Abstract
In order to associate a transition system with an event structure, it is customary to use configurations, constructing a transition system by repeatedly adding executable events. It is also possible to use residuals, constructing a transition system by repeatedly deleting non-executable events. The present paper proposes a systematic investigation of how the two methods are interrelated. The focus will be on asymmetric versions of prime, bundle, and dual event structures. For each of them, configuration-based and residual-based transition system semantics will be defined. The pairwise bisimilarity of the resulting transition systems will be proved, considering interleaving, multiset, and pomset semantics.
E. Best, N. Gribovskaya and I. Virbitskaite—Supported by DFG (German Research Foundation) and by RFBR (Russian Foundation for Basic Research) through the grant CAVER (Be 1267/14-1 and 14-01-91334, respectively).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For a set \(Y \subseteq X\) and a relation \(r\subseteq X\times X\), \(r_Y\) denotes the restriction of r to Y.
- 2.
A relation \(r\subseteq X\times X\) is acyclic if it has no “cycles” of the form \(e_0\ r\ e_1\ r\ \ldots \ r\ e_n\ r\ e_0\), with \(n\ge 1\) and \(e_i\in X\) for all \(0\le i\le n\).
- 3.
A relation \(r\subseteq X\times X\) is well-founded if it has no infinite descending chains, i.e., \(\langle e_i\rangle _{i\in \mathbb {N}}\) such that \(e_{i+1}\ r\ e_i\), \(e_i\ne e_{i+1}\), for all \(i\in \mathbb {N}\).
- 4.
Stability ensures that two distinct events of a bundle set are in mutual disabling.
- 5.
Notice that in [8], for asymmetric bundle/dual event structures the removal operator has been defined in a different way, without removing conflict sets. All the events in a trace t and bundles \(W\mapsto e\) such that \(W\cap \overline{t}\ne \emptyset \) are removed. However, the events conflicting with some event in t are retained simply making them impossible by adding empty bundles. There, the removal operator has been formally defined as follows: \(\mathcal {E}{\setminus } [t]=(E'\), \(\mapsto '\), \(\rightsquigarrow \cap (E'\times E')\), L, \(l\mid _{E'})\), where \(E'= E{\setminus }\overline{t}\) and \(\mapsto '= \big (\!\!\mapsto {\setminus }\{(W,e)\in \ \mapsto \mid W\cap \overline{t}\ne \emptyset \}\big )\) \(\cup \) \(\{(\emptyset ,e)\mid e\in E'\), \(e\rightsquigarrow e'\), for some \(e'\in \overline{t}\}\). We say in advance that the “residual” transition systems constructed on the base of the removal operator from [8] and our removal operator are isomorphic. This implies that all bisimilarity results obtained in our paper are valid for event structures treated within the process algebra PA in the work [8].
- 6.
We allow a single arrow between two states to denote multiple transitions. For instance, the arrow from \(\mathcal {E}^p\) to \(\mathcal {E}_{\emptyset }\) in \( TE _{pom}(\mathcal {E}^p)\) (Fig. 3) denotes two transitions.
References
Baldan, P., Corradini, A., Montanari, U.: Contextual petri nets, asymmetric event structures, and processes. Inf. Comput. 171(1), 1–49 (2001)
Boudol, G., Castellani, I.: Concurrency and atomicity. Theor. Comput. Sci. 59, 25–84 (1989)
van Glabbeek, R.J.: History preserving process graphs (1995). http://boole.stanford.edu/~rvg/pub/history.draft.dvi
van Glabbeek, R.J., Goltz, U.: Refinement of actions and equivalence notions for concurrent systems. Acta Informatica 37, 229–327 (2001)
van Glabbeek, R.J., Plotkin, G.D.: Configuration structures, event structures and petri nets. Theor. Comput. Sci. 410(41), 4111–4159 (2009)
van Glabbeek, R.J., Vaandrager, F.: Bundle event structures and CCSP. In: Amadio, R., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 57–71. Springer, Heidelberg (2003). doi:10.1007/978-3-540-45187-7_4
Gutierrez, J., Wooldridge, M.: Equilibria of concurrent games on event structures. In: Proceedings of CSL-LICS 2014, pp. 46:1–46:10 (2014)
Katoen, J.-P.: Quantitative and qualitative extensions of event structures. Ph.D. thesis. Twente University (1996)
Langerak, R.: Bundle event structures: a non-interleaving semantics for LOTOS. In: Formal Description Techniques V, IFIP Transactions, C-10 (1993)
Langerak, R., Brinksma, E., Katoen, J.-P.: Causal ambiguity and partial orders in event structures. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 317–331. Springer, Heidelberg (1997). doi:10.1007/3-540-63141-0_22
Loogen, R., Goltz, U.: Modelling nondeterministic concurrent processes with event structures. Fundamenta Informatica XIV, 39–74 (1991)
Majster-Cederbaum, M., Roggenbach, M.: Transition systems from event structures revisited. Inf. Process. Lett. 67(3), 119–124 (1998)
Nielsen, M., Plotkin, G., Winskel, G.: Petri nets, event structures and domains. Theor. Comput. Sci. 13(1), 85–108 (1981)
Nielsen, M., Thiagarajan, P.S.: Regular event structures and finite petri nets: the conflict-free case. In: Esparza, J., Lakos, C. (eds.) ICATPN 2002. LNCS, vol. 2360, pp. 335–351. Springer, Heidelberg (2002). doi:10.1007/3-540-48068-4_20
Pinna, G.M., Poigné, A.: On the nature of events: another perspectives in concurrency. Theor. Comput. Sci. 138(2), 425–454 (1995)
Winskel, G.: Events in computation. Ph.D. thesis. University of Edinburgh (1980)
Winskel, G.: Event structures. In: Brauer, W., Reisig, W., Rozenberg, G. (eds.) ACPN 1986. LNCS, vol. 255, pp. 325–392. Springer, Heidelberg (1987). doi:10.1007/3-540-17906-2_31
Winskel, G., Nielsen, M.: Models for concurrency. In: Handbook of Logic in Computer Science, vol. 4 (1995)
Winskel, G.: Distributed probabilistic and quantum strategies. Electron. Notes Theor. Comput. Sci. 298, 403–425 (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Best, E., Gribovskaya, N., Virbitskaite, I. (2017). Configuration- and Residual-Based Transition Systems for Event Structures with Asymmetric Conflict. In: Steffen, B., Baier, C., van den Brand, M., Eder, J., Hinchey, M., Margaria, T. (eds) SOFSEM 2017: Theory and Practice of Computer Science. SOFSEM 2017. Lecture Notes in Computer Science(), vol 10139. Springer, Cham. https://doi.org/10.1007/978-3-319-51963-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-51963-0_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-51962-3
Online ISBN: 978-3-319-51963-0
eBook Packages: Computer ScienceComputer Science (R0)