# Combinatorial rewriting on traces

## Abstract

There are two main problems in working with replacement systems over free partially commutative monoids: For finite noetherian systems confluence is undecidable, in general, and the known algorithm to compute irreducible normal forms need time square in the derivation length instead of linear. We first give a decidable and sufficient condition for finite noetherian systems such that confluence becomes decidable. This condition is weaker than the known ones before. Then we give a decidable and sufficient condition such that irreducible normal forms are computable in time linear to the derivation length. Furthermore, we prove that the first condition is implied by the second. We also present a new uniform algorithm for computing normal forms using Zielonka's theory of asynchronous automata.

## Keywords

Word Problem Turing Machine Replacement System Derivation Step Commutative Monoid## Preview

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## References

- [AR88]I.J. Aalbersberg and G. Rozenberg. Theory of traces.
*Theoret. Comput. Sci.*, 60:1–82, 1988.CrossRefGoogle Scholar - [BD88]L. Bachmair and N. Dershowitz. Critical pair criteria for completion.
*J. Symbolic Computation*, 6:1–18, 1988.Google Scholar - [BL87]R. Book and H.-N. Liu. Rewriting systems and word problems in a free partially commutative monoid.
*Inform. Proc. Letters*, 26:29–32, 1987.CrossRefGoogle Scholar - [Boo82]R. Book. Confluent and other types of Thue systems.
*J. Assoc. for Comp. Mach.*, 29:171–182, 1982.Google Scholar - [CM87]R. Cori and Y. Métivier. Approximation d' une trace, automates asynchrones et ordre des evenement dans les systemes repartis. Technical Report 1-8708, UER de Mathematiques et d' Informatique, Université de Bordeaux I, 1987.Google Scholar
- [Die87]V. Diekert. On the Knuth-Bendix completion for concurrent processes. In Th. Ottmann, editor,
*Proc. of the 14th International Colloquium on Automata Languages and Programming, Karlsruhe 1987, (ICALP'87)*, number 267 in Lect. Notes in Comp. Sci., pages 42–53. Springer, 1987. Appeared also in a revised version in Theoret. Comp. Science 66:117–136, 1989.Google Scholar - [Die89]V. Diekert. Word problems over traces which are solvable in linear time. In B. Monien et al., editors,
*Proceedings of the 6th Annual Symposium on Theoretical Aspects of Computer Science (STACS'89), Paderborn 1989*, number 349 in Lect. Notes in Comp. Sci., pages 168–180. Springer, 1989. To appear in revised version in Theoret. Comp. Science.Google Scholar - [Jou83]J.P. Jouannaud. Confluent and coherent equational term rewriting systems applications to proofs in abstract data types. In Ausiello G. et al., editors,
*Proceeding of the conference of Trees in Algebra and Programming (CAAP'83)*, number 159 in Lect. Notes in Comp. Sci., pages 269–283. Springer, 1983.Google Scholar - [KMN88]D. Kapur, D. Musser, and P. Narendran. Only prime superposition need be considered in the Knuth-Bendix completion procedure.
*J. Symbolic Computation*, 6:19–36, 1988.Google Scholar - [Maz77]A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.Google Scholar
- [Maz87]A. Mazurkiewicz. Trace theory. In W. Brauer et al., editors,
*Petri Nets, Applications and Relationship to other Models of Concurrency*, number 255 in Lect. Notes in Comp. Sci., pages 279–324. Springer, 1987.Google Scholar - [NO88]P. Narendran and F. Otto. Preperfectness is undecidable for Thue systems containing only length-reducing rules and a single commutation rule.
*Information Proc. Letters*, 29:125–130, 1988.CrossRefGoogle Scholar - [Ott89]F. Otto. On deciding confluence of finite string rewriting systems modulo partial commutativity.
*Theoret. Comput. Sci.*, 67:19–36, 1989.CrossRefGoogle Scholar - [Per89]D. Perrin. Partial commutations. In
*Proc. of the 16th International Colloquium on Automata, Languages and Programming (ICALP '89), Stresa 1989, Italy*, number 372 in Lect. Notes in Comp. Sci., pages 637–651. Springer, 1989.Google Scholar - [WB83]F. Winkler and B. Buchberger. A criterion for eleminating unnecessary reductions in the knuth-Bendix algorithm. In
*Proc. Coll. on Algebra, Combinatorics and Logic in Computer Science, Györ, Hungary*, 1983.Google Scholar - [Wra88]C. Wrathall. The word problem for free partially commutative groups.
*J. Symbolic Computation*, 6:99–104, 1988.Google Scholar - [Zie87]W. Zielonka. Notes on finite asynchronous automata.
*R.A.I.R.O.-Informatique théorique et Application*, 21:99–135, 1987.Google Scholar