Algebraic Properties of Symmetric and Partially Symmetric Boolean Functions
TL;DR: A canonical form is derived for ?
Abstract: Symmetric and partially symmetric functions are studied from an algebraic point of view. Tests are given for detecting these properties. A more general approach involving the concept of ?-symmetric functions is given. A canonical form is derived for ?-symmetric functions which leads to synthesis procedures that improve results of Shannon.
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TL;DR: This procedure of detecting invariance is directly applied for the identification of total or partial symmetry of a switching function whose variables of symmetry may be either all unprimed (or all primed), mixed, or of multiform nature.
Abstract: This note presents a method for identifying total or partial symmetry of switching functions based on the application of the principle of residue test by numerical methods. The invariance of a switching function under a single interchange of two variables can be readily detected from the equality of some of the residues of expansion about these two variables. This procedure of detecting invariance is directly applied for the identification of total or partial symmetry of a switching function whose variables of symmetry may be either all unprimed (or all primed), mixed, or of multiform nature.
36 citations
TL;DR: The number of realizable functions is greatly increased when the input restriction is relaxed, and it is shown that the fraction of n-variable functions that are realizable becomes vanishingly small as n grows large.
Abstract: Cascades of two-input, one-output general function cells have been studied previously with the restriction that no external variable may drive more than one cell. Upon relaxing this restriction, it is shown that a variable that drives more than one cell in a cascade need never drive more than one cell that is not an EXCLUSIVE-OR cell, and that cell must be the first cell driven by the variable in the cascade. From this it follows that there exists a canonical form for cellular cascades with repeated inputs. Furthermore, the length of a cascade required to produce an n-variable function is bounded by (n2+n-4)/2, and there are functions that meet this bound for all n. Derived from knowledge of the canonical form, a realizability and synthesis algorithm is described that is an extension of previously described algorithms. The algorithm has been used to test the 402 Harvard functions, and a table of minimal-length cascades of realizable functions is included in the paper. Although the number of realizable functions is greatly increased when the input restriction is relaxed, it is shown that the fraction of n-variable functions that are realizable becomes vanishingly small as n grows large. In particular, for example, of the 2n+1 symmetric functions of n-variables, precisely 12 are realizable for all n?3.
33 citations
TL;DR: New algorithms for generating a regular two-dimensional layout representation for multi-output, incompletely specified Boolean functions, called, Pseudo-Symmetric Binary Decision Diagrams (PSBDDs), are presented and show that symmetrization of reallife benchmark functions can be done efficiently.
Abstract: New algorithms for generating a regular two-dimensional layout representation for multi-output, incompletely specified Boolean functions, called, Pseudo-Symmetric Binary Decision Diagrams (PSBDDs), are presented. The regular structure of the function representation allows accurate prediction of post-layout areas and delays before the layout is physically generated. It simplifies power estimation on the gate level and allows for more accurate power optimization. The theoretical background of the new diagrams, which are based on ideas from contact networks, and the form of decision diagrams for symmetric functions is discussed. PSBDDs are especially well suited for deep sub-micron technologies where the delay of interconnections limits the device performance. Our experimental results are very good and show that symmetrization of reallife benchmark functions can be done efficiently.
21 citations
Cites background or methods from "Algebraic Properties of Symmetric a..."
...The above theorem was presented for the first time by Arnold and Harrison [15] for a total of 2 variables for the n-variable function....
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...THEOREM Every Boolean function can be made totally symmetric (symmetricized) by repeating some of its variables [15]....
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IBM1
TL;DR: It is important to point out that only the symmetric switching function devices offer rewrite-ability to eliminate the part number problem, and accommodation for a large number of inputs to ease interconnection and delay equalization problems.
Abstract: Although the literature on the bubble logic devices is limited, the concepts and device configurations are diverse. In conductor-access devices, logic can be performed by bubble transfer operations. In field-access devices, logic can be performed by providing alternative paths which are selected by interaction between bubbles. Examples include the conjugate logic gates, the resident-bubble cellular logic, and the chevron 3-3 circuits. Logic can also be performed by counting bubbles, such as in the symmetric switching function implementation. The various mechanisms for implementing bubble logic are all described by truth tables. To assess their efficiency, they are compared in terms of space and delay when they are used to implement the same logic element - a full adder. They are all comparable except for the resident-bubble cellular logic which requires excessive space and delay. However, it is important to point out that only the symmetric switching function devices offer rewrite-ability to eliminate the part number problem, and accommodation for a large number of inputs to ease interconnection and delay equalization problems.
21 citations
01 Apr 2019
TL;DR: The symbolic reliability analysis of a commodity-supply system is completed successfully herein, yielding results that have been checked symbolically, and also were shown to agree numerically with those obtained earlier.
Abstract: Symmetric switching functions (SSFs) play a prominent role in the reliability analysis of a binary k-out-of-n: G system, which is a dichotomous system that is successful if and only if at least k out of its n components are successful. The aim of this paper is to extend the utility of SSFs to the reliability analysis of a multi-state k-out-of-n: G system, which is a multi-state system whose multi-valued success is greater than or equal to a certain value j (lying between 1 (the lowest output level) and M (the highest output level)) whenever at least km components are in state m or above for all m such that 1 ≤ m ≤ j. This paper is devoted to the analysis of non-repairable multi-state k-out-of-n: G systems with independent non-identical components. The paper utilizes algebraic techniques of multiple-valued logic (together with known properties of SSFs) to evaluate each of the multiple levels of the system output as an individual binary or propositional function of the system multi-valued inputs. The formula of each of these levels is then written as a probability–ready expression, thereby allowing its immediate conversion, on a one-to-one basis, into a probability or expected value. The symbolic reliability analysis of a commodity-supply system (which serves as a standard gold example of a multi-state k-out-of-n: G system) is completed successfully herein, yielding results that have been checked symbolically, and also were shown to agree numerically with those obtained earlier.
21 citations
References
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TL;DR: It will be shown that several of the well-known theorems on impedance networks have roughly analogous theorem in relay circuits, including the delta-wye and star-mesh transformations, and the duality theorem.
Abstract: In the control and protective circuits of complex electrical systems it is frequently necessary to make intricate interconnections of relay contacts and switches Examples of these circuits occur in automatic telephone exchanges, industrial motor-control equipment, and in almost any circuits designed to perform complex operations automatically In this article a mathematical analysis of certain of the properties of such networks will be made Particular attention will be given to the problem of network synthesis Given certain characteristics, it is required to find a circuit incorporating these characteristics The solution of this type of problem is not unique and methods of finding those particular circuits requiring the least number of relay contacts and switch blades will be studied Methods will also be described for finding any number of circuits equivalent to a given circuit in all operating characteristics It will be shown that several of the well-known theorems on impedance networks have roughly analogous theorems in relay circuits Notable among these are the delta-wye (δ-Y) and star-mesh transformations, and the duality theorem
922 citations
TL;DR: A basic part of the general synthesis problem is the design of a two-terminal network with given operating characteristics, and this work shall consider some aspects of this problem.
Abstract: THE theory of switching circuits may be divided into two major divisions, analysis and synthesis. The problem of analysis, determining the manner of operation of a given switching circuit, is comparatively simple. The inverse problem of finding a circuit satisfying certain given operating conditions, and in particular the best circuit is, in general, more difficult and more important from the practical standpoint. A basic part of the general synthesis problem is the design of a two-terminal network with given operating characteristics, and we shall consider some aspects of this problem.
774 citations