An Operational Semantics of Graph Transformation Systems Using Symmetric Nets

Abstract

Graph transformation systems (GTS) have been successfully proposed as a general, theoretically sound model for concurrency. Petri nets (PN), on the other side, are a central and intuitive formalism for concurrent or distributed systems, well supported by a number of analysis techniques/tools. Some PN classes have been shown to be instances of GTS. In this paper, we change perspective presenting an operational semantics of GTS in terms of Symmetric Nets, a well-known class of Coloured Petri nets featuring a structured syntax that outlines model symmetries. Some practical exploitations of the proposed operational semantics are discussed. In particular, a recently developed structural calculus for SN is used to validate graph rewriting rules in a symbolic way.An Operational Semantics of GTS the GreatSPN package [1]; an operational interleaving semantics for GTS is provided in a natural way building the state-transition system of a SN; a compact state-transition system -called symbolic reachability graph [7], in which states (markings) representing isomorphic graphs are folded, can be directly derived once an initial symbolic graph encoding is set; some recent advances in SN (symbolic) structural analysis [4], [3], implemented in the SNExpression tool (www.di.unito.it/~depierro/SNex) may be exploited to check some conditions ensuring rule well-definiteness, validate rules, and verify their potential concurrency; in particular, a fully automated calculus of symbolic structural relations in SN models may be profitably used. All these concepts are instantiated on a few, though significant, examples of graph rewriting rules, and a simple GTS. All the examples used in the paper are available in GreatSPN format at https://github.com/lgcapra/GTS-SN.The GTS formalization based on SN may be considered as an alternative to classical approaches, in particular the algebraic ones based on single and double pushout. The strengths of this new proposal are a more intuitive definition of derivations, and the availability of well established tools for the editing/validation/analysis of models. The relationship between SN rules and single/double pushout derivations, however, is not treated in this paper, and deserves further investigations.The balance of the paper is as follows: Section 2 introduces SN and related background notions; Section 3 presents the encoding of a GTS as a SN, and its operational semantics; symbolic structural conditions for rule well-definiteness are also set up; Section 4 shows an application of SN structural calculus for verifying rule concurrency in a GTS; finally, Section 5 contains the conclusion and describes ongoing work

A structural approach to graph transformation based on symmetric Petri nets

“…It would be impossible to supply a complete list of these proposals, let us refer to [15] (and included references) for the earliest, and to [16,17] for more recent ones. This paper extends [18] and is somewhat related to most of those mentioned above. [18] considers the relationship between GTS and PN from a completely new perspective, by proposing a formalization of Graph Transformation Rules in terms of Symmetric Nets (SN) [19].…”

“…This paper extends [18] and is somewhat related to most of those mentioned above. [18] considers the relationship between GTS and PN from a completely new perspective, by proposing a formalization of Graph Transformation Rules in terms of Symmetric Nets (SN) [19]. SN are a subclass of Coloured Petri nets [20,21] featuring a particular syntax that outlines model symmetries and allows for efficient, both state-space based and structural, model analysis.…”

“…The first part of this paper significantly extends and consolidates the formalization of Graph Transformation Rules in terms of SN introduced in [18]. We consider a more general class of graphs, edge-labelled multi-graphs, and provide a sound structural characterization of SN rules by using the symbolic calculus implemented in SNexpression.…”

“…SN are a subclass of Coloured Petri nets [20,21] featuring a particular syntax that outlines model symmetries and allows for efficient, both state-space based and structural, model analysis. The idea in [18] is to map a Graph Transformation Rule (GTR) to a SN transition connected to a couple of places whose state (marking) encodes a graph. [18] only deals with directed simple graphs.…”

“…The idea in [18] is to map a Graph Transformation Rule (GTR) to a SN transition connected to a couple of places whose state (marking) encodes a graph. [18] only deals with directed simple graphs.…”