Towards Critical Pair Analysis for the Graph Programming Language GP 2

Part 4: Regular PapersInternational audienceWe present the foundations of critical pair analysis for the graph programming language GP 2. Our goal is to develop a static checker that can prove or refute confluence (functional behaviour) for a large class of graph programs. In this paper, we introduce symbolic critical pairs of GP 2 rule schemata, which are labelled with expressions, and establish the completeness and finiteness of the set of symbolic critical pairs over a finite set of rule schemata. We give a procedure for their construction

Confluence Analysis for a Graph Programming Language

GP 2 is a high-level domain-specific language for programming with graphs. Users write a set of graph transformation rules and organise them with imperative-style control constructs to perform a desired computation on an input graph. As rule selection and matching are non-deterministic, there might be different graphs resulting from program execution. Confluence is a property that establishes the global determinism of a computation despite possible local non-determinism. Conventional confluence analysis is done via so-called critical pairs, which are conflicts in minimal context. A key challenge is extending critical pairs to the setting of GP 2. This thesis concerns the development of confluence analysis for GP 2. First, we extend the notion of conflict to GP 2 rules, and prove that non-conflicting rule applications commute. Second, we define symbolic critical pairs and establish their properties, namely that there are only finitely many of them and that they represent all possible conflicts. We give an effective procedure for their construction. Third, we solve the problem of unifying GP 2 list expressions, which arises during the construction of critical pairs, by giving a unification procedure which terminates with a finite and complete set of unifiers (under certain restrictions). Last but not least, we specify a confluence analysis algorithm based on symbolic critical pairs, and show its soundness by proving the Local Confluence Theorem. Several existing programs are analysed for confluence to demonstrate how the analysis handles several GP 2 features at the same time, and to demonstrate the merit of the used techniques

A Unification Algorithm for GP 2

The graph programming language GP 2 allows to apply sets of ruleschemata (or “attributed” rules) non-deterministically. To analyse conflicts of pro-grams statically, graphs labelled with expressisons are overlayed to construct criticalpairs of rule applications. Each overlay induces a system of equations whose solu-tions represent different conflicts. We present a rule-based unification algorithm forGP expressions that is terminating, sound and complete. For every input equation,the algorithm generates a finite set of substitutions. Soundness means that each ofthese substitutions solves the input equation. Since GP labels are lists constructed byconcatenation, unification modulo associativity and unit law is required. This prob-lem, which is also known as word unification, is infinitary in general but becomesfinitary due to GP’s rule schema syntax and the assumption that rule schemata areleft-linear. Our unification algorithm is complete in that every solution of an inputequation is an instance of some substitution in the generated set