citations
TL;DR: An algorithm is described for the optimal evaluation of monotonic Boolean functions, in which the memory blocks employed are of minimal size in Shannon's sense.
Abstract: An algorithm is described for the optimal evaluation of monotonic Boolean functions, in which the memory blocks employed are of minimal size in Shannon's sense A version of the algorithm is given, suitable for the case when the results are read out in sequence as the evaluation proceeds
10 citations
TL;DR: This work defines two measures, γ and c, of complexity for Boolean functions, which have a more combinatorial flavor and it is easy to show that both of them are close to 0 for literally all “meaningful” Boolean functions of many variables.
TL;DR: An algorithm devised to learn concept descriptions in this form, where the input is a set of objects which are instances of the concept to be learnt.
TL;DR: In this paper, it is shown that n+1 coefficients only are necessary to define a much wider class of Boolean functions, namely all Boolean functions which can be realized from a threshold-logic core function with pre-and postlinear-translation operations.
Abstract: It is well known that only n+1 spectral coefficients, the Chow or modified-Chow parameters, are necessary to uniquely define any given linearly separable (threshold) function. It is here shown that n+1 coefficients only are necessary to define a much wider class of Boolean functions, namely all Boolean functions which can be realised from a threshold-logic core function with pre- and postlinear-translation operations. The use of n+1 spectral coefficients as a fault signature for all such functions is therefore possible.
TL;DR: In this paper, a cycle of papers is based on the concept of generalized Bolean functions introduced by the author in the first article of the series, where every generalized Boolean function f:B^n->B can be written in a manner similar to the canonical disjunctive form using some function defined on AxB, where A is a finite subset of B containing 0 and 1.
TL;DR: In this article, a new approach to the solution of problems in recognition theory is noted within the framework of the more general problem of the appearance of regularities in the data flow, and of the problem of converting the latter into knowledge.
Abstract: A new approach to the solution of problems in recognition theory is noted within the framework of the more general problem of the appearance of regularities in the data flow, and of the problem of converting the latter into knowledge, is some adequate model of the class of objects under investigation that operationally permit finding the solution of diverse problems of pattern recognition, classification, empirical prediction, filling in the blanks in experimental tables, etc. This approach is developed in application to the case of binary criteria, which permits efficient utilization of the apparatus of Boolean function theory, but allows extension also of the case of multi-valued criteria. A method is proposed for the determination, by a training sample, of the general properties of a single class of real objects representable by appropriate points of the Boolean space m of all criteria. The method relies on a unique a priori hypothesis about the preference of regularities connecting minimal groups of criteria. The expediency of the appearance of sufficiently strong regularities of the type elementary exclusions that yield implicative relations between criteria and the construction of differentiated prediction procedures on their basis, which permit extrapolation of partially assigned properties of objects notmore » in the training sample with a foundation will follow logically from this hypothesis. 12 references.« less
Journal Article•
TL;DR: In this paper, the development of an algorithm for minimising Boolean polynomials is proposed, which operates in two phases: a rapid search is conducted for ways to reduce the polynomial to a smaller form.
Abstract: The development of an algorithm for minimising Boolean polynomials is proposed. The algorithm operates in two phases. In the first phase a rapid search is conducted for ways to reduce the polynomial to a smaller form. In the second phase a deep search is conducted to further reduce the polynomial and eventually obtain the minimal form.
01 Sep 1982
TL;DR: In this paper, the power of negation in combinatorial complexity theory has been investigated and it has been shown that negation can be exponentially powerful in the more restricted setting of projections among families of Boolean functions.
Abstract: The power of negation in combinatorial complexity theory has for a long time been an intriguing question. In this paper we demonstrate that in the more restricted setting of projections among families of Boolean functions, negation can be exponentially powerful.
TL;DR: In this article, it was shown that the syndrome of a Boolean function F (for diagnostics) can be nicely modeled to be F in a special pseudo-Boolean algebra at that point in state space where all variables are equal to 1 2 or as the probability of F being 1 in case all its variables appear with probability 1 2.