Reflexive Transitive Closure Research Articles

On graph compatible Splittings of M-matrices

Let A ϵ R nn . Then A = M − N is called a regular splitting of A if M -1 ⩾ 0 and N ⩾ 0, and A = M − N is called a graph compatible splitting of A if ⊼[Ggr]( A)⊼ ⊇ [Ggr]( M), where [Ggr]( A) is the graph of A and ⊼[Ggr]( A)⊼ its reflexive transitive closure. We answer in the negative a question raised by H. Schneider whether a regular splitting A = M − N of an M-matrix is graph compatible if all diagonal elements of A (or M) are positive.

Micro-store width optimization through bit steering

This paper presents a bit optimization procedure based on the bit steering concept. A relation R, called companion relation, has been defined over the fields in a microinstruction, and the reflexive-transitive closure R ∗ of this relation has been used for examining the microinstructions of a microprogram for possible bit dimension reduction. An algorithm that computes R ∗ and attempts bit reduction on a given microprogram has been presented. The algorithm (implemented on DEC-10) when input with the sample microprogram has been found to yield further bit reduction over and above the earlier reported reductions.

A simplified proof of the characterization theorem for Gröbner-bases

In /2/ a certain type of bases ("Gröbner-Bases") for polynomial ideals has been introduced whose usefulness stems from the fact that a number of important computability problems in the theory of polynomial ideals are reducible to the construction of bases of this type. The key to an algorithmic construction of Gröbner-bases is a characterization theorem for Gröbner-bases whose proof in /2/is rather complex.In this paper a simplified proof is given. The simplification is based on two new lemmas that are of some interest in themselves. The first lemma characterizes the congruence relation modulo a polynomial ideal as the reflexive-transitive closure of a particular reduction relation ("M-reduction") used in the definition of Gröbner-bases and its inverse. The second lemma is a lemma on general reduction relations, which allows to guarantee the Church-Rosser property under very weak assumptions.

Characterization of recursively enumerable sets

AbstractLet N, O and S denote the set of nonnegative integers, the graph of the constant 0 function and the graph of the successor function respectively. For sets P, Q, R ⊆ N2 operations of transposition, composition, and bracketing are defined as follows: P∪ = {〈x, y〉 ∣ 〈y, x〉 ∈ P}, PQ = {〈x, z〉 ∣ ∃y〈x, y〉 ∈ P & 〈y, z〉 ∈ Q}, and [P, Q, R] = ⋃n ∈ M(Pn Q Rn).Theorem. The class of recursively enumerable subsets of N2 is the smallest class of sets with O and S as members and closed under transposition, composition, and bracketing.This result is derived from a characterization by Julia Robinson of the class of general recursive functions of one variable in terms of function composition and “definition by general recursion.” A key step in the proof is to show that if a function F is defined by general recursion from functions A, M, P and R then F = [P∪, A∪M, R].The above definitions of the transposition, composition, and bracketing operations on subsets of N2 can be generalized to subsets of X2 for an arbitrary set X. In this abstract setting it is possible to show that the bracket operation can be defined in terms of K, L, transposition, composition, intersection, and reflexive transitive closure where K: X → X and L: X → X are functions for decoding pairs.