Sunil Ranjan Das

On Control Memory Minimization in Microprogrammed Digital Computers

The problem of minimizing the bit dimension of control memories in microprogrammed digital computers is considered in this paper. We start essentially with the same basic formulation as that of Grasselli and Montanari [2]. However, in order to minimize the computational requirements, we start directly with the set of maximum compatibility classes of microcommands whose number is usually small, and readily obtain near-minimal irredundant solutions. A minimal solution is then obtained from the irredundant solutions.

Some Studies on Connected Cover Term Matrices of Switching Functions

ABSTRACT 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.

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

ABSTRACT 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...

Direct Determination of all the Minimal Prime Implicant Covers of Switching Functions

Abstract 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.

Multiform Partial Symmetry and Linearity

It is known from a paper by Das that multiform total symmetry is equivalent to linearity. In this correspondence the results of Das are generalized and a simple proof is given to show that even for multiform partial symmetries the corresponding subfunctions are linear subfunctions.

Minimal Third-order Expressions of Boolean Unate Functions

ABSTRACT A simple and straightforward procedure for finding absolute minimal third-order expressions (in the ‘ sum-of-product-of-sum’ forms) of a special class of Boolean functions called unate functions is suggested in the paper. The central idea developed through the procedure involves a decomposition of the assigned Boolean function first into a group of sub-functions called maximal uniliteral sub-functions (MTJLs) each of which is realizable in a minimal second-order ‘ product-of-sum ’ form and then a selection of an appropriate sub-set of these maximal uniliteral sub-functions or MULs (or their sub-functions) in order to cover all the prime implicants of the function minimally.

Some Further Studies on Determination of all the Minimal Prime Implicant Covers of Switching Functions

ABSTRACT A method of determination of all the minimal prime implicant covers of switching functions by utilizing their connected cover term matrices is suggested in the paper. The method presented is an extension of the technique suggested by the authors in a previous paper (Choudhury and Das 1964). When majority of the prime implicants of a switching function will occur in more than two different columns of the cover table, it is shown that the determination of all the minimal prime implicant covers can often be greatly facilitated by initially dividing the cover table into more than two suitable sub-tables.

On a Method of Determination of one of the Minimal Solutions of Switching Functions by utilizing their Connected Cover Term Matrices

ABSTRACT A method of determination of one of the minimal prime implicant covers of switching functions by utilizing the properties of the connected cover term matrices has been presented in the paper. The essential prime implicants of the function have been recognized by making a table (called cover table) in which all the minterms of the function appear in a row and below each of the minterms the set of prime implicants covering it occurs in a column. After the selection of the essential prime implicants, the remaining columns of the table have been arranged in a number of open connected cover term matrices from which one of the minimal prime implicant covers of the function has been obtained readily. The number of enumerations involved in finding the minimal solutions of functions having cyclic tables has been reduced in the method.

On a Method of Simplification of Multiple-output Switching Functions†

ABSTRACT A method for finding out the simplest two-stage expression for a given multiple-output switching function requiring minimum number of AND-blocks is presented in this paper. The different component functions of the given multiple-output function are plotted on a chart. A knowledge of the weights of the minterms, prime implicants and the essential prime implicants of the individual component functions enables one to determine easily from the chart the multiple-output prime implicants (or Mopis( which are essential. In the next step, those Mopis are selected which satisfy the condition for being included in one of the minimal covers. Finally, other Mopis are selected by trial for the coverage of the still uncovered terms. A procedure has also been suggested for finding out a cover requiring minimum number of diodes for two-level circuit realization of the given multiple-output function starting from a minimal Mopi cover. The method can be suitably modified for simplification of functions incorpor...

Standard Test Set for Testing and Realization of Threshold Functions

ABSTRACT A method for testing and zero-order integral minimal realization of threshold functions using a reduced set of the total number of inequalities is suggested in the present paper. The paper utilizes the concept of essential second-order incremental weights and other basic ideas developed in an earlier paper by the authors (Choudhury et al. 1966). It is first shown that the information regarding the essentiality of the second-order incremental weights and some of the ordering relations necessary for testing linear separability of a function can be obtained from a knowledge of the ordering relations existing between only some specific pairs of coefficient combinations called the standard test set, which relations can, in turn, be readily obtained from a consideration of only a sub-set of the set of total number of inequalities of the function. On the basis of this, it is next shown that a zero-order integral minimal realization of the function can be obtained by using only a reduced set of the total...