Skip to main content
SpringerLink
Log in

Moment Semantics for Reversible Rule-Based Systems

  • Conference paper
  • First Online:
Reversible Computation (RC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9138))

Included in the following conference series:

Abstract

We develop a notion of stochastic rewriting over marked graphs – i.e. directed multigraphs with degree constraints. The approach is based on double-pushout (DPO) graph rewriting. Marked graphs are expressive enough to internalize the ‘no-dangling-edge’ condition inherent in DPO rewriting. Our main result is that the linear span of marked graph occurrence-counting functions – or motif functions – form an algebra which is closed under the infinitesimal generator of (the Markov chain associated with) any such rewriting system. This gives a general procedure to derive the moment semantics of any such rewriting system, as a countable (and recursively enumerable) system of differential equations indexed by motif functions. The differential system describes the time evolution of moments (of any order) of these motif functions under the rewriting system. We illustrate the semantics using the example of preferential attachment networks; a well-studied complex system, which meshes well with our notion of marked graph rewriting. We show how in this case our procedure obtains a finite description of all moments of degree counts for a fixed degree.

This work was sponsored by the European Research Council (ERC) under the grants DOPPLER (587327) and RULE (320823).

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. The Preferential Attachment ODE Generator (2015). https://github.com/sstucki/pa-ode-gen/

  2. Bapodra, M., Heckel, R.: From graph transformations to differential equations. ECEASST 30 (2010)

    Google Scholar 

  3. Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)

    Article  MathSciNet  Google Scholar 

  4. Barabási, A.L., Albert, R., Jeong, H.: Mean-field theory for scale-free random networks. Physica A: Statistical Mechanics and its Applications 272(1), 173–187 (1999)

    Article  Google Scholar 

  5. Chaput, P., Danos, V., Panangaden, P., Plotkin, G.D.: Approximating Markov processes by averaging. Journal of the ACM 61(1), 5 (2014)

    Article  MathSciNet  Google Scholar 

  6. 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)

    Chapter  Google Scholar 

  7. 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 Transformation, pp. 163–245 (1997)

    Google Scholar 

  8. Danos, V., Heindel, T., Honorato-Zimmer, R., Stucki, S.: Approximations for stochastic graph rewriting. In: Merz, S., Pang, J. (eds.) ICFEM 2014. LNCS, vol. 8829, pp. 1–10. Springer, Heidelberg (2014)

    Google Scholar 

  9. Danos, V., Honorato-Zimmer, R., Jaramillo-Riveri, S., Stucki, S.: Deriving rate equations for site graph rewriting systems. In: SASB (2013)

    Google Scholar 

  10. Danos, V., Honorato-Zimmer, R., Jaramillo-Riveri, S., Stucki, S.: Coarse-graining the dynamics of ideal branched polymers. In: Electronic Notes in Theoretical Computer Science, Workshop on Static Analysis and Systems Biology, SASB 2012, Deauville, pp. 47–64, April 2015

    Google Scholar 

  11. Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks with aging of sites. Phys. Rev. E 62, 1842–1845 (2000)

    Google Scholar 

  12. Dorogovtsev, S.N., Mendes, J.F.F., Samukhin, A.N.: Structure of growing networks with preferential linking. Phys. Rev. Lett. 85, 4633–4636 (2000)

    Article  Google Scholar 

  13. Durrett, R., Gleeson, J.P., Lloyd, A.L., Mucha, P.J., Shi, F., Sivakoff, D., Socolar, J.E., Varghese, C.: Graph fission in an evolving voter model. Proceedings of the National Academy of Sciences 109(10), 3682–3687 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ehrig, H., Ehrig, K., Habel, A., Pennemann, K.H.: Theory of constraints and application conditions: From graphs to high-level structures. Fundamenta Informaticae 74(1), 135–166 (2006)

    MATH  MathSciNet  Google Scholar 

  15. Ehrig, H., Heckel, R., Korff, M., Löwe, M., Ribeiro, L., Wagner, A., Corradini, A.: Algebraic approaches to graph transformation. Part II: Single pushout approach and comparison with double pushout approach. In: Rozenberg, G. (ed.) Handbook of Graph Grammars and Computing by Graph Transformation, pp. 247–312. World Scientific, River Edge (1997)

    Chapter  Google Scholar 

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

    Google Scholar 

  17. Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley (1986)

    Google Scholar 

  18. Evans, M.R., Ferrari, P.A., Mallick, K.: Matrix representation of the stationary measure for the multispecies TASEP. Journal of Statistical Physics 135(2), 217–239 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fages, F., Soliman, S.: Formal cell biology in Biocham. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 54–80. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  20. Feret, J., Danos, V., Harmer, R., Krivine, J., Fontana, W.: Internal coarse-graining of molecular systems. PNAS 106(16), 6453–6458 (2009)

    Article  Google Scholar 

  21. Gleeson, J.P.: High-accuracy approximation of binary-state dynamics on networks. Physical Review Letters 107(6), 068701 (2011)

    Article  Google Scholar 

  22. Harmer, R., Danos, V., Feret, J., Krivine, J., Fontana, W.: Intrinsic information carriers in combinatorial dynamical systems. Chaos 20(3) (2010)

    Google Scholar 

  23. Hayman, J., Heindel, T.: Pattern graphs and rule-based models: the semantics of Kappa. In: Pfenning, F. (ed.) FOSSACS 2013 (ETAPS 2013). LNCS, vol. 7794, pp. 1–16. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  24. Heckel, R.: DPO transformation with open maps. In: Ehrig, H., Engels, G., Kreowski, H.-J., Rozenberg, G. (eds.) ICGT 2012. LNCS, vol. 7562, pp. 203–217. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  25. Heckel, R., Lajios, G., Menge, S.: Stochastic graph transformation systems. Fundam. Inform. 74(1), 63–84 (2006)

    MATH  MathSciNet  Google Scholar 

  26. Heckel, R., Wagner, A.: Ensuring consistency of conditional graph grammars - a constructive approach. Electronic Notes in Theoretical Computer Science 2(0), 118–126 (1995)

    Article  Google Scholar 

  27. van Kampen, N.: Stochastic processes in physics and chemistry, 3rd edition, North-Holland (2007)

    Google Scholar 

  28. Lack, S., Sobociński, P.: Adhesive categories. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 273–288. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  29. Lack, S., Sobocinski, P.: Adhesive and quasiadhesive categories. Theoretical Informatics and Applications 39(2), 522–546 (2005)

    MathSciNet  Google Scholar 

  30. Lopez, C.F., Muhlich, J.L., Bachman, J.A., Sorger, P.K.: Programming biological models in Python using PySB. Molecular Systems Biology 9(1) (2013)

    Google Scholar 

  31. Löwe, M.: Algebraic Approach to Single-Pushout Graph Transformation. Theoretical Computer Science 109(1&2), 181–224 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lynch, J.F.: A logical characterization of individual-based models. In: 23rd Annual IEEE Symposium on Logic in Computer Science, LICS 2008, pp. 379–390. IEEE (2008)

    Google Scholar 

  33. Norris, J.R.: Markov chains. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press (1998)

    Google Scholar 

  34. Shkarin, S.A.: Some results on solvability of ordinary linear differential equations in locally convex spaces. Mathematics of the USSR-Sbornik 71(1), 29 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  35. Stukalin, E.B., Phillips III, H., Kolomeisky, A.B.: Coupling of two motor proteins: a new motor can move faster. Physical Review Letters 94(23), 238101 (2005)

    Article  Google Scholar 

  36. Thomas, P., Matuschek, H., Grima, R.: Intrinsic noise analyzer: a software package for the exploration of stochastic biochemical kinetics using the system size expansion. PloS ONE 7(6), e38518 (2012)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département d’Informatique, École Normale Supérieure, Paris, France

    Vincent Danos

  2. School of Informatics, University of Edinburgh, Edinburgh, UK

    Tobias Heindel & Ricardo Honorato-Zimmer

  3. Programming Methods Laboratory, EPFL, Lausanne, Switzerland

    Sandro Stucki

Authors
  1. Vincent Danos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tobias Heindel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ricardo Honorato-Zimmer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sandro Stucki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vincent Danos .

Editor information

Editors and Affiliations

  1. CNRS and Université Paris Diderot, Paris, France

    Jean Krivine

  2. Inria, St. Ismier Cedex, France

    Jean-Bernard Stefani

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Danos, V., Heindel, T., Honorato-Zimmer, R., Stucki, S. (2015). Moment Semantics for Reversible Rule-Based Systems. In: Krivine, J., Stefani, JB. (eds) Reversible Computation. RC 2015. Lecture Notes in Computer Science(), vol 9138. Springer, Cham. https://doi.org/10.1007/978-3-319-20860-2_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-20860-2_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20859-6

  • Online ISBN: 978-3-319-20860-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation