Abstract
Mazurkiewicz trace theory is not powerful enough to describe concurrency paradigms as, for instance, the “Producer / Consumer”. We propose in this paper a generalization of Mazurkiewicz trace monoids which allows to model such problems. We consider quotients of the free monoids by congruences which preserve the commutative images of words. An equivalence class in the quotient monoid consists of all the sequential observations of a distributed computation. In order to characterize congruences which do model concurrency, we study the relationship of this approach and the classical representation of distributed computations with partial orders. We show that the only congruences for which the classes can be represented by partial orders and the concatenation transfers modularly to partial orders are congruences generated by commutations, that is trace congruences. We prove necessary conditions and sufficient conditions on congruences so that their classes can be represented by partial orders. In particular, an important sufficient condition covers both trace congruences and the “Producer / Consumer” congruence.
This research has been supported by the ESPRIT Basic Research Actions No. 6317 ASMICS II.
Preview
Unable to display preview. Download preview PDF.
References
I.J. Aalbersberg and G. Rozenberg. Theory of traces. Theoretical Computer Science, 60:1–82, 1988.
I. Biermann and B. Rozoy. Context traces and transition systems. In S. Kuril, M.U. Caglayan, E. Gelembe, H.L. Akin, and C. Ersoy, editors, Proceedings of the 9th International Symposium on Computer and Information Science ISCIS IX, pages 301–309. Bogazici University Printhouse, Turkey, 1994.
M. Clerbout and M. Latteux. Semi-Commutations. Information and Computation, 73:59–74, 1987.
M. Clerbout, M. Latteux, and Y. Roos. Semi-commutations. In G. Rozenberg and V. Diekert, editors, The Book of Traces, pages 487–552. World Scientific, Singapore, 1995.
V. Diekert. Combinatorics on Traces. Number 454 in Lecture Notes in Computer Science. Springer Verlag, 1990.
V. Diekert and G. Rozenberg, editors. Book of Traces. World Scientific, Singapore, 1995. to appear.
P.W. Hoogers, H.C.M. Kleijn, and P.S. Thiagarajan. A trace semantics for petri nets. In W. Kuich, editor, Proceedings of the 19th International Colloquium on Automata Languages and Programming (ICALP'92), number 623 in Lecture Notes in Computer Science, pages 595–604. Springer Verlag, 1992.
J. Lacaze. Parties reconnaissables de monoÏdes définis par générateurs et relations. R.A.I.R.O. — Informatique Théorique et Applications, 26:541–552, 1992.
A. Mazurkiewicz. Concurrent program schemes and their interpretations. Tech. rep. DAIMI PB 78, Aarhus University, 1977.
A. Mazurkiewicz. Trace theory. In W. Brauer et al., editors, Advances in Petri Nets'86, number 255 in Lecture Notes in Computer Science, pages 279–324. Springer Verlag, 1987.
M. Nielsen and G. Winskel. Trace structures and other models for concurrency. In G. Rozenberg and V. Diekert, editors, The Book of Traces, pages 271–306. World Scientific, Singapore, 1995.
D. Perrin. Partial commutations. In G. Ausiello et al., editors, Proceedings of the 16th International Colloquium on Automata, Languages and Programming (ICALP'89), number 372 in Lecture Notes in Computer Science, pages 637–651. Springer Verlag, 1989.
G. Winskel. An introduction to event structures. In J.W. de Bakker, W.-P. de Roever, and G. Rozenberg, editors, Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency, number 354 in Lecture Notes in Computer Science, pages 123–172. Springer Verlag, 1988.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bauget, S., Gastin, P. (1995). On congruences and partial orders. In: Wiedermann, J., Hájek, P. (eds) Mathematical Foundations of Computer Science 1995. MFCS 1995. Lecture Notes in Computer Science, vol 969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60246-1_149
Download citation
DOI: https://doi.org/10.1007/3-540-60246-1_149
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60246-0
Online ISBN: 978-3-540-44768-9
eBook Packages: Springer Book Archive