A Progress Report on New Decision Algorithms for Finitely Presented Abelian Groups

Abstract

We report on the current state of our development of new decision algorithms for finitely presented Abelian groups (FPAG) based upon commutative-associative (C-A) term rewriting system methods. We show that the uniform word problem is solvable by a completion algorithm which generates Church-Rosser, Noetherian, C-A term rewriting systems. The raw result is theoretical, and few would contemplate implementing it directly because of the incredible amount of trash which would be generated. Much of this trash can be obviated by a different approach which achieves the same end. First, the uniform identity problem is solved by a modified C-A completion algorithm which generates Church-Rosser bases, and then the desired complete set can be computed directly from the Church-Rosser bases. Computer generated examples of the first part of this two part procedure are given. The second part is still under development. Our computer experiences suggest that Church-Rosser bases may often contain large numbers of rules, even for simple presentations. So we were naturally interested in finding better Church-Rosser basis algorithms. With some minor changes the method of [1966] can be used to generate Church-Rosser bases. The Smith basis algorithm appears promising because computer experiments suggest that the number of rules grows slowly. However, small examples with small coefficients can give rise to bases with very large coefficients which exceed machine capacity so we are not entirely satisfied with this approach.

This work was supported in part by NSF Grant MCS-8209143 and a Louisiana Tech University research grant.