Skip to main content
SpringerLink
Log in
Book cover

Federation of International Conferences on Software Technologies: Applications and Foundations

STAF 2016: Software Technologies: Applications and Foundations pp 101–111Cite as

On the Definition of Parallel Independence in the Algebraic Approaches to Graph Transformation

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9946))

Abstract

Parallel independence between transformation steps is a basic and well-understood notion of the algebraic approaches to graph transformation, and typically guarantees that the two steps can be applied in any order obtaining the same resulting graph, up to isomorphism. The concept has been redefined for several algebraic approaches as variations of a classical “algebraic” condition, requiring that each matching morphism factorizes through the context graphs of the other transformation step. However, looking at some classical papers on the double-pushout approach, one finds that the original definition of parallel independence was formulated in set-theoretical terms, requiring that the intersection of the images of the two left-hand sides in the host graph is contained in the intersection of the two interface graphs. The relationship between this definition and the standard algebraic one is discussed in this position paper, both in the case of left-linear and non-left-linear rules.

This is a preview of subscription content, 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

Learn about institutional subscriptions

Notes

  1. 1.

    The definitions of parallel independence based on Conditions 1 or 2 date back to the mid seventies of last century. Besides of [8] they also appear in [12]. In [17] parallel independence is defined set-theoretically (see diagram (3)) as \(m_1(L_1) \subseteq g_2(D_2) \wedge m_2(L_2) \subseteq g_1(D_1)\), a variant of Condition 2.

  2. 2.

    The conditions for parallel independence for non-linear rules in the context of RSqPO, presented in [7], are even stronger. First, besides Condition 2, making reference to diagram (4) it is required that \((\rho _2, h_1 \circ m_{2d})\) and \((\rho _1, h_2 \circ m_{1d})\) are (RSqPO-) redexes. Furthermore, and more interestingly for the present discussion, since productions can also be non-right-linear, besides Condition 4 it is also required that the squares in (9) are pullbacks.

    (9)

    .

References

  1. Cockett, J., Lack, S.: Restriction categories II: partial map classification. Theor. Comput. Sci. 294(1–2), 61–102 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Corradini, A., Duval, D., Echahed, R., Prost, F., Ribeiro, L.: AGREE – algebraic graph rewriting with controlled embedding. In: Parisi-Presicce, F., Westfechtel, B. (eds.) ICGT 2015. LNCS, vol. 9151, pp. 35–51. Springer, Heidelberg (2015). doi:10.1007/978-3-319-21145-9_3

    Chapter  Google Scholar 

  3. Corradini, A., Duval, D., Prost, F., Ribeiro, L.: Parallelism in AGREE transformations. In: Echahed, R., Minas, M. (eds.) ICGT 2016. LNCS, vol. 9761, pp. 37–53. Springer, Heidelberg (2016). doi:10.1007/978-3-319-40530-8_3

    Chapter  Google Scholar 

  4. Corradini, A., Ehrig, H., Löwe, M., Montanari, U., Rossi, F.: Abstract graph derivations in the double pushout approach. In: Schneider, H.J., Ehrig, H. (eds.) Graph Transformations in Computer Science. LNCS, vol. 776, pp. 86–103. Springer, Heidelberg (1994). doi:10.1007/3-540-57787-4_6

    Chapter  Google Scholar 

  5. Corradini, A., Heindel, T., Hermann, F., König, B.: Sesqui-Pushout rewriting. In: Corradini, A., Ehrig, H., Montanari, U., Ribeiro, L., Rozenberg, G. (eds.) ICGT 2006. LNCS, vol. 4178, pp. 30–45. Springer, Heidelberg (2006). doi:10.1007/11841883_4

    Chapter  Google Scholar 

  6. Corradini, A., Montanari, U., Rossi, F., Ehrig, H., Heckel, R., Löwe, M.: Algebraic approaches to graph transformation - part I: basic concepts and double pushout approach. In: Handbook of Graph Grammars and Computing by Graph Transformations, Foundations, vol. 1, pp. 163–246 (1997)

    Google Scholar 

  7. Danos, V., Heindel, T., Honorato-Zimmer, R., Stucki, S.: Reversible Sesqui-Pushout rewriting. In: Giese, H., König, B. (eds.) ICGT 2014. LNCS, vol. 8571, pp. 161–176. Springer, Heidelberg (2014). doi:10.1007/978-3-319-09108-2_11

    Google Scholar 

  8. Ehrig, H., Rosen, B.: Commutativity of Independent Transformations on Complex Objects. IBM Thomas J. Watson Research Division (1976)

    Google Scholar 

  9. Ehrig, H.: Introduction to the algebraic theory of graph grammars (a survey). In: Claus, V., Ehrig, H., Rozenberg, G. (eds.) Graph Grammars 1978. LNCS, vol. 73, pp. 1–69. Springer, Heidelberg (1979). doi:10.1007/BFb0025714

    Chapter  Google Scholar 

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

    MATH  Google Scholar 

  11. Ehrig, H., Habel, A., Kreowski, H., Parisi-Presicce, F.: Parallelism and concurrency in high-level replacement systems. Math. Struct. Comput. Sci. 1(3), 361–404 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ehrig, H., Kreowski, H.-J.: Parallelism of manipulations in multidimensional information structures. In: Mazurkiewicz, A. (ed.) MFCS 1976. LNCS, vol. 45, pp. 284–293. Springer, Heidelberg (1976). doi:10.1007/3-540-07854-1_188

    Chapter  Google Scholar 

  13. Ehrig, H., Pfender, M., Schneider, H.J.: Graph-grammars: an algebraic approach. In: 14th Annual Symposium on Switching and Automata Theory, Iowa City, Iowa, USA, 15–17 October 1973, pp. 167–180. IEEE Computer Society (1973)

    Google Scholar 

  14. Kreowski, H.: Manipulation von Graphmanipulationen. Ph.D. thesis, Technische Universität, Berlin (1977)

    Google Scholar 

  15. Lack, S., Sobocinski, P.: Adhesive and quasiadhesive categories. Theor. Inf. Appl. 39(3), 511–545 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Löwe, M.: Graph rewriting in span-categories. In: Ehrig, H., Rensink, A., Rozenberg, G., Schürr, A. (eds.) ICGT 2010. LNCS, vol. 6372, pp. 218–233. Springer, Heidelberg (2010). doi:10.1007/978-3-642-15928-2_15

    Chapter  Google Scholar 

  17. Rosen, B.K.: A Church-Rosser theorem for graph grammars. ACM SIGACT News 7(3), 26–31 (1975)

    Article  Google Scholar 

Download references

Acknowledgments

The idea of spelling out the relationship between the standard algebraic and the pullback-based definitions of parallel independence maturated during stimulating discussions with Dominque Duval, Frédéric Prost, Rachid Echahed and Leila Ribeiro, during the work on the AGREE approach to GT. Hans-Jörg Kreowski provided me some references to the early literature on parallel independence. During the workshop where this work was presented, Michael Löwe suggested several technical improvements, including a new version of the last part of the proof of Proposition 2 that does not need partial maps classifiers: this will be presented in a forthcoming report.

Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Università di Pisa, Pisa, Italy

    Andrea Corradini

Authors
  1. Andrea Corradini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrea Corradini .

Editor information

Editors and Affiliations

  1. Dipartimento di Informatica, Universita di Pisa, Pisa, Italy

    Paolo Milazzo

  2. Budapest University of Technology and Economics, Budapest, Hungary

    Dániel Varró

  3. Vienna University of Technology, Vienna, Austria

    Manuel Wimmer

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this paper

Cite this paper

Corradini, A. (2016). On the Definition of Parallel Independence in the Algebraic Approaches to Graph Transformation. In: Milazzo, P., Varró, D., Wimmer, M. (eds) Software Technologies: Applications and Foundations. STAF 2016. Lecture Notes in Computer Science(), vol 9946. Springer, Cham. https://doi.org/10.1007/978-3-319-50230-4_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-50230-4_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50229-8

  • Online ISBN: 978-3-319-50230-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation