Structured Operational Semantics for Graph Rewriting

📝 Abstract

Process calculi and graph transformation systems provide models of reactive systems with labelled transition semantics. While the semantics for process calculi is compositional, this is not the case for graph transformation systems, in general. Hence, the goal of this article is to obtain a compositional semantics for graph transformation system in analogy to the structural operational semantics (SOS) for Milner’s Calculus of Communicating Systems (CCS). The paper introduces an SOS style axiomatization of the standard labelled transition semantics for graph transformation systems. The first result is its equivalence with the so-called Borrowed Context technique. Unfortunately, the axiomatization is not compositional in the expected manner as no rule captures “internal” communication of sub-systems. The main result states that such a rule is derivable if the given graph transformation system enjoys a certain property, which we call “complementarity of actions”. Archetypal examples of such systems are interaction nets. We also discuss problems that arise if “complementarity of actions” is violated.

💡 Analysis

Process calculi and graph transformation systems provide models of reactive systems with labelled transition semantics. While the semantics for process calculi is compositional, this is not the case for graph transformation systems, in general. Hence, the goal of this article is to obtain a compositional semantics for graph transformation system in analogy to the structural operational semantics (SOS) for Milner’s Calculus of Communicating Systems (CCS). The paper introduces an SOS style axiomatization of the standard labelled transition semantics for graph transformation systems. The first result is its equivalence with the so-called Borrowed Context technique. Unfortunately, the axiomatization is not compositional in the expected manner as no rule captures “internal” communication of sub-systems. The main result states that such a rule is derivable if the given graph transformation system enjoys a certain property, which we call “complementarity of actions”. Archetypal examples of such systems are interaction nets. We also discuss problems that arise if “complementarity of actions” is violated.

📄 Content

Bliudze, S., Bruni, R., Carbone, M., Silva, A. (Eds.); ICE 2011 EPTCS 59, 2011, pp. 37–51, doi:10.4204/EPTCS.59.4 Structured Operational Semantics for Graph Rewriting∗ Andrei Dorman Dip. di Filosofia, Universit`a Roma Tre LIPN – UMR 7030, Universit´e Paris 13 andrei.dorman@lipn.univ-paris13.fr Tobias Heindel LIPN – UMR 7030, Universit´e Paris 13 tobias.heindel@lipn.univ-paris13.fr Process calculi and graph transformation systems provide models of reactive systems with labelled transition semantics. While the semantics for process calculi is compositional, this is not the case for graph transformation systems, in general. Hence, the goal of this article is to obtain a compositional semantics for graph transformation system in analogy to the structural operational semantics (SOS) for Milner’s Calculus of Communicating Systems (CCS). The paper introduces an SOS style axiomatization of the standard labelled transition semantics for graph transformation systems. The first result is its equivalence with the so-called Borrowed Context technique. Unfortunately, the axiomatization is not compositional in the expected manner as no rule captures “internal” communication of sub-systems. The main result states that such a rule is derivable if the given graph transformation system enjoys a certain property, which we call “complementarity of actions”. Archetypal examples of such systems are interaction nets. We also discuss problems that arise if “complementarity of actions” is violated. Key words: process calculi, graph transformation, structural operational semantics, compositional methods 1 Introduction Process calculi remain one of the central tools for the description of interactive systems. The archetypal example of process calculi are Milner’s π-calculus and the even more basic calculus of communication systems (CCS). The semantics of these calculi is given by labelled transition systems (LTS), which in fact can be given as a structural operational semantics (SOS). An advantage of SOS is their potential for combination with compositional methods for the verification of systems (see e.g. [17]). Fruitful inspiration for the development of LTS semantics for other “non-standard” process calculi originates from the area of graph transformation where techniques for the derivation of LTS semantics from “reaction rules” have been developed [16, 7]. The strongest point of these techniques is the context independence of the resulting behavioral equivalences, which are in fact congruences. Moreover, these techniques have lead to original LTS-semantics for the ambient calculus [15, 3], which are also given as SOS systems. Already in the special case of ambients, the SOS-style presentation goes beyond the standard techniques of label derivation in [16, 7]. An open research challenge is the development of a general technique for the canonical derivation of SOS-style LTS-semantics. The problem is the “monolithic” character of the standard LTS for graph transformation systems. In the present paper, we set out to develop a partial solution to the problem for what we shall call CCS-like graph transformation systems. The main idea is to develop an analogy to CCS where each action α has a co-action α that can synchronize to obtain a silent transition; this is the so-called communication rule. In analogy, one can restrict attention to graph transformation systems with rules that allow to assign to each (hyper-)edge a unique co-edge. Natural examples of such systems are interaction nets as ∗This work was partially supported by grants from Agence Nationale de la Recherche, ref. ANR-08-BLANC-0211-01 (COMPLICE project) and ref. ANR-09-BLAN-0169 (PANDA project). 38 SOS for Graph Rewriting introduced by Lafont [11, 1]. In fact, one of the motivations of the paper is to derive SOS semantics for interaction nets. Structure and contents of the paper We first introduce the very essentials of graph transformation and the so-called Borrowed Context (BC) technique [7] for the special case of (hyper-)graph transformation in Section 2. To make the analogy between CCS and BC as formal as possible, we introduce the system SOSBC in Section 3, which is meant to provide the uninitiated reader with a new perspective on the BC technique. Moreover, the system SOSBC emphasizes the “local” character of graph transformations as every transition can be decomposed into a “basic” action in some context. In particular, we do not have any counterpart to the communication rule of CCS, which shall be addressed in Section 4. We illustrate why it is not evident when and how two labeled transitions of two states that share their interface can be combined into a single synchronized action. However, we will be able to describe sufficient conditions on (hyper-)graph transformation systems that allow to derive the counterpart of the communication rule of CCS in the system SOSBC. Systems of this kind have a natural notion of “complementarity of actions” in the LTS. 2 Preliminaries

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