Abstract
In this work, we study the interplay between monadic second order logic and the partial order theory of bounded place/transition-nets. First, we show that the causal behavior of any bounded p / t-net can be compared with respect to inclusion with the set of partial orders specified by a given MSO sentence \(\varphi \). Subsequently, we address the synthesis of Petri nets from MSO specifications. More precisely, we show that given any MSO sentence \(\varphi \), one can automatically construct a bounded Petri net whose behaviour minimally includes the set of partial orders specified by \(\varphi \). Combining this synthesis result with the comparability results we study three problems in the realm of automated correction of faulty Petri nets, and show that these problems are decidable.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A Petri net is pure if no transition consumes and produces a token at the same place.
- 2.
In the execution semantics the fact that an event \(v_1\) is smaller than an event \(v_2\) indicates that \(v_2\) occurs after \(v_1\), but \(v_1\) need not be necessarily the cause of \(v_2\). If N is a net and p is a place that is a linear combination of places in N, then N and \(N\cup \{p\}\) have the same execution behavior. This is not true for the causal semantics.
- 3.
A DAG has slicewidth c if it can be decomposed into a unit decomposition over \(\overrightarrow{{\varSigma }}(c,T)\).
References
Avellaneda, F., Morin, R.: Checking partial-order properties of vector addition systems with states. In: Proceedings of ACSD 2013, pp. 100–109. IEEE (2013)
Badouel, E., Darondeau, P.: On the synthesis of general Petri nets. Technical Report PI-1061, IRISA (1996)
Badouel, E., Darondeau, P.: Theory of regions. In: Reisig, W., Rozenberg, G. (eds.) Lectures on Petri Nets I: Basic Models. LNCS, vol. 1491, pp. 529–586. Springer, Heidelberg (1998)
Bauderon, M., Courcelle, B.: Graph expressions and graph rewritings. Math. Syst. Theor. 20(2–3), 83–127 (1987)
Bergenthum, R., Desel, J., Lorenz, R., Mauser, S.: Synthesis of Petri nets from infinite partial languages. In: Proceedings of ACSD 2008, pp. 170–179. IEEE (2008)
Brandenburg, F.-J., Skodinis, K.: Finite graph automata for linear and boundary graph languages. Theor. Comput. Sci. 332(1–3), 199–232 (2005)
Bruggink, H.S., König, B.: On the recognizability of arrow and graph languages. In: Ehrig, H., Heckel, R., Rozenberg, G., Taentzer, G. (eds.) Graph Transformations. LNCS, vol. 5214, pp. 336–350. Springer, Heidelberg (2008)
Büchi, J.R.: Weak second order arithmetic and finite automata. Z. Math. Logik Grundl. Math. 6, 66–92 (1960)
Courcelle, B.: The monadic second-order logic of graphs I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)
Darondeau, P.: Deriving unbounded Petri nets from formal languages. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR’98 Concurrency Theory. LNCS, vol. 1466, pp. 533–548. Springer, Heidelberg (1998)
Darondeau, P.: Region based synthesis of P/T-nets and its potential applications. In: Nielsen, M., Simpson, D. (eds.) ICATPN 2000. LNCS, vol. 1825, p. 16. Springer, Heidelberg (2000)
de Oliveira Oliveira, M.: Hasse diagram generators and Petri nets. Fundam. Inf. 105(3), 263–289 (2010)
de Oliveira Oliveira, M.: Canonizable partial order generators. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 445–457. Springer, Heidelberg (2012)
de Oliveira Oliveira, M.: Subgraphs satisfying MSO properties on z-topologically orderable digraphs. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 123–136. Springer, Heidelberg (2013)
Elgot, C.C.: Decision problems of finite automata and related arithmetics. Trans. Am. Math. Soc. 98, 21–52 (1961)
Engelfriet, J., et al.: Context-free graph grammars and concatenation of graphs. Acta Inf. 34, 773–803 (1997)
Goltz, U., Reisig, W.: Processes of place/transition-nets. In: Diaz, J. (ed.) Automata, Languages and Programming. LNCS, vol. 154, pp. 264–277. Springer, Heidelberg (1983)
Madhusudan, P., Thiagarajan, P.S.: Distributed controller synthesis for local specifications. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, p. 396. Springer, Heidelberg (2001)
Petri, C.A.: Fundamentals of a theory of asynchronous information flow. Proc. of IFIP Congr. 62, 166–168 (1962). Munchen
Place, T., Zeitoun, M.: Separating regular languages with first-order logic. In: Proceedings of CSL/LICS, p. 75. ACM (2014)
Thomas, W.: Finite-state recognizability of graph properties. Theor. des Automates et Appl. 172, 147159 (1992)
Thomas, W.: Languages, automata, and logic. Handb. Formal Lang. Theor. 3, 389–455 (1997)
Vogler, Walter (ed.): Modular Construction and Partial Order Semantics of Petri Nets. LNCS, vol. 625. Springer, Heidelberg (1992)
von Essen, C., Jobstmann, B.: Program repair without regret. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 896–911. Springer, Heidelberg (2013)
Acknowledgements
The author gratefully acknowledges financial support from the European Research Council, ERC grant agreement 339691, within the context of the project Feasibility, Logic and Randomness (FEALORA).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
de Oliveira Oliveira, M. (2015). MSO Logic and the Partial Order Semantics of Place/Transition-Nets. In: Leucker, M., Rueda, C., Valencia, F. (eds) Theoretical Aspects of Computing - ICTAC 2015. ICTAC 2015. Lecture Notes in Computer Science(), vol 9399. Springer, Cham. https://doi.org/10.1007/978-3-319-25150-9_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-25150-9_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25149-3
Online ISBN: 978-3-319-25150-9
eBook Packages: Computer ScienceComputer Science (R0)