Cross-sections for finitely presented monoids with decidable word problems
A finitely presented monoid has a decidable word problem if and only if it has a recursive cross-section if and only if it can be presented by some left-recursive convergent string-rewriting system. However, regular cross-sections or even context-free cross-sections do not suffice. This is shown by presenting examples of finitely presented monoids with decidable word problems that do not admit regular cross-sections, and that, hence, cannot be presented by left-regular convergent stringrewriting systems. Also examples of finitely presented monoids with decidable word problems are presented that do not even admit context-free cross-sections. On the other hand, it is shown that each finitely presented monoid with a decidable word problem has a finite presentation that admits a cross-section which is a Church-Rosser language.
Unable to display preview. Download preview PDF.
- [Adi79]A.I. Adian. The Burnside Problem and Identities in Groups. Springer-Verlag, Berlin, 1979.Google Scholar
- [Ave86]J. Avenhaus. On the descriptive power of term rewriting systems. J. Symbolic Computation, 2:109–122, 1986.Google Scholar
- [Ber79]J. Berstel. Transductions and Context-free Languages. Teubner Studienbücher. Teubner-Verlag, 1979.Google Scholar
- [BO93]R.V. Book and F. Otto. String-Rewriting Systems. Springer-Verlag, New York, 1993.Google Scholar
- [BO95]G. Buntrock and F. Otto. Growing context-sensitive languages and Church-Rosser languages. In E.W. Mayr and C. Puech, editors, Proc. of STACS 95, Lecture Notes in Computer Science 900, pages 313–324. Springer-Verlag, Berlin, 1995.Google Scholar
- [Bun96]G. Buntrock. Wachsende kontext-sensitive Sprachen. Habilitationsschrift, Fakultät für Mathematik und Informatik, Universität Würzburg, 1996.Google Scholar
- [Gil87]R.H. Gilman. Groups with a rational cross-section. In S.M. Gersten and J.R. Stallings, editors, Computational Group Theory and Topology, pages 175–183. Princeton, 1987.Google Scholar
- [Kob95]Y. Kobayashi. A finitely presented monoid which has solvable word problem but has no regular complete presentation. Theoretical Computer Science, 146:321–329, 1995.Google Scholar
- [MNO88]R. McNaughton, P. Narendran, and F. Otto. Church-Rosser Thue systems and formal languages. Journal Association Computing Machinery, 35:324–344, 1988.Google Scholar
- [SK91]A. Sattler-Klein. Divergence phenomena during completion. In R.V. Book, editor, Rewriting Techniques and Applications, pages 374–385. Springer-Verlag, Berlin, 1991. Lecture Notes in Computer Science 488.Google Scholar
- [Squ87]C.C. Squier. Word problems and a homological finiteness condition for monoids. J. Pure Applied Algebra, 49:201–217, 1987.Google Scholar