Applications of Bar Code to involutive divisions and a greedy algorithm for complete sets

TL;DR

This paper introduces a method using Bar Code applications to involutive divisions, presenting a greedy algorithm to find variable orderings that make a set of terms complete, improving efficiency over brute-force approaches.

Contribution

It develops a novel greedy algorithm that constructs Bar Codes to determine variable orderings for set completeness, avoiding exhaustive permutation checks.

Findings

The algorithm successfully finds variable orderings for completeness.

Bar Code application effectively detects set completeness.

The method reduces computational complexity in ordering problems.

Abstract

In this paper, we describe how to get Janet decomposition for a finite set of terms and detect completeness of that set by means of the associated Bar Code. Moreover, we explain an algorithm to find a variable ordering (if it exists) s.t. a given set of terms is complete according to that ordering. The algorithm is greedy and constructs a Bar Code from the maximal to the minimal variable, adjusting the variable ordering with a sort of backtracking technique, thus allowing to construct the desired ordering without trying all the n! possible orderings