Showing papers on "Boolean function published in 1976"
TL;DR: A non-constructive argument (not based on the construction of an “oracle”) settles this question for d a prime power: at least v216 entries of the adjacency matrix of a v-vertex undirected graph G must be examined in the worst case to determine if G has any given non-trivial monotone graph property.
192 citations
03 May 1976
TL;DR: It is shown that there is a combinational (acyclic) Boolean circuit of complexity 0(slog s), that can be made to compute any Boolean function of complexity s by setting its specially designated set of control inputs to appropriate fixed values.
Abstract: We show that there is a combinational (acyclic) Boolean circuit of complexity 0(slog s), that can be made to compute any Boolean function of complexity s by setting its specially designated set of control inputs to appropriate fixed values. We investigate the construction of such “universal circuits” further so as to exhibit directions in which refinements of the asymptotic multiplicative constant factor in the complexity bound can be found. In this pursuit useful detailed guidance is provided by available lower bound arguments. In the final section we discuss some other problems in computational complexity that can be related directly to the graph-theoretic ideas behind our constructions. For motivation we start by illustrating some of the applications of universal circuits themselves.
174 citations
TL;DR: An exact and an approximate method for calculating the probability of occurrence of the output event from priority-AND (sequential) failure logic is given and can be used during fault-tree analysis without modification to existing quantitative evaluation techniques.
Abstract: An exact and an approximate method for calculating the probability of occurrence of the output event from priority-AND (sequential) failure logic is given. The approximate method can be used during fault-tree analysis without modification to existing quantitative evaluation techniques. Assumptions made include s-independent, exponentially distributed, non-repairable basic events as input to the priority-AND failure logic.
87 citations
IBM1
TL;DR: A property of the truth table of a symmetric Boolean function is given from which one can infer a lower bound on the minimal number of 2-ary Boolean operations that are necessary to compute the function.
Abstract: A property of the truth table of a symmetric Boolean function is given from which one can infer a lower bound on the minimal number of 2-ary Boolean operations that are necessary to compute the function. For certain functions ofn arguments, lower bounds between roughly 2n and 5n/2 can be obtained. In particular, for eachm ≥ 3, a lower bound of 5n/2 −O(1) is established for the function ofn arguments that assumes the value 1 iff the number of arguments equal to 1 is a multiple ofm. Fixingm = 4, this lower bound is the best possible to within an additive constant.
57 citations
IBM1
TL;DR: The complexity of approximately realizing a partially specified Boolean function, in whose table a fractiond of the entries are unspecified and a fractionp of the specified entries are l's with errors allowed in a fraction not more thane of thespecified entries, is less by the factor (1 −d) [H(p) − H(e]].
Abstract: This paper explores the connections between two areas pioneered by Shannon: the transmission of information with a fidelity criterion, and the realization of Boolean functions by networks and formulae. We study three phenomena:
1.
The effect of the relative number of O's and l's in a function's table on its complexity.
2.
The effect of the number of unspecified entries in a partially specified function's table on its complexity.
3.
The effect of the number of errors allowed in the realization of a function on its complexity.
55 citations
TL;DR: This class of examples shows that an algorithm given in [STAN-CS-75-512] for computing functions defined on paths in trees is optimum to within a constant factor.
Abstract: Let $F_1$, $F_2$,..., $F_m$ be a set of Boolean functions of the form $F_i$ = $\wedge$ {x$\in X_i$}, where $\wedge$ denotes conjunction and each $X_i$ is a subset of a set X of n Boolean variables. We study the size of monotone Boolean networks for computing such sets of functions. We exhibit anomalous sets of conujunctions whose smallest monotone networks contain disjunctions. We show that if |$F_i$| is sufficiently small for all i, such anomalies cannot happen. We exhibit sets of m conjunctions in n unknowns which require $c_2$m$\alpha$(m,n) binary conjunctions, where $\alpha$(m,n) is a very slowly growing function related to a functional inverse of Ackermann''s function. This class of examples shows that an algorithm given in [STAN-CS-75-512] for computing functions defined on paths in trees is optimum to within a constant factor.
34 citations
TL;DR: A Boolean expression wilth n literals, i.e., n distinct appearances of variables, can be evaluated by a parallel processing system in at most 1.81 log2n steps, or, equivalently, by a network constructed with two-input AND and OR gates.
Abstract: A Boolean expression wilth n literals, i.e., n distinct appearances of variables, can be evaluated by a parallel processing system in at most 1.81 log2n steps, or, equivalently, by a network constructed with two-input AND and OR gates and having at most 1.81 log2n levels.
26 citations
06 Sep 1976
TL;DR: The paper attempts to present descriptional complexity as an identifiable part of the theory of complexity incorporating many diverse areas of research and discusses relation between descriptional and computational complexity.
Abstract: The paper attempts (i) to present descriptional complexity as an identifiable part of the theory of complexity incorporating many diverse areas of research, (ii) to formulate basic problems and to survey some results (especially those concerning languages) in descriptional complexity, (iii) to discuss relation between descriptional and computational complexity.
23 citations
TL;DR: This work relates the implementation of several Boolean functions using a threshold gate and XOR gates to a previous paper by Kaplan and Winder, which derived the result, and to a mathematical study of Lawson, which yields a synthesis procedure with very general applicability.
Abstract: In the above paper1Edwards shows the implementation of several Boolean functions using a threshold gate and XOR gates. An unstated, but implicit, result is that any given Boolean function can be synthesized in this way. This result is of some historical interest. We wish to relate it to a previous paper by Kaplan and Winder [1], which in effect derived the result, and then to a mathematical study of Lawson [2], [3], which yields a synthesis procedure with very general applicability.
22 citations
TL;DR: It is proved that there exist many structurally different optimal computations for Eq (n) and the smallest number of arbitrary binary Boolean operations that are used in any Boolean computation for F is L(Eq(n) = 2n −3, L (and(n), nor(n))=2n−2.
21 citations
01 Aug 1976
TL;DR: The three complexities of circuit size, formula size and depth are considered, and some of the principal results concerning their relationships and estimations are presented, with outlined proofs for some of ths simpler theorems.
Abstract: : The complexity of a finite Boolean function may be defined with respect to its computation by networks of logical elements in a variety of ways. The three complexities of circuit size, formula size and depth are considered, and some of the principal results concerning their relationships and estimations are presented, with outlined proofs for some of ths simpler theorems. This survey is restricted to networks in which all two-argument logical functions may be used.
TL;DR: This paper considers the reduction in algorithmic complexity that can be achieved by permitting approximate answers to computational problems, and shows that partial sorting of N items, insisting on matching any nonzero fraction of the terms with their correct successors, requires O (N \log N) comparisons.
Abstract: This paper considers the reduction in algorithmic complexity that can be achieved by permitting approximate answers to computational problems. It is shown that Shannon's rate-distortion function could, under quite general conditions, provide lower bounds on the mean complexity of inexact computations. As practical examples of this approach, we show that partial sorting of N items, insisting on matching any nonzero fraction of the terms with their correct successors, requires O (N \log N) comparisons. On the other hand, partial sorting in linear time is feasible (and necessary) if one permits any finite fraction of pairs to remain out of order. It is also shown that any error tolerance below 50 percent can neither reduce the state complexity of binary N -sequences from the zero-error value of O(N) nor reduce the combinational complexity of N -variable Boolean functions from the zero-error level of O(2^{N}/N) .
TL;DR: It is shown that for large n
Abstract: A solution to the problem of counting the number of fanout-free Boolean functions of n variables is presented. The relevant properties of fanout-free functions and circuits are summarized. The AND and OR ranks of a fanout-free function are defined. Recursive formulas for determining the number of distinct functions of specified rank are derived. Based on these, expressions are obtained for
01 Sep 1976
TL;DR: In this paper, the problem of multiple-fault analysis through the use of the Boolean difference was considered and suitable expressions were derived that give the set of test-input codes for the detection of all possible multiple faults of combinational circuits.
Abstract: The Boolean difference has proved to be an elegant mathematical concept in the study of single faults of a stuck-at nature in combinational logic circuits. Recently Ku and Masson have extended this tool of analysis to cover all possible multiple-fault situations of logic circuits as well. In this letter, we have considered the problem of multiple-fault analysis through the use of the Boolean difference and derived suitable expressions that give the set of test-input codes for the detection of all possible multiple faults of combinational circuits. The developed expressions are compact and simpler, in general, and require much less computation to mire at the test vectors, particularly in cases where the multiple faults of interest are specified. These expressions, like those of Ku and Masson, are useful when only k simultaneous faults or all faults up to and including k faults are to be considered.
TL;DR: A more general arrangement—a tree—structured arrangement of the trees is considered and it is shown to give no improvement in the problem of finding an optimal sequential order of scanning the trees for evaluation of F.
01 Jan 1976
TL;DR: This report contains the information needed to use the WAM-BAM computer codes, developed to evaluate probabilistically, systems modeled with complex Boolean algebra.
Abstract: This report contains the information needed to use the WAM-BAM computer codes. These codes were developed to evaluate probabilistically, systems modeled with complex Boolean algebra. A common example is a fault tree, whose gates, with WAM-BAM, can be any Boolean function (i.e., AND, OR, NAND, NOR, NOT). The current version operates on the CDC 6600 and CDC 7600 computers and requires 170000(base 8) words of storage. (GRA)
TL;DR: A theory of decomposition of Boolean functions having the following properties is presented: ease of understanding and application without previous decomposition theory background, flexibility of application to meet varying logic design criteria, and suitability of both numerical calculation and chart techniques.
Abstract: The object of this paper is the presentation of a theory of decomposition of Boolean functions having the following properties: 1) ease of understanding and application without previous decomposition theory background; 2) flexibility of application to meet varying logic design criteria; 3) suitability of both numerical calculation and chart techniques; and 4) reduction in the number and complexity of calculations required to achieve decomposition.
03 May 1976
TL;DR: The complexity of realizing a monotone but otherwise arbitrary Boolean function is studied by means of networks and formulae and it is shown that a more economical realization may be possible if basis functions that are not themselvesmonotone are used.
Abstract: In this paper we study the complexity of realizing a monotone but otherwise arbitrary Boolean function. We consider realizations by means of networks and formulae. In both cases the possibility exists that although a monotone function can always be realized in terms of monotone basis functions, a more economical realization may be possible if basis functions that are not themselves monotone are used. Thus, we have four cases, namely:1. The cost of realizing a monotone function with a network over a universal basis.2. The cost of realizing a monotone function with a network over a monotone basis.3. The cost of realizing a monotone function with a formula over a universal basis.4. The cost of realizing a monotone function with a formula over a monotone basis.For the first case, we obtain a complete solution to the problem. For the other three cases, we obtain improvements over previous results and come within a logarithmic factor or two of a complete solution.
TL;DR: A square root of an isotone Boolean function f with respect to a variable xi was defined by Reischer and Simovici as a Boolean function s such that holds identically.
Abstract: A square root of an isotone Boolean function f with respect to a variable xi was defined by Reischer and Simovici [10] as a Boolean function s such that holds identically. More generally, given a partition (T,Y,Z) of the set X = (x1,···,xn) of variables, we may be interested in finding a functional decomposition of the form
TL;DR: This note presents an alternate approach to solving the inverse problem of logic by applying a decomposition technique based on Boolean equations.
Abstract: This note presents an alternate approach to solving the inverse problem of logic by applying a decomposition technique based on Boolean equations.
TL;DR: A lattice difference operator is defined for discrete functions which, together with the concept of extended vector, allows to derive new attractive algorithms for obtaining the prime blocks and antiblocks of a discrete function.
TL;DR: A simple refinement of Morreale's method yields a useful criterion to identify and cancel the redundant subfunctions generated in the application of the Pi-algorithm.
Abstract: A simple refinement of Morreale's method yields a useful criterion to identify and cancel the redundant subfunctions generated in the application of the Pi-algorithm. Use of such a criterion improves the efficiency of the algorithm. Extension to multiterminal Boolean functions is also presented.
TL;DR: A new normal form of Boolean functions based on the sum, product and negation is presented and a transformation of Galois normal form into the present normal form is shown.