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MSO Logic and the Partial Order Semantics of Place/Transition-Nets

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9399)

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.

Keywords

  • Monadic second order logic
  • Petri nets
  • Slice languages

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Notes

  1. 1.

    A Petri net is pure if no transition consumes and produces a token at the same place.

  2. 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. 3.

    A DAG has slicewidth c if it can be decomposed into a unit decomposition over \(\overrightarrow{{\varSigma }}(c,T)\).

References

  1. Avellaneda, F., Morin, R.: Checking partial-order properties of vector addition systems with states. In: Proceedings of ACSD 2013, pp. 100–109. IEEE (2013)

    Google Scholar 

  2. Badouel, E., Darondeau, P.: On the synthesis of general Petri nets. Technical Report PI-1061, IRISA (1996)

    Google Scholar 

  3. 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)

    CrossRef  Google Scholar 

  4. Bauderon, M., Courcelle, B.: Graph expressions and graph rewritings. Math. Syst. Theor. 20(2–3), 83–127 (1987)

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. 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)

    Google Scholar 

  6. Brandenburg, F.-J., Skodinis, K.: Finite graph automata for linear and boundary graph languages. Theor. Comput. Sci. 332(1–3), 199–232 (2005)

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. 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)

    CrossRef  Google Scholar 

  8. Büchi, J.R.: Weak second order arithmetic and finite automata. Z. Math. Logik Grundl. Math. 6, 66–92 (1960)

    MathSciNet  CrossRef  MATH  Google Scholar 

  9. Courcelle, B.: The monadic second-order logic of graphs I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. 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)

    CrossRef  Google Scholar 

  11. 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)

    CrossRef  Google Scholar 

  12. de Oliveira Oliveira, M.: Hasse diagram generators and Petri nets. Fundam. Inf. 105(3), 263–289 (2010)

    MathSciNet  MATH  Google Scholar 

  13. 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)

    CrossRef  Google Scholar 

  14. 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)

    CrossRef  Google Scholar 

  15. Elgot, C.C.: Decision problems of finite automata and related arithmetics. Trans. Am. Math. Soc. 98, 21–52 (1961)

    MathSciNet  CrossRef  MATH  Google Scholar 

  16. Engelfriet, J., et al.: Context-free graph grammars and concatenation of graphs. Acta Inf. 34, 773–803 (1997)

    MathSciNet  CrossRef  MATH  Google Scholar 

  17. 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)

    CrossRef  Google Scholar 

  18. 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)

    CrossRef  Google Scholar 

  19. Petri, C.A.: Fundamentals of a theory of asynchronous information flow. Proc. of IFIP Congr. 62, 166–168 (1962). Munchen

    Google Scholar 

  20. Place, T., Zeitoun, M.: Separating regular languages with first-order logic. In: Proceedings of CSL/LICS, p. 75. ACM (2014)

    Google Scholar 

  21. Thomas, W.: Finite-state recognizability of graph properties. Theor. des Automates et Appl. 172, 147159 (1992)

    Google Scholar 

  22. Thomas, W.: Languages, automata, and logic. Handb. Formal Lang. Theor. 3, 389–455 (1997)

    MathSciNet  CrossRef  Google Scholar 

  23. Vogler, Walter (ed.): Modular Construction and Partial Order Semantics of Petri Nets. LNCS, vol. 625. Springer, Heidelberg (1992)

    MATH  Google Scholar 

  24. 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)

    CrossRef  Google Scholar 

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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).

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Correspondence to Mateus de Oliveira Oliveira .

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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

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  • DOI: https://doi.org/10.1007/978-3-319-25150-9_22

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