Skip to main content

Reachability in Graph Transformation Systems and Slice Languages

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9151)

Abstract

In this work we show that the reachability problem for graph transformation systems is in the complexity class XP when parameterized with respect to the depth of derivations and the cutwidth of the source graph. More precisely, we show that for any set \(\mathcal {R}\) of graph transformation rules, one can determine in time \(f(c,d)\cdot |G|\cdot |H|^{g(c,d)}\) whether a graph G of cutwidth c can be transformed into a graph H in depth at most d by the application of graph transformation rules from \(\mathcal {R}\). In particular, our algorithm runs in polynomial time when c and d are constants. On the other hand, we show that the problem becomes NP-hard if we allow \(c=O(|G|)\) and \(d=5\). In the case in which all transformation rules are monotone we get an algorithm running in time \(f(c,d)\cdot |G|^{O(c)}\cdot |H|\). To prove our main theorems we will establish an interesting connection between graph transformation systems and regular slice languages. More precisely, we show that if \(\mathcal {A}\) is a slice automaton representing a set \({\mathcal {L}}_{{\mathcal {G}}}(\mathcal {A})\) of graphs, then one can construct in time linear in \(|\mathcal {A}|\) a slice automaton \(\mathcal {N}(\mathcal {A})\) representing the set of all graphs that can be obtained from graphs in \({\mathcal {L}}_{{\mathcal {G}}}(\mathcal {A})\) by the application of one layer of transformation rules in \(\mathcal {R}\).

Keywords

  • Graph transformation systems
  • Reachability
  • Slice languages

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-21145-9_8
  • Chapter length: 17 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   44.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-21145-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   59.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.

Notes

  1. 1.

    XP is the class of problems that can be solved in time \(f(\overline{p})\cdot n^{g(\overline{p})}\) where n is the size of the input, f and g are computable functions, and \(\overline{p}\) is a list of parameters.

References

  1. Andries, M., Engels, G., Habel, A., Hoffmann, B., Kreowski, H.-J., Kuske, S., Plump, D., Schürr, A., Taentzer, G.: Graph transformation for specification and programming. Sci. Comput. Program. 34(1), 1–54 (1999)

    CrossRef  MATH  Google Scholar 

  2. Baldan, P., Corradini, A., König, B.: A framework for the verification of infinite-state graph transformation systems. Inf. Comput. 206(7), 869–907 (2008)

    CrossRef  MATH  Google Scholar 

  3. Baresi, L., Heckel, R.: Tutorial introduction to graph transformation: a software engineering perspective. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenberg, G. (eds.) ICGT 2004. LNCS, vol. 3256, pp. 431–433. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

  5. Bertrand, N., Delzanno, G., König, B., Sangnier, A., Stückrath, J.: On the decidability status of reachability and coverability in graph transformation systems. In: Rewriting Techniques and Applications, vol. 12, pp. 101–116 (2012)

    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.) ICGT 2008. LNCS, vol. 5214, pp. 336–350. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  8. Corradini, A., Montanari, U., Rossi, F.: Graph processes. Fundamenta Informaticae 26(3), 241–265 (1996)

    MathSciNet  MATH  Google Scholar 

  9. de Oliveira Oliveira, M.: Hasse diagram generators and petri nets. Fundamenta Informaticae 105(3), 263–289 (2010)

    MathSciNet  MATH  Google Scholar 

  10. 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 

  11. 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 

  12. Downey, R.G., Fellows, M.R.: Fixed parameter tractability and completeness. In: Complexity Theory: Current Research, pp. 191–225 (1992)

    Google Scholar 

  13. Ehrig, H., Ehrig, K., Prange, U., Taentzer, G.: Fundamentals of Algebraic Graph Transformation. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)

    Google Scholar 

  14. Ehrig, H., Pfender, M., Schneider, H.J.: Graph-grammars: an algebraic approach. In: Switching and Automata Theory, pp. 167–180. IEEE Computer Society (1973)

    Google Scholar 

  15. Ehrig, H., Rosen, B.K.: Parallelism and concurrency of graph manipulations. Theoret. Comput. Sci. 11(3), 247–275 (1980)

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

  17. Poskitt, C.M., Plump, D.: Verifying total correctness of graph programs. Electron. Commun. EASST 61, 1–20 (2013)

    Google Scholar 

  18. Rensink, A.: Explicit state model checking for graph grammars. In: Degano, P., De Nicola, R., Meseguer, J. (eds.) Concurrency, Graphs and Models. LNCS, vol. 5065, pp. 114–132. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  19. Rozenberg, G., Ehrig, H.: Handbook of graph grammars and computing by graph transformation, vol. 1. World Scientific Publishing, Singapore (1999)

    Google Scholar 

  20. Thilikos, D.M., Serna, M., Bodlaender, H.L.: Cutwidth I: a linear time fixed parameter algorithm. J. Algorithms 56(1), 1–24 (2005)

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Thomas, W.: Finite-state recognizability of graph properties. Theorie des Automates et Applications 176, 147–159 (1992)

    Google Scholar 

Download references

Acknowledgements

I gratefully acknowledge 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

Authors

Corresponding author

Correspondence to Mateus de Oliveira Oliveira .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

de Oliveira Oliveira, M. (2015). Reachability in Graph Transformation Systems and Slice Languages. In: Parisi-Presicce, F., Westfechtel, B. (eds) Graph Transformation. ICGT 2015. Lecture Notes in Computer Science(), vol 9151. Springer, Cham. https://doi.org/10.1007/978-3-319-21145-9_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-21145-9_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21144-2

  • Online ISBN: 978-3-319-21145-9

  • eBook Packages: Computer ScienceComputer Science (R0)