Are Two Binary Operators Necessary to Obtain a Finite Axiomatisation of Parallel Composition?
Authors: 
Luca Aceto, Valentina Castiglioni, 
Wan Fokkink, Anna Ingólfsdóttir, Bas LuttikAuthors Info & Claims
Luca Aceto, Valentina Castiglioni, Wan Fokkink, Anna Ingólfsdóttir, Bas LuttikAuthors Info & ClaimsArticle No.: 22, Pages 1 - 56
Abstract
Bergstra and Klop have shown that bisimilarity has a finite equational axiomatisation over ACP/CCS extended with the binary left and communication merge operators. Moller proved that auxiliary operators are necessary to obtain a finite axiomatisation of bisimilarity over CCS, and Aceto et al. showed that this remains true when Hennessy’s merge is added to that language. These results raise the question of whether there is one auxiliary binary operator whose addition to CCS leads to a finite axiomatisation of bisimilarity. We contribute to answering this question in the simplified setting of the recursion-, relabelling-, and restriction-free fragment of CCS. We formulate three natural assumptions pertaining to the operational semantics of auxiliary operators and their relationship to parallel composition and prove that an auxiliary binary operator facilitating a finite axiomatisation of bisimilarity in the simplified setting cannot satisfy all three assumptions.
References
[1]
Luca Aceto. 2003. Some of my favourite results in classic process algebra. Bull. EATCS 81 (2003), 90–108.
[2]
Luca Aceto, Elli Anastasiadi, Valentina Castiglioni, Anna Ingólfsdóttir, and Bas Luttik. 2021. In search of lost time: Axiomatising parallel composition in process algebras. In Proceedings of LICS’21. IEEE, 1–14. DOI:
[3]
Luca Aceto, Bard Bloom, and Frits W. Vaandrager. 1994. Turning SOS rules into equations. Inf. Comput. 111, 1 (1994), 1–52. DOI:
[4]
Luca Aceto, Valentina Castiglioni, Wan J. Fokkink, Anna Ingólfsdóttir, and Bas Luttik. 2020. Are two binary operators necessary to obtain a finite axiomatisation of parallel composition? CoRR abs/2010.01943 (2020).
[5]
Luca Aceto, Valentina Castiglioni, Wan J. Fokkink, Anna Ingólfsdóttir, and Bas Luttik. 2021. Are two binary operators necessary to finitely axiomatise parallel composition? In Proceedings of CSL’21(LIPIcs, Vol. 183). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 8:1–8:17. DOI:
[6]
Luca Aceto, Taolue Chen, Anna Ingólfsdóttir, Bas Luttik, and Jaco van de Pol. 2011. On the axiomatizability of priority II. Theor. Comput. Sci. 412, 28 (2011), 3035–3044. DOI:
[7]
Luca Aceto, Zoltán Ésik, and Anna Ingólfsdóttir. 2002. Equational axioms for probabilistic bisimilarity. In Proceedings of AMAST’02(Lecture Notes in Computer Science, Vol. 2422). Springer, 239–253. DOI:
[8]
Luca Aceto, Wan Fokkink, Anna Ingólfsdóttir, and Bas Luttik. 2005. CCS with Hennessy’s merge has no finite-equational axiomatization. Theor. Comput. Sci. 330, 3 (2005), 377–405. DOI:
[9]
Luca Aceto, Wan Fokkink, Anna Ingólfsdóttir, and Bas Luttik. 2005. Finite equational bases in process algebra: Results and open questions. In Processes, Terms and Cycles: Steps on the Road to Infinity, Essays Dedicated to Jan Willem Klop, on the Occasion of His 60th Birthday(Lecture Notes in Computer Science, Vol. 3838). Springer, 338–367. DOI:
[10]
Luca Aceto, Wan Fokkink, Anna Ingólfsdóttir, and Bas Luttik. 2009. A finite equational base for CCS with left merge and communication merge. ACM Trans. Comput. Log. 10, 1 (2009), 6:1–6:26. DOI:
[11]
Luca Aceto, Wan Fokkink, Anna Ingólfsdóttir, and Mohammad Reza Mousavi. 2010. Lifting non-finite axiomatizability results to extensions of process algebras. Acta Inf. 47, 3 (2010), 147–177. DOI:
[12]
Luca Aceto, Wan Fokkink, Anna Ingólfsdóttir, and Sumit Nain. 2006. Bisimilarity is not finitely based over BPA with interrupt. Theor. Comput. Sci. 366, 1–2 (2006), 60–81. DOI:
[13]
Luca Aceto, Wan Fokkink, and Chris Verhoef. 2001. Structural operational semantics. In Handbook of Process Algebra. North-Holland/Elsevier, 197–292. DOI:
[14]
Luca Aceto, Anna Ingólfsdóttir, Bas Luttik, and Paul van Tilburg. 2008. Finite equational bases for fragments of CCS with restriction and relabelling. In Proceedings of IFIP TCS 2008(IFIP, Vol. 273). Springer, 317–332. DOI:
[15]
Suzana Andova, Jos C. M. Baeten, and Tim A. C. Willemse. 2006. A complete axiomatisation of branching bisimulation for probabilistic systems with an application in protocol verification. In Proceedings of CONCUR 2006(Lecture Notes in Computer Science, Vol. 4137). Springer, 327–342. DOI:
[16]
Suzana Andova and Sonja Georgievska. 2009. On compositionality, efficiency, and applicability of abstraction in probabilistic systems. In Proceedings of SOFSEM’09(Lecture Notes in Computer Science, Vol. 5404). Springer, 67–78. DOI:
[17]
Jos C. M. Baeten, Jan A. Bergstra, and Scott A. Smolka. 1995. Axiomatizing probabilistic processes: ACP with generative probabilities. Inf. Comput. 121, 2 (1995), 234–255. DOI:
[18]
Jan A. Bergstra and Jan Willem Klop. 1984. The algebra of recursively defined processes and the algebra of regular processes. In Proceedings of ICALP’11(Lecture Notes in Computer Science, Vol. 172). Springer, 82–94. DOI:
[19]
Jan A. Bergstra and Jan Willem Klop. 1984. Process algebra for synchronous communication. Inf. Contr. 60, 1–3 (1984), 109–137. DOI:
[20]
Jan A. Bergstra and Jan Willem Klop. 1985. Algebra of communicating processes with abstraction. Theor. Comput. Sci. 37 (1985), 77–121. DOI:
[21]
Bard Bloom, Sorin Istrail, and Albert R. Meyer. 1995. Bisimulation can’t be traced. J. ACM 42, 1 (1995), 232–268. DOI:
[22]
Valentina Castiglioni, Ruggero Lanotte, and Simone Tini. 2019. Fully syntactic uniform continuity formats for bisimulation metrics. In The Art of Modelling Computational Systems: A Journey from Logic and Concurrency to Security and Privacy - Essays Dedicated to Catuscia Palamidessi on the Occasion of Her 60th Birthday(Lecture Notes in Computer Science, Vol. 11760). Springer, 293–312. DOI:
[23]
Valentina Castiglioni, Michele Loreti, and Simone Tini. 2020. The metric linear-time branching-time spectrum on nondeterministic probabilistic processes. Theor. Comput. Sci. 813 (2020), 20–69. DOI:
[24]
Valentina Castiglioni and Simone Tini. 2020. Probabilistic divide & congruence: Branching bisimilarity. Theor. Comput. Sci. 802 (2020), 147–196. DOI:
[25]
Valentina Castiglioni and Simone Tini. 2020. Raiders of the lost equivalence: Probabilistic branching bisimilarity. Inf. Process. Lett. 159–160 (2020), 105947. DOI:
[26]
Pedro R. D’Argenio, Daniel Gebler, and Matias David Lee. 2014. Axiomatizing bisimulation equivalences and metrics from probabilistic SOS rules. In Proceedings of FOSSACS’14(Lecture Notes in Computer Science, Vol. 8412). Springer, 289–303. DOI:
[27]
Robert de Simone. 1985. Higher-level synchronising devices in Meije-SCCS. Theor. Comput. Sci. 37 (1985), 245–267. DOI:
[28]
Yuxin Deng. 2005. Axiomatisations and Types for Probabilistic and Mobile Processes. (Axiomatisations et types pour des processus probabilistes et mobiles). Ph.D. Dissertation. Mines ParisTech, France. Retrieved from https://tel.archives-ouvertes.fr/tel-00155225.
[29]
Jan Friso Groote. 1990. A new strategy for proving omega-completeness applied to process algebra. In Proceedings of CONCUR’90(Lecture Notes in Computer Science, Vol. 458). Springer, 314–331. DOI:
[30]
Jan Friso Groote and Erik P. de Vink. 2019. An axiomatization of strong distribution bisimulation for a language with a parallel operator and probabilistic choice. In From Software Engineering to Formal Methods and Tools, and Back - Essays Dedicated to Stefania Gnesi on the Occasion of Her 65th Birthday(Lecture Notes in Computer Science, Vol. 11865). Springer, 449–463. DOI:
[31]
Matthew Hennessy. 1988. Axiomatising finite concurrent processes. SIAM J. Comput. 17, 5 (1988), 997–1017. DOI:
[32]
Matthew Hennessy and Robin Milner. 1985. Algebraic laws for nondeterminism and concurrency. J. ACM 32, 1 (1985), 137–161. DOI:
[33]
Robert M. Keller. 1976. Formal verification of parallel programs. Commun. ACM 19, 7 (1976), 371–384. DOI:
[34]
Kim Guldstrand Larsen and Arne Skou. 1992. Compositional verification of probabilistic processes. In Proceedings of CONCUR’92(Lecture Notes in Computer Science, Vol. 630). Springer, 456–471. DOI:
[35]
Bas Luttik and Vincent van Oostrom. 2005. Decomposition orders another generalisation of the fundamental theorem of arithmetic. Theor. Comput. Sci. 335, 2–3 (2005), 147–186. DOI:
[36]
Cornelis A. Middelburg. 2020. Probabilistic process algebra and strategic interleaving. Sci. Ann. Comput. Sci. 30, 2 (2020), 205–243. DOI:
[37]
Robin Milner. 1980. A Calculus of Communicating Systems. (Lecture Notes in Computer Science, Vol. 92.)Springer. DOI:
[38]
Robin Milner. 1989. Communication and Concurrency. Prentice Hall.
[39]
Robin Milner and Faron Moller. 1993. Unique decomposition of processes. Theor. Comput. Sci. 107, 2 (1993), 357–363. DOI:
[40]
Faron Moller. 1989. Axioms for Concurrency. Ph.D. Dissertation. Department of Computer Science, University of Edinburgh. Report CST-59-89.
[41]
Faron Moller. 1990. The importance of the left merge operator in process algebras. In Proceedings of ICALP’90(Lecture Notes in Computer Science, Vol. 443). Springer, 752–764. DOI:
[42]
Faron Moller. 1990. The nonexistence of finite axiomatisations for CCS congruences. In Proceedings of LICS’90. IEEE Computer Society, 142–153. DOI:
[43]
David M. R. Park. 1981. Concurrency and automata on infinite sequences. In Proceedings of GI-Conference(Lecture Notes in Computer Science, Vol. 104). Springer, 167–183. DOI:
[44]
Gordon D. Plotkin. 1981. A Structural Approach to Operational Semantics. Report DAIMI FN-19. Computer Science Department, Aarhus University.
[45]
Eugene W. Stark and Scott A. Smolka. 2000. A complete axiom system for finite-state probabilistic processes. In Proof, Language, and Interaction, Essays in Honour of Robin Milner. The MIT Press, 571–596.
[46]
Walter Taylor. 1977. Equational logic. In Contributions to Universal Algebra. North-Holland, 465–501. DOI:
[47]
Ferry Timmers and Jan Friso Groote. 2020. A complete axiomatisation for probabilistic trace equivalence. Sci. Ann. Comput. Sci. 30, 1 (2020), 69–104. DOI:
[48]
Rob J. van Glabbeek. 1993. The linear time–branching time spectrum II. In Proceedings of CONCUR’93(Lecture Notes in Computer Science, Vol. 715). Springer, 66–81. DOI:
[49]
Rob J. van Glabbeek, Jan Friso Groote, and Erik P. de Vink. 2019. A complete axiomatization of branching bisimilarity for a simple process language with probabilistic choice (extended abstract). In The Art of Modelling Computational Systems: A Journey from Logic and Concurrency to Security and Privacy - Essays Dedicated to Catuscia Palamidessi on the Occasion of Her 60th Birthday(Lecture Notes in Computer Science, Vol. 11760). Springer, 139–162. DOI:
[50]
Rob J. van Glabbeek, Scott A. Smolka, Bernhard Steffen, and Chris M. N. Tofts. 1990. Reactive, generative, and stratified models of probabilistic processes. In Proceedings of LICS’90. IEEE Computer Society, 130–141. DOI:
Index Terms
- Are Two Binary Operators Necessary to Obtain a Finite Axiomatisation of Parallel Composition?
Recommendations
CCS with Hennessy's merge has no finite-equational axiomatization
Expressiveness in concurrencyThis paper confirms a conjecture of Bergstra and Klop's from 1984 by establishing that the process algebra obtained by adding an auxiliary operator proposed by Hennessy in 1981 to the recursion free fragment of Milner's Calculus of Communicating Systems ...
Sound and Complete Equational Reasoning over Comodels
Comodels of Lawvere theories, i.e. models in Setop, model state spaces with algebraic access operations. Standard equational reasoning is known to be sound but incomplete for comodels. We give two sound and complete calculi for equational reasoning over ...
Impossibility Results in the Equational Logic of Processes
This talk offers a survey of negative results on the existence of finite equational axiomatizations for bisimulation equivalence over fragments of algebraic process calculi.
Comments
Information & Contributors
Information
Published In
October 2022
279 pages
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 20 October 2022
Online AM: 20 April 2022
Accepted: 29 March 2022
Revised: 04 March 2022
Received: 31 May 2021
Published in TOCL Volume 23, Issue 4
Check for updates
Author Tags
Qualifiers
- Research-article
- Refereed
Funding Sources
- Icelandic Research Fund
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 118Total Downloads
- Downloads (Last 12 months)18
- Downloads (Last 6 weeks)1
Reflects downloads up to 04 Oct 2024
Other Metrics
Citations
View Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign inFull Access
View options
View or Download as a PDF file.
PDFeReader
View online with eReader.
eReaderFull Text
View this article in Full Text.
Full TextHTML Format
View this article in HTML Format.
HTML Format