We present an equational framework for term graph rewriting with cycles. The usual notion of homomorphism is phrased in terms of the notion of bisimulation, which is well-known in process algebra and concurrency theory. Specifically, a homomorphism is a functional bisimulation. We prove that the bisimilarity class of a term graph, partially ordered by functional bisimulation, is a complete lattice. It is shown how Equational Logic induces a notion of copying and substitution on term graphs, or systems of recursion equations, and also suggests the introduction of hidden or nameless nodes in a term graph. Hidden nodes can be used only once. The general framework of term graphs with copying is compared with the more restricted copying facilities embodied in the µ-rule, and translations are given between term graphs and µ-expressions. Using these, a proof system is given for µ-expressions that is complete for the semantics given by infinite tree unwinding. Next, orthogonal term graph rewrite ...

Reduction Systems Definition 2.2. An Abstract Reduction System (short: ARS) consists of a set A and a sequence ! i of binary relations on A, labelled by some set I. We often drop the label if I is a singleton. We write A j= P if the ARS A = (A; ! i ; : : : ); i 2 I has the property P . Further we write A j= P Q iff A j= P and A j= Q. An ARS A = (A; !) has the diamond property , A j= \Sigma, iff /;! ` !;/. It has the Church-Rosser property (is confluent), A j= CR, iff (A; !!) j= \Sigma. Given an ARS A = (A; !), we write CR(t) as shorthand for (fu j t !! ug; !) j= CR. 4 Stefan Kahrs Under most circumstances, confluence is a useful property of ARSs, mainly because: if (A; !) j= CR, and if two elements x; y 2 A are equivalent w.r.t. the smallest equivalence containing !, then there is a z 2 A such that x!! z //y. Roughly: the ARS decides the equivalence. An ARS A = (A; ! a ; ! b ) commutes directly , A j= CD, iff / a ; ! b ` ! b ; / a . To prove confluence of an ARS, it is sometimes...

Graphs and graph transformations are natural means to describe systems of processes. Graphs represent structure of the system, and graph rewriting rules model dynamic behaviour. In this chapter, we illustrate the technique by describing Petri nets, statecharts, parallel logic programming, and systems of processes. Whereas description of Petri nets is based on usual graphs, statecharts lead us to hierarchical graphs, and parallel logic programming needs jungles. Finally, we combine different approaches to describe systems of processes. Topological structure is represented by a hypergraph. Local states and communication channels correspond to nodes that are labelled with parts of a global jungle playing the role of a shared data structure. The formal model takes advantage of comma-category approach allowing to change both the structure of graph and the contents of nodes consistently and to treat different graph structures as well as different labelling mechanisms in

We present a survey of confluence properties of (acyclic) term graph rewriting. Results and counterexamples are given for different kinds of term graph rewriting: besides plain applications of rewrite rules, extensions with the operations of collapsing and copying, and with both operations together are considered. Collapsing and copying together constitute bisimilarity of term graphs. We establish sufficient conditions for---and counterexamples to---confluence, confluence modulo bisimilarity and the Church-Rosser property modulo bisimilarity. Moreover, we address rewriting modulo bisimilarity, that is, rewriting of bisimilarity classes of term graphs. 1991 Computing Reviews Classification System: F.1.1, F.4.1, F.4.2 Keywords and Phrases: term graph rewriting, bisimilarity, confluence, Church-Rosser property Note: Work carried out under project SEN2.2, Data Manipulation. Part of the research of the third author was performed while he was on leave at CWI by a grant of the HCM ne...

This is a survey of the main aims and results of the ESPRIT Basic Research Working Group COMPUTING BY GRAPH TRANSFORMATION II, 1992 - 1996, following up the first phase of COMPUGRAPH, 1989 - 1992. The research goals and main results are presented within the following three research areas: Foundations, Concurrency, and Graph Transformations for Specification and Programming. 1 Introduction The research area of graph grammars or graph transformations is a relatively young discipline of computer science. Its origins date back to the early seventies. Nevertheless, methods, techniques, and results from the area of graph transformations have already been studied and applied in many fields of computer science such as formal language theory, pattern recognition and generation, compiler construction, software engineering, concurrent and distributed systems modeling, database design and theory, logical and functorial programming, AI etc. This wide applicability is due to the fact that graphs ar...

. For efficiency reasons, term rewriting is usually implemented by graph rewriting. It is known that graph rewriting is a sound and complete implementation of (almost) orthogonal term rewriting systems; see [BEG + 87]. In this paper, we extend the result to properly oriented orthogonal conditional systems with strict equality. In these systems extra variables are allowed in conditions and right-hand sides of rules. 1 Introduction Attempts to combine the functional and logic programming paradigms have recently been receiving increasing attention; see [Han94b] for an overview of the field. It has been argued in [Han95] that strict equality is the only sensible notion of equality for possibly nonterminating programs. In this paper, we adopt this point of view--so every functional logic program is regarded as an orthogonal conditional term rewriting system (CTRS) with strict equality. The standard operational semantics for functional (or equational) logic programming is conditional narr...

The concept of graph substitution recently introduced by the authors is applied to term graphs, yielding a uniform framework for unification, rewriting, and narrowing on term graphs. The notion of substitution allows definitions of these concepts that are close to the corresponding definitions in the term world. The rewriting model obtained in this way is equivalent to "collapsed tree rewriting" and hence is complete for equational deduction. For term graph narrowing, a completeness result is established which corresponds to Hullot's classical result for term narrowing. The general motivation for using term graphs instead of terms is to improve efficiency: sharing common subterms saves space and avoids the repetition of computations. 1 Introduction In [8] we introduced graph substitutions by using hyperedges as graph variables and hyperedge replacement to substitute graphs for variables. Graph unification, graph matching and substitution-based graph rewriting were introduced and studi...

Introduction It is well-known that termination is not modular for disjoint term rewriting systems (TRSs). In Toyama's counterexample the combination of the terminating systems R 1 = fF (0; 1; x) ! F (x; x; x)g, and R 2 = fg(x; y) ! x; g(x; y) ! yg yields a non-terminating system because there is the cyclic rewrite derivation 0 1 g 1 0 + 0 1 g 0 1 g g 0 1 0 1 g 0 1 g g 0 1 F F F In the last decade, many sufficient criteria for the modularity of termination have been given; see [Mid90, Ohl94, Gra96] for an overview. For instance, termination is modular for the classes of non-collapsing and non-duplicat

A relationship between double-pushout rewriting of total and partial hypergraphs is established by means of free completions. Given a double-pushout transformation rule P of total hypergraphs, a corresponding double-pushout transformation rule P 0 of partial subhypergraphs can be found a priori such that double-pushout derivations by the application of P are free completions of double-pushout derivations by the application of P 0 . It allows a more efficient double-pushout rewriting of total hypergraphs by rewriting suitable partial hypergraphs and freely completing the resulting partial hypergraphs. The main results in this paper are actually established in the more general setting of hierarchical unary algebras. 1 Introduction The algebraic approach to graph transformation, both in double-pushout and single-pushout form, has evolved in the last few years from the transformation of total structures (i.e., graphs, hypergraphs, algebras) to the transformation of partial structures ...

The aim of this paper is to implement the beta-reduction in the lambda-calculus with a hypergraph rewriting mechanism called collapsed lambda-tree rewriting. It turns out that collapsed lambda-tree rewriting is sound with respect to beta-reduction and complete with respect to the Gross-Knuth strategy. As a consequence, there exists a normal form for a collapsed lambda-tree if and only if there exists a normal form for the represented lambda-term.