International Colloquium on Theoretical Aspects of Computing

Theoretical Aspects of Computing - ICTAC 2015 pp 368-387 | Cite as

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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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 

Notes

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).

References

  1. 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. 2.
    Badouel, E., Darondeau, P.: On the synthesis of general Petri nets. Technical Report PI-1061, IRISA (1996)Google Scholar
  3. 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)CrossRefGoogle Scholar
  4. 4.
    Bauderon, M., Courcelle, B.: Graph expressions and graph rewritings. Math. Syst. Theor. 20(2–3), 83–127 (1987)MathSciNetCrossRefMATHGoogle Scholar
  5. 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. 6.
    Brandenburg, F.-J., Skodinis, K.: Finite graph automata for linear and boundary graph languages. Theor. Comput. Sci. 332(1–3), 199–232 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 8.
    Büchi, J.R.: Weak second order arithmetic and finite automata. Z. Math. Logik Grundl. Math. 6, 66–92 (1960)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Courcelle, B.: The monadic second-order logic of graphs I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. 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) CrossRefGoogle Scholar
  12. 12.
    de Oliveira Oliveira, M.: Hasse diagram generators and Petri nets. Fundam. Inf. 105(3), 263–289 (2010)MathSciNetMATHGoogle Scholar
  13. 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) CrossRefGoogle Scholar
  14. 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) CrossRefGoogle Scholar
  15. 15.
    Elgot, C.C.: Decision problems of finite automata and related arithmetics. Trans. Am. Math. Soc. 98, 21–52 (1961)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Engelfriet, J., et al.: Context-free graph grammars and concatenation of graphs. Acta Inf. 34, 773–803 (1997)MathSciNetCrossRefMATHGoogle Scholar
  17. 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) CrossRefGoogle Scholar
  18. 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) CrossRefGoogle Scholar
  19. 19.
    Petri, C.A.: Fundamentals of a theory of asynchronous information flow. Proc. of IFIP Congr. 62, 166–168 (1962). MunchenGoogle Scholar
  20. 20.
    Place, T., Zeitoun, M.: Separating regular languages with first-order logic. In: Proceedings of CSL/LICS, p. 75. ACM (2014)Google Scholar
  21. 21.
    Thomas, W.: Finite-state recognizability of graph properties. Theor. des Automates et Appl. 172, 147159 (1992)Google Scholar
  22. 22.
    Thomas, W.: Languages, automata, and logic. Handb. Formal Lang. Theor. 3, 389–455 (1997)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Vogler, Walter (ed.): Modular Construction and Partial Order Semantics of Petri Nets. LNCS, vol. 625. Springer, Heidelberg (1992) MATHGoogle Scholar
  24. 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) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of MathematicsAcademy of Sciences of the Czech RepublicPragueCzech Republic

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