The compatibility relation occurs in many different disciplines in science and engineering. When a compatibility relation exists between pairs of elements in a set, an important problem is to derive the collection of aU those elements that form maximal compatibles. If the set of elements with the compatibility relation can be visualized as a compatibility graph of which the different nodes represent the elements of the set, the only edges of the graph being the nonoriented lines joining pairs of elements with the compatibility relation, then the problem of deriving the maximal compatibles becomes identical to the graph theory problem of finding all the maximal complete subgraphs in a symmetric graph. Recently, in connection with simplifying incompletely specified sequential machines, where a kind of compatibility relation also exists between pairs of internal states, Das and Sheng proposed a method for deriving the different maximal compatibles through finding all of the modified cut-sets of the incompatibility graph of the machine. This paper, without confining itself to only incompletely specified machines, considers the problem involving the compatibility relation in a broader perspective and suggests a new approach for finding aU the modified cut-sets of the incompatibility graph of a set having a compatibility relation between its different pairs of elements.

On a New Approach for Finding All the Modified Cut-Sets in an Incompatibility Graph

This note describes an iterative procedure for generating the prime implicants of switching functions by utilizing a new tabular mode of functional representation called clause-column table. The procedure generates all the prime implicants and can be applied equally well to functions given in the sum-of-products or in the product-of-sums froms, both canonical and noncanonical. The procedure can also be readily adapted to determine the prime implicants of functions having a large number of unspecified or DON'T CARE terms.

Clause-Column Table Approach for Generating All the Prime Implicants of Switching Functions

There exist a number of methods for finding minimal or nonminimal irredundant solutions of switching functions. Recently in a paper by Marcovitz and Shub, an improved algorithm for simplifying switching functions using a Karnaugh map is suggested. The present study extends the basic idea of the authors in finding minimal and nonminimal irredundant solutions by using the cover table representation of switching functions instead of the map.

An Approach for Simplifying Switching Functions by Utilizing the Cover Table Representation

Proceedings of an International Symposium on the Theory of Switching. 2 volumes. Pp. 650. 120s. 1959. (Harvard University Press)

The synthesis of space-efficient support hardware for built-in self-testing (BIST) is of critical importance in the design and manufacture of VLSI circuits, and a number of efficient algorithms have been proposed. The paper reports new techniques that facilitate designing such space-efficient BIST support circuits using knowledge of compact test sets, with the target objective of minimizing the storage requirements for the circuit under test (CUT), while retaining the fault coverage information. The suggested techniques take advantage of some well-known concepts of conventional switching theory, particularly those of cover table and frequency ordering, as commonly utilized in the minimization of switching functions, in conjunction with a new measure of failure probability in case of stochastic dependence of line errors, besides knowledge of Hamming distance, sequence weights, and derived sequences as previously used by the authors in sequence characterization, in the selection of specific logic gates for merger of an arbitrary number of output bit streams from the CUT. The outputs coming out of the space compactor may eventually be fed into a time compactor to derive the CUT signatures. The techniques give good design with high fault coverage for single stuck-line faults, with low CPU simulation time, and acceptable area overhead. Design algorithms are provided, and the simplicity and ease of their implementations are demonstrated. In particular, the paper gives results on extensive simulation runs on the ISCAS 85 combinational benchmark circuits with ATALANTA, FSIM, and COMPACTEST programs.

Data compression in space under generalized mergeability based on concepts of cover table and frequency ordering

The central idea developed in the present paper involves the decomposition of the prime irnplicant covering problems of switching functions into the number of readily tractable sub-problems. It is shown that the minimizing function (Boolean representation of the prime implicant table) of a switching function can suitably be split up into a number of sub-functions such that the sum terms of each of these sub-functions can be arranged in any of the four possible distinct matrices called connected cover term matrices. The irredundant solutions of the sub-problem corresponding to each of the connected cover term matrices can be readily obtained, without the generation of a single redundant or duplicate solution, by following a systematic procedure suggested in the paper. The paper is concerned with the detailed study of the properties of the connected cover term matrices.

Some Studies on Connected Cover Term Matrices of Switching Functions

A method of direct determination of all the minimal prime implicant covers of switching functions has been presented in the paper. It has been shown that some of the difficulties encountered in finding directly all the minimal prime implicant covers of the function for which the columns of the cover table cannot be arranged in a single connected cover term matrix or in a number of connected cover term matrices with mutually disjoint sots of prime implicants can be overcome by first dividing the cover table into a number of sub-tables such that the columns of one of the sub-tables can be arranged to form a connected cover term matrix by ignoring the presence of some of the prime implicants from some of its columns. Next by associating the different irredundant covers of the other sub-tables with this connected cover term matrix, all the minimal prime implicant covers of the function can be found out.

Direct determination of all the minimal prime implicant covers of switching functions

A method for testing and realization of threshold functions through classification of inequalities is suggested in the present paper. The standard procedure of testing and realization of threshold functions consists in solving a system of linear inequalities in which the unknowns are the different weights to be assigned to the variables of the functions. In the paper it is first shown that when the different inequalities involving the weights of the variables are represented in terms of their subscripts only, thon, depending on the number and on the sum of the subscripts appearing on either side, the entire set of inequalities of any function can be classified into nine distinct types. The sot of inequalities is next expressed in terms of the least weight and the other different incremental weights, the knowledge of which along with that of the types of inequalities, furnishes information regarding I-realizability of the function and on the assignments of values to the different weights for integ...

A Method for Testing and Realization of Threshold Functions through Classification of Inequalities

A method of simplification of switching functions involving a very large number of ‘ don't care’ states is suggested in the present paper. First a tabular technique is suggested which generates all the prime implicants starting from the maxterm type expressions of switching functions, avoiding generation of the prime implicants formed of ‘don't care’ states only. The technique presented is simple and iterative. Next it is suggested how the knowledge of the sets of prime implicants thus obtained can be utilized for finding minimal or other irredundant sums of switching functions.

Simplification of Switching Functions Involving a very Large Number of ‘Don't Care’ States†

A method for finding out one of the minimal forms of a switching function is presented in this paper. The given switching function is decomposed into a number of component functions of smaller number of variables by the application of the expansion theorem. Theso component functions are plotted on a chart. From a knowledge of tho weight of each term, prime implicants and the essential prime implicants of the component functions, the essentia! prime implicants for the coverage of the originally given function can be readily determined with the help of the chart. The chart also makes it possible to select oasily those prime implicants which are suro to occur in one of the minimal covers. Most economic coverage for the terms still remaining uncovered is next found out by a systematic procedure. Tho method is applicable to tho cases of functions incorporating a certain number of ‘ don't care ’ states.

Sunil Ranjan Das

Papers

Some Studies on Connected Cover Term Matrices of Switching Functions

Direct determination of all the minimal prime implicant covers of switching functions

A Method for Testing and Realization of Threshold Functions through Classification of Inequalities

Simplification of Switching Functions Involving a very Large Number of ‘Don't Care’ States†

A Chart Method for the Determination of one of the Minimal Forms of a Switching Function