# Axiomatizing flat iteration

## Abstract

*P*

^{*}Q, obtained by restricting the first argument to be a sum of atomic actions. It generalizes prefix iteration, in which the first argument is a single action. Complete finite equational axiomatizations are given for five notions of bisimulation congruence over basic CCS with flat iteration, viz. strong congruence, branching congruence,

*η*-congruence, delay congruence and weak congruence. Such axiomatizations were already known for prefix iteration and are known not to exist for general iteration. The use of flat iteration has two main advantages over prefix iteration:

- 1.
The current axiomatizations generalize to full CCS, whereas the prefix iteration approach does not allow an elimination theorem for an asynchronous parallel composition operator.

- 2.
The greater expressiveness of flat iteration allows for much shorter completeness proofs.

In the setting of prefix iteration, the most convenient way to obtain the completeness theorems for *η*-, delay, and weak congruence was by reduction to the completeness theorem for branching congruence. In the case of weak congruence this turned out to be much simpler than the only direct proof found. In the setting of flat iteration on the other hand, the completeness theorems for delay and weak (but not *η*-) congruence can equally well be obtained by reduction to the one for strong congruence, without using branching congruence as an intermediate step. Moreover, the completeness results for prefix iteration can be retrieved from those for flat iteration, thus obtaining a second indirect approach for proving completeness for delay and weak congruence in the setting of prefix iteration.

## Preview

Unable to display preview. Download preview PDF.

### References

- [1]L. Aceto and J.F. Groote (1995),
*A complete equational axiomatization for MPA with string iteration*, BRICS Research Report RS-95-28, Department of Mathematics and Computer Science, Aalborg University. Available by anonymous ftp from ftp.daimi.aau.dk in the directory pub/BRICS/RS/95/28.Google Scholar - [2]L. Aceto, W. Fokkink, R. van Glabbeek and A. Ingólfsdóttir (1996),
*Axiomatizing prefix iteration with silent steps*, I&C 127(1), pp. 26–40.Google Scholar - [3]L. Aceto and A. Ingólfsdóttir (1996),
*An equational axiomatization of observation congruence for prefix iteration*, in Proc. AMAST '96, Munich, Germany, M. Wirsing and M. Nivat, eds., LNCS 1101, Springer-Verlag, pp. 195–209.Google Scholar - [4]J. Baeten, J. Bergstra and J. Klop (1987),
*On the consistency of Koomen's fair abstraction rule*, TCS 51, pp. 129–176.CrossRefGoogle Scholar - [5]T. Basten (1996),
*Branching bisimilarity is an equivalence indeed!*, IPL 58(3), pp. 141–147.Google Scholar - [6]J. Bergstra, I. Bethke and A. Ponse (1994),
*Process algebra with iteration and nesting*, Computer Journal 37, pp. 243–258. Originally appeared as report P9314, Programming Research Group, University of Amsterdam, 1993.Google Scholar - [7]J. Bergstra and J. Klop (1984),
*The algebra of recursively defined processes and the algebra of regular processes*, in Proceedings 11th ICALP, Antwerpen, J. Paredaens, ed., LNCS 172, Springer-Verlag, pp. 82–95.Google Scholar - [8]F. Corradini, R. De Nicola and A. Labella (1995),
*Fully abstract models for nondeterministic Kleene algebras (extended abstract)*, in Proc. CONCUR 95, Philadelphia, I. Lee and S. Smolka, eds., LNCS 962, Springer-Verlag, pp. 130–144.Google Scholar - [9]W. Fokkink (1994),
*A complete equational axiomatization for prefix iteration*, IPL 52, pp. 333–337.Google Scholar - [10]W. Fokkink (1996),
*A complete axiomatization for prefix iteration in branching bisimulation*, Fundamenta Informaticae 26, pp. 103–113.Google Scholar - [11]W. Fokkink and H. Zantema (1994),
*Basic process algebra with iteration: Completeness of its equational axioms*, Computer Journal 37, pp. 259–267.Google Scholar - [12]R. V. Glabbeek (1995),
*Branching bisimulation as a tool in the analysis of weak bisimulation*. Available at ftp://boole.stanford.edu/pub/DVI/tool.dvi.gz.Google Scholar - [13]R. V. Glabbeek and W. Weijland (1996),
*Branching time and abstraction in bisimulation semantics*, JACM 43(3), pp. 555–600.Google Scholar - [14]M. Hennessy and R. Milner (1985),
*Algebraic laws for nondeterminism and concurrency*, JACM 32, pp. 137–161.CrossRefGoogle Scholar - [15]S. Kleene (1956),
*Representation of events in nerve nets and finite automata*, in Automata Studies, C. Shannon and J. McCarthy, eds., Princeton University Press, pp. 3–41.Google Scholar - [16]
- [17]R. Milner (1989),
*A complete axiomatisation for observational congruence of finite-state behaviours*, I&C 81, pp. 227–247.Google Scholar - [18]E.-R. Olderog and C.A.R. Hoare (1986),
*Specification-oriented semantics for communicating processes*, Acta Informatica 23, pp. 9–66.Google Scholar - [19]P. Sewell (1994),
*Bisimulation is not finitely (first order) equationally axiomatisable*, in Proc. 9th LICS, Paris, IEEE Computer Society Press, pp. 62–70.Google Scholar