Disjunctive Form Research Articles

Stability theorems for infinitely constrained mathematical programs

The primary concern of this paper is to investigate stability conditions for the mathematical program: findx ∈En that maximizesf(x):gj(x)≦0 for somej ∈J, wheref is a real scalarvalued function and eachg is a real vector-valued function of possibly infinite dimension. It should be noted that we allow, possibly infinitely many, disjunctive forms. In an earlier work, Evans and Gould established stability theorems wheng is a continuous finite-dimensional real-vector function andJ=1. It is pointed out that the results of this paper reduce to the Evans-Gould results under their assumptions. Furthermore, since we use a slightly more general definition of lower and upper semicontinuous point-to-set mappings, we can dispense with the continuity ofg (except in a few instances where it is implied by convexity assumptions).

Comments on " ANew Algorithm for Generating Prime Implicants"

One of the major areas in switching theory research has been concerned with obtaining suitable algorithrns for the minimization of Boolean functions in connection with the general problem of their economic realization. A solution of the minimization problem, in general, involves consideration of two distinct phases. In the first phase all the prime implicants of the function are found, while in the second phase, from this set of all the prime implicants, a minimal subset (according to some criterion of minimality) of prime implicants is selected such that their disjunction is equivalent to the function and from which none of the prime implicants can be dropped without sacrificing equivalence. Many different algorithms exist for solving both the first and the second phase of this minimization problem. In a recent paper,' Slagle et al. describe a new algorithm for the generation of all the prime implicants of a Boolean function. As claimed by the authors, this algorithm is different from those previously given in the literature. The algorithm is efficient, does not generate the same prime implicant more than once (though the algorithm sometimes generates some non-prime implicants), and does not need large capacity of memory for implementation on a digital computer. The algorithm works equally well with either the conjunctive or the disjunctive (both canonical and noncanonical) form of the function.

Generation of Prime Implicants by Direct Multiplication

This correspondence shows that the Necula [1] and Slagle, Chang. and Lee [2] methods for determining prime implicants are a mechanization of a method first given by Nelson [3], i. e., generating of all prime implicants by direct expansion of a conjunctive form to a disjunctive form.

An Algorithm for Threshold Element Analysis

A majority or threshold element is a device with a finite number of inputs and a single output. The output, F n is a Boolean function of the n binary valued variables (x 1 , x 2 , ..., x n ) applied to the inputs. The analysis of a threshold element is the determination of the output Boolean function, F n when the input variables and their multiplicities are specified. The multiplicity of an input variable x i is the number of inputs to which the variable is applied. The fundamental problem in analysis is the determination of F n when the number of inputs is moderately large. An algorithm is given which permits a simple straightforward determination of F n in reduced, disjunctive form.

M. J. Ghazala Gazalé. Irredundant disjunctive and conjunctive forms of a Boolean function. IBM journal of research and development, vol. 1 (1957), pp. 171–176. - T. Rado. Comments on the presence function of Gazalé. IBM journal of research and development, vol. 6 (1962), pp. 268–269.

M. J. Ghazala Gazalé. Irredundant disjunctive and conjunctive forms of a Boolean function. IBM journal of research and development, vol. 1 (1957), pp. 171–176. - T. Rado. Comments on the presence function of Gazalé. IBM journal of research and development, vol. 6 (1962), pp. 268–269.