An Algorithm for Solving Boolean Equations
About: This article is published in IEEE Transactions on Electronic Computers.The article was published on 1963-10-01. It has received 24 citations till now. The article focuses on the topics: Boolean expression & Product term.
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TL;DR: It is demonstrated that a multiple-output incompletely specified switching function is reaeized if a ≤ relation is satisfied between the corresponding charchteristic functions, which leads to a new unified outlook on functional decomposition as used in modular synthesis procedures.
Abstract: A methodology based on the theory of Boolean equations has been developed which permits a unified approach to the analysis and synthesis of combinational logic circuits. The type of circuits covered by the approach includes both the classical loopless combinational networks as well as those that contain closed feedback loops and thus have internally a sequential character. To that end, a general multiple-output circuit represented by a Mealy-type machine is studied using characteristic equations (functions) that describe its internal structure. It is shown how behavioral properties of the circuit are reflected through the sosutions of these equations. Moreover, it is demonstrated that a multiple-output incompletely specified switching function is reaeized if a ≤ relation is satisfied between the corresponding charchteristic functions. This leads to a new unified outlook on functional decomposition as used in modular synthesis procedures. Although the building modules are allowed to be sequential circuits, it is shown under which conditions the feedback loops are redundant with respect to the realization of a given output characteristic function, and thus the existence conditions of nondegenerate combinational circuits with loops are stated.
70 citations
TL;DR: An efficient tabular method is presented to solve switching equations based on the use of a Boolean algebra, and the compactness and simplicity of the method are outstanding, and it is straightforward to implement for computer processing.
Abstract: Several problems in switching theory such as automatic test pattern generation, can be exactly and elegantly investigated by using switching equations. An efficient tabular method is presented to solve these switching equations. The solutions of a given equation are compacted into a table, and a Boolean algebra is defined for these tables. The proposed procedure is based on the use of this Boolean algebra. The compactness and simplicity of the method are outstanding, and it is straightforward to implement for computer processing. The complexity of the procedure is computed, and some experimental results for a set of benchmark equations are provided in order to point out the effectiveness of this method. >
21 citations
TL;DR: This paper describes a method for constructing reduced solutions, i.e., general solutions involving the fewest parameters possible for some equations, using fewer than n parameters.
Abstract: The family of solutions for a Boolean equation is commonly represented in a single formula involving arbitrary Boolean parameters. It is well known that n parameters suffice to construct a general solution for an equation in n unknowns. For some equations, however, a general solution may be constructed using fewer than n parameters. This paper describes a method for constructing reduced solutions, i.e., general solutions involving the fewest parameters possible.
20 citations
TL;DR: An on-line testing procedure for constructing a test set for identifying a specific fault in a circuit to within an equivalence class is outlined, which eliminates the need for precalculating a fault dictionary.
Abstract: This paper deals with the problem of identifying multiple stuck-type hardware failures in combinational switching networks. Our work is an extension of that of Poage, and Bossen and Hong, and we employ the cause-effect equation for representing faulty circuit behavior. We introduce the concept of solving simultaneous equations over check point variables. These check point solutions are studied in detail. From the solutions one can calculate the function realized by a faulty circuit. We outline an on-line testing procedure for constructing a test set for identifying a specific fault in a circuit to within an equivalence class. This procedure eliminates the need for precalculating a fault dictionary, which, in many instances, can be quite advantageous. We also outline how to apply these techniques to the following problems: 1) identifying redundancy; 2) determining the set of faults not detected by an arbitrary test set; and 3) constructing a complete fault dictionary.
19 citations
Cites background from "An Algorithm for Solving Boolean Eq..."
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TL;DR: In this paper some aspects of the relation between the state of the art and the way of teaching in the field of switching theory are described.
Abstract: In this paper some aspects of the relation between the state of the art and the way of teaching in the field of switching theory are described. Some critical comments and suggestions concerning the contemporary way of teaching switching theory are presented.
10 citations
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15 citations
01 Mar 1962
TL;DR: In this article, a Veitch chart-based approach is presented to solve a special type of Boolean functional equation, in which the arguments Ai's and Xi's are binary variables and the a1's are implicit functions of the Ai's only.
Abstract: Typically, simultaneous Boolean equations are expressed by a set of relations Fi = fi, i = 1,2 ?, where the Fi and fi are switching functions of N binary variables. In many logical design problems, a special type of Boolean equation is often encountered; namely, F(A0, A1 ?, An?1, X0, X1 ?, Xm?1) = f(a1, a2 ?, ak, X0, X1 ?, Xm?1), in which the arguments Ai's and Xi's are binary variables and the a1's are implicit functions of the Ai's only. This paper presents a somewhat new approach to solving such special types of Boolean functional equations. The solution function (see reference 1) can be formulated and mapped on a Veitch chart, especially tailored to the present problem, and all possible sets of solutions of the implicit functions ai's which satisfy the original functional equation are obtained simultaneously. Other possible applications and extensions of this method are discussed also.
4 citations
01 Jan 1960
TL;DR: A new method for reducing logical problems of higher complexity based on matrix logic without iterative simplifications is presented, resulting in savings in design effort and hardware for the eventual circuit.
Abstract: A new method for reducing logical problems of higher complexity based on matrix logic is presented. Sets of propositions related to a set of constraints are transformed into minimum form without iterative simplifications. The savings in design effort and hardware for the eventual circuit are demonstrated in a reference problem.
4 citations
2 citations