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Boolean function

About: Boolean function is a research topic. Over the lifetime, 10089 publications have been published within this topic receiving 201604 citations. The topic is also known as: Boolean operation.


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Journal ArticleDOI
TL;DR: It is shown that a family of surfaces having d degrees of freedom has a natural separating capacity of 2d pattern vectors, thus extending and unifying results of Winder and others on the pattern-separating capacity of hyperplanes.
Abstract: This paper develops the separating capacities of families of nonlinear decision surfaces by a direct application of a theorem in classical combinatorial geometry. It is shown that a family of surfaces having d degrees of freedom has a natural separating capacity of 2d pattern vectors, thus extending and unifying results of Winder and others on the pattern-separating capacity of hyperplanes. Applying these ideas to the vertices of a binary n-cube yields bounds on the number of spherically, quadratically, and, in general, nonlinearly separable Boolean functions of n variables. It is shown that the set of all surfaces which separate a dichotomy of an infinite, random, separable set of pattern vectors can be characterized, on the average, by a subset of only 2d extreme pattern vectors. In addition, the problem of generalizing the classifications on a labeled set of pattern points to the classification of a new point is defined, and it is found that the probability of ambiguous generalization is large unless the number of training patterns exceeds the capacity of the set of separating surfaces.

1,995 citations

Journal ArticleDOI
Akers1
TL;DR: This paper describes a method for defining, analyzing, testing, and implementing large digital functions by means of a binary decision diagram that provides a complete, concise, "implementation-free" description of the digital functions involved.
Abstract: This paper describes a method for defining, analyzing, testing, and implementing large digital functions by means of a binary decision diagram. This diagram provides a complete, concise, "implementation-free" description of the digital functions involved. Methods are described for deriving these diagrams and examples are given for a number of basic combinational and sequential devices. Techniques are then outlined for using the diagrams to analyze the functions involved, for test generation, and for obtaining various implementations. It is shown that the diagrams are especially suited for processing by a computer. Finally, methods are described for introducing inversion and for directly "interconnecting" diagrams to define still larger functions. An example of the carry look-ahead adder is included.

1,813 citations

Proceedings ArticleDOI
24 Jun 1990
TL;DR: A package for manipulating Boolean functions based on the reduced, ordered, binary decision diagram (ROBDD) representation is described, based on an efficient implementation of the if-then-else (ITE) operator.
Abstract: Efficient manipulation of Boolean functions is an important component of many computer-aided design tasks This paper describes a package for manipulating Boolean functions based on the reduced, ordered, binary decision diagram (ROBDD) representation The package is based on an efficient implementation of the if-then-else (ITE) operator A hash table is used to maintain a strong canonical form in the ROBDD, and memory use is improved by merging the hash table and the ROBDD into a hybrid data structure A memory function for the recursive ITE algorithm is implemented using a hash-based cache to decrease memory use Memory function efficiency is improved by using rules that detect when equivalent functions are computed The usefulness of the package is enhanced by an automatic and low-cost scheme for recycling memory Experimental results are given to demonstrate why various implementation trade-offs were made These results indicate that the package described here is significantly faster and more memory-efficient than other ROBDD implementations described in the literature

1,252 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the theory of Boolean rings is mathematically equivalent to the topological theory of locally-bicompact totally-disconnected topological spaces.
Abstract: In an earlier paperf we have developed an abstract theory of Boolean algebras and their representations by algebras of classes. We now relate this theory to the study of general topology. The first part of our discussion is devoted to showing that the theory of Boolean rings is mathematically equivalent to the theory of locally-bicompact totally-disconnected topological spaces. In R we have already prepared the way for a topological treatment of the perfect representation of an arbitrary Boolean ring. Continuing in this way, we find that the perfect representation is converted by the introduction of a suitable topology into a space of the indicated type. We have no difficulty in inverting this result, proving that every locally-bicompact totally-disconnected topological space arises by the same procedure from a suitable Boolean ring.' It is thus convenient to call the spaces corresponding in this manner to Boolean rings, Boolean spaces. The algebraic properties of Boolean rings can, of course, be correlated in detail with the topological properties of the corresponding Boolean spaces. A simple instance of the correlation is the theorem that the Boolean rings with unit are characterized as those for which the corresponding Boolean spaces are bicompact. A familiar example of a bicompact Boolean space is the Cantor discontinuum or ternary set, which we discuss at the close of Chapter I. Having established this direct connection between Boolean rings and topology, we proceed in the second part of the discussion to considerations of a yet more general nature. We propose the problem of representing an arbitrary TVspace by means of maps in bicompact Boolean spaces. Our solution of this problem embodies an explicit construction of such maps, which we shall now describe briefly. In a given TVspace dt, the open sets and the nowhere dense sets generate a Boolean ring, with 9Î as unit, which characterizes the topological structure of 9Î. Those subrings which contain 9Î and which are so large that the interiors of their member sets constitute bases for 9î, also char-

1,166 citations

Journal ArticleDOI
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.
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

1,103 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023177
2022351
2021275
2020351
2019369
2018342