Skip to main content
SpringerLink
Log in

Towards a Navigational Logic for Graphical Structures

  • Chapter
  • First Online:
Book cover Graph Transformation, Specifications, and Nets

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10800))

Abstract

One of the main advantages of the Logic of Nested Conditions, defined by Habel and Pennemann, for reasoning about graphs, is its generality: this logic can be used in the framework of many classes of graphs and graphical structures. It is enough that the category of these structures satisfies certain basic conditions.

In a previous paper [14], we extended this logic to be able to deal with graph properties including paths, but this extension was only defined for the category of untyped directed graphs. In addition it seemed difficult to talk about paths abstractly, that is, independently of the given category of graphical structures. In this paper we approach this problem. In particular, given an arbitrary category of graphical structures, we assume that for every object of this category there is an associated edge relation that can be used to define a path relation. Moreover, we consider that edges have some kind of labels and paths can be specified by associating them to a set of label sequences. Then, after the presentation of that general framework, we show how it can be applied to several classes of graphs. Moreover, we present a set of sound inference rules for reasoning in the logic.

This work has been partially supported by funds from the Spanish Ministry for Economy and Competitiveness (MINECO) and the European Union (FEDER funds) under grant COMMAS (ref. TIN2013-46181-C2-1-R, TIN2013-46181-C2-2-R) and from the Basque Project GIU15/30, and grant UFI11/45.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    But in Petri Nets we may also consider that both places and transitions play the role of the nodes in a graph and that the edges in a Petri Net are the arrows in the graphical representation of the net going from places to transitions or from transitions to places.

  2. 2.

    Even if we may consider that empty paths are not really paths, assuming that every node is connected to itself through an empty path provides some technical simplifications.

  3. 3.

    That is \(\underline{\mathtt{Struct}} \) is embedded in \(\underline{\mathtt{Patterns}}\) via the functor Pattern.

  4. 4.

    Notice that \(L_1\) or \(L_2\) may just consist of the empty string, in which case \(n=n_1\) or \(n'=n_2\), respectively.

  5. 5.

    In particular, we may consider that in a grounded symbolic graph G we have \(\Phi _G \equiv (x_1 = v_1 \wedge \dots \wedge x_k = v_k)\), for some values \(v_1, \dots , v_k\) .

  6. 6.

    In [18, 19] it is proved that we can transform any condition into a clausal form.

  7. 7.

    Split Introduction rule may seem not very useful, however in [14] it was needed to achieve completeness. A morphism \(a:P\rightarrow Q\) is a split mono if it has a left inverse, that is, if there is a morphism \(a^{-1}\) such that \(a^{-1}\circ a =id_P\).

References

  1. Barceló, P.: Querying graph databases. In: Proceedings of the 32nd ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2013, New York, 22–27 June 2013, pp. 175–188 (2013), https://doi.org/10.1145/2463664.2465216

  2. Barceló, P., Libkin, L., Reutter, J.L.: Querying graph patterns. In: Proceedings of the 30th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2011, 12–16 June 2011, Athens, pp. 199–210 (2011). https://doi.org/10.1145/1989284.1989307

  3. Courcelle, B.: The expression of graph properties and graph transformations in monadic second-order logic. In: Rozenberg, G. (ed.) Handbook of Graph Grammars, pp. 313–400. World Scientific, River Edge (1997). https://dl.acm.org/citation.cfm?id=278918.278932

    Google Scholar 

  4. Cruz, I.F., Mendelzon, A.O., Wood, P.T.: A graphical query language supporting recursion. In: Proceedings SIGMOD 1987 Annual Conference, San Francisco, 27–29 May 1987, pp. 323–330 (1987). https://doi.org/10.1145/38713.38749

  5. Ehrig, H., Ehrig, K., Prange, U., Taentzer, G.: Fundamentals of Algebraic Graph Transformation. Springer, Heidelberg (2006)

    MATH  Google Scholar 

  6. Ehrig, H., Habel, A.: Graph grammars with application conditions. In: Rozenberg, G., Salomaa, A. (eds.) The Book of L, pp. 87–100. Springer, Heidelberg (1986)

    Chapter  Google Scholar 

  7. Flick, N.E.: On correctness of graph programs relative to recursively nested conditions. In: Workshop on Graph Computation Models (GCM 2015), vol. 1403, pp. 97–112 (2015). http://ceur-ws.org/Vol-1403/

  8. Habel, A., Heckel, R., Taentzer, G.: Graph grammars with negative application conditions. Fundamenta Informaticae 26, 287–313 (1996). https://doi.org/10.3233/FI-1996-263404

    MathSciNet  MATH  Google Scholar 

  9. Habel, A., Pennemann, K.H.: Correctness of high-level transformation systems relative to nested conditions. Math. Struct. Comput. Sci. 19(2), 245–296 (2009). https://doi.org/10.1017/S0960129508007202

    Article  MathSciNet  MATH  Google Scholar 

  10. Habel, A., Radke, H.: Expressiveness of graph conditions with variables. ECEASST, 30 (2010) https://doi.org/10.14279/tuj.eceasst.30.404

  11. Heckel, R., Wagner, A.: Ensuring consistency of conditional graph rewriting - a constructive approach. Electr. Notes Theor. Comput. Sci. 2, 118–126 (1995). https://doi.org/10.1016/S1571-0661(05)80188-4

    Article  MATH  Google Scholar 

  12. Lambers, L., Orejas, F.: Tableau-Based Reasoning for Graph Properties. In: Giese, H., König, B. (eds.) ICGT 2014. LNCS, vol. 8571, pp. 17–32. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09108-2_2

    Google Scholar 

  13. Libkin, L., Vrgoc, D.: Regular path queries on graphs with data. In: 15th International Conference on Database Theory, ICDT 2012, Berlin, 26–29 March 2012, pp. 74–85 (2012). https://doi.org/10.1145/2274576.2274585

  14. Navarro, M., Orejas, F., Pino, E., Lambers, L.: A logic of graph conditions extended with paths. In: Workshop on Graph Computation Models (GCM 2016), Vienna (2016)

    Google Scholar 

  15. Orejas, F.: Attributed graph constraints. In: Ehrig, H., Heckel, R., Rozenberg, G., Taentzer, G. (eds.) ICGT 2008. LNCS, vol. 5214, pp. 274–288. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-87405-8_19

    Chapter  Google Scholar 

  16. Orejas, F.: Symbolic graphs for attributed graph constraints. J. Symb. Comput. 46(3), 294–315 (2011). https://doi.org/10.1016/j.jsc.2010.09.009

    Article  MathSciNet  MATH  Google Scholar 

  17. Orejas, F., Ehrig, H., Prange, U.: Reasoning with graph constraints. Formal Asp. Comput. 22(3–4), 385–422 (2010). https://doi.org/10.1007/s00165-009-0116-9

    Article  MATH  Google Scholar 

  18. Pennemann, K.-H.: Resolution-like theorem proving for high-level conditions. In: Ehrig, H., Heckel, R., Rozenberg, G., Taentzer, G. (eds.) ICGT 2008. LNCS, vol. 5214, pp. 289–304. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-87405-8_20

    Chapter  Google Scholar 

  19. Pennemann, K.H.: Development of Correct Graph Transformation Systems, PhD Thesis. Department of Computing Science, University of Oldenburg (2009)

    Google Scholar 

  20. Poskitt, C.M., Plump, D.: Verifying monadic second-order properties of graph programs. In: Giese, H., König, B. (eds.) ICGT 2014. LNCS, vol. 8571, pp. 33–48. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09108-2_3

    Google Scholar 

  21. Rensink, A.: Representing first-order logic using graphs. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenberg, G. (eds.) ICGT 2004. LNCS, vol. 3256, pp. 319–335. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30203-2_23

    Chapter  Google Scholar 

  22. Trakhtenbrot, B.A.: The impossibility of an algorithm for the decision problem on finite classes (In Russian). Doklady Akademii Nauk SSSR, 70:569–572, 1950. English translation. In: Nine Papers on Logic and Quantum Electrodynamics, AMS Transl. Ser. 2(23), 1–5 (1963)

    Google Scholar 

  23. Wood, P.T.: Query languages for graph databases. SIGMOD Rec. 41(1), 50–60 (2012). https://doi.org/10.1145/2206869.2206879

    Article  Google Scholar 

Download references

Acknowledgements

We are grateful to the anonymous reviewers for their comments that have contributed to improve the paper.

Author information

Authors and Affiliations

  1. Hasso Plattner Institut, University of Potsdam, Potsdam, Germany

    Leen Lambers

  2. Universidad del País Vasco (UPV/EHU), San Sebastián, Spain

    Marisa Navarro

  3. Universitat Politècnica de Catalunya, Barcelona, Spain

    Fernando Orejas & Elvira Pino

Authors
  1. Leen Lambers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marisa Navarro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fernando Orejas
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Elvira Pino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elvira Pino .

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Leicester, Leicester, United Kingdom

    Reiko Heckel

  2. Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Gabriele Taentzer

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Lambers, L., Navarro, M., Orejas, F., Pino, E. (2018). Towards a Navigational Logic for Graphical Structures. In: Heckel, R., Taentzer, G. (eds) Graph Transformation, Specifications, and Nets. Lecture Notes in Computer Science(), vol 10800. Springer, Cham. https://doi.org/10.1007/978-3-319-75396-6_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-75396-6_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-75395-9

  • Online ISBN: 978-3-319-75396-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation