Moment Semantics for Reversible Rule-Based Systems

  • Vincent DanosEmail author
  • Tobias Heindel
  • Ricardo Honorato-Zimmer
  • Sandro Stucki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9138)


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.


Stochastic processes Moment semantics Reversible Computing Graph rewriting Rule-based systems 


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Vincent Danos
    • 1
    Email author
  • Tobias Heindel
    • 2
  • Ricardo Honorato-Zimmer
    • 2
  • Sandro Stucki
    • 3
  1. 1.Département d’InformatiqueÉcole Normale SupérieureParisFrance
  2. 2.School of InformaticsUniversity of EdinburghEdinburghUK
  3. 3.Programming Methods LaboratoryEPFLLausanneSwitzerland

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