The issue of confluence is of major importance for the successful application of attributed graph transformation, such as automated translation of UML models into semantic domains. Whereas termination is undecidable in general and must be established by carefully designing the rules, local confluence can be shown for term rewriting and graph rewriting using the concept of critical pairs. In this paper, we discuss typed attributed graph transformation using a new simplified notion of attribution. For this kind of attributed graph transformation systems we establish a definition of critical pairs and prove a critical pair lemma, stating that local confluence follows from confluence of all critical pairs.

In this paper the concurrent semantics of double-pushout (DPO) graph rewriting, which is classically defined in terms of shift-equivalence classes of graph derivations, is axiomatised via the construction of a free monoidal bi-category. In contrast to a previous attempt based on 2-categories, the use of bi-categories allows to define rewriting on concrete graphs. Thus, the problem of composition of isomorphism classes of rewriting sequences is avoided. Moreover, as a first step towards the recovery of the full expressive power of the formalism via a purely algebraic description, the concept of disconnected rules is introduced, i.e., rules whose interface graphs are made of disconnected nodes and edges only. It is proved that, under reasonable assumptions, rewriting via disconnected rules enjoys similar concurrency properties like in the classical approach.

Several formal concurrent semantics have been proposed for graph rewriting, a powerful formalism for the specification of concurrent and distributed systems which generalizes P/T Petri nets. In this paper we relate two such semantics recently proposed for the algebraic doublepushout approach to graph rewriting, namely the derivation trace and the graph process semantics. The notion of concatenable graph process is

Petri nets are widely accepted as a specification formalism for concurrent and distributed systems. One of the reasons of their success is the fact that they are equipped with a rich theory, including well-understood concurrent semantics; they also provide an interesting benchmark for tools and techniques for the description of concurrent systems. Graph grammars can be regarded as a proper generalization of Petri nets, where the current state of a system is described by a graph instead as by a collection of tokens. In this tutorial paper I will review some basic definitions and constructions concerning the concurrent semantics of nets, and I will show to what extent corresponding notions have been developed for graph grammars. Most of such results come out from a joint research by the Berlin and Pisa COMPUGRAPH groups. 1 Introduction The nets which owe their name to Carl Adam Petri [28,29] have been the first formal tool proposed for the specification of the behaviour of systems which...

Jungle evaluation is an approach to define term rewriting with sharing based on graph grammars. This approach preserves important properties of term rewriting like termination, and confluence for terminating systems (under mild restrictions). In this paper, term rewriting with sharing is further accelerated, by memoization known from functional programming languages: The result of evaluating a function with some arguments is tabulated so that it can be looked up later on when the function is re-applied to the same arguments. We show that term rewriting with sharing and memoization is correct and complete w.r.t. jungle evaluation if the rules are non-overlapping and non-looping. Redundant re-evaluation of functions is avoided, independent of a particular strategy for applying evaluation rules. 1 Introduction Term rewriting is a basis for prototyping algebraic specifications of abstract data types, and a foundation of functional programming languages (see [DJ90] and [Klo90] for overvie...