A Progress Report on New Decision Algorithms for Finitely Presented Abelian Groups
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  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.
KeywordsWord Problem Nilpotent Group Decision Algorithm Commutative Semigroup Critical Pair
Unable to display preview. Download preview PDF.
- Cardoza, E. Computational complexity of the word problem for commutative semigroups. M.Sc. thesis, MIT, Cambridge, MA, Aug. 1978; and MAC Tech. Memo. 67, Oct. 1975.Google Scholar
- Fages, F. (some results of Fages presented by J. P. Jouannaud at a GE term rewriting system conference, Sept. 1983)Google Scholar
- Jacobson, N. Lectures in Abstract Algebra, II Linear Algebra, Van Nostrand, 1953.Google Scholar
- Kapur, D. (comparison of computer generated examples at a GE term rewriting system conference, Sept. 1983)Google Scholar
- Lankford, D. On proving term rewriting systems are Noetherian. Louisiana Tech U., Math. Dept., report MTP-3, May 1979.Google Scholar
- Lankford, D. and Ballantyne, A. Decision procedures for simple equational theories with commutative-associative axioms: complete sets of commutative-associative reductions. U. of Texas, Math. Dept., ATP project, report ATP-39, Aug. 1977.Google Scholar
- Lankford, D. and Ballantyne, A. The refutation completeness of blocked permutative narrowing and resolution. Proc. Fourth Workshop on Automated Deduction, Austin, Texas, Feb. 1979, W. Joyner, ed., 168–174.Google Scholar
- Peterson, G. and Stickel, J. Complete sets of reductions for some equational theories. JACM 28,2 (Apr. 1981), 233–264.Google Scholar
- Smith, D. A basis algorithm for finitely generated Abelian groups. Math. Algorithms 1,1 (Jan. 1966), 13–26.Google Scholar
- Stickel, M. A complete unification algorithm for associative-commutative functions. Advance Papers of the Fourth International Conference on Artificial Intelligence, AI Lab, MIT, Aug. 1975, 71–76.Google Scholar
- Reidemeister, K. Einführung in die kombinatorische Topologie, Braunschweig, 1932, 50–56.Google Scholar