Term Graph Rewriting Research Articles

Semantics-Preserving DPO-Based Term Graph Rewriting

Term graph rewriting is important as "conceptual implementation" of the execution of functional programs, and of data-flow optimisations in compilers. One way to define term graph transformation rule application is via the well-established and intuitively accessible double-pushout (DPO) approach; we present a new result proving semantics preservation for such DPO-based term graph rewriting.

Convergence in infinitary term graph rewriting systems is simple

Term graph rewriting provides a formalism for implementing term rewriting in an efficient manner by emulating duplication via sharing. Infinitary term rewriting has been introduced to study infinite term reduction sequences. Such infinite reductions can be used to modelnon-strict evaluation. In this paper, we unify term graph rewriting and infinitary term rewriting thereby addressing both components oflazy evaluation: non-strictness and sharing. In contrast to previous attempts to formalise infinitary term graph rewriting, our approach is based on a simple and natural generalisation of the modes of convergence of infinitary term rewriting. We show that this new approach is better suited for infinitary term graph rewriting as it is simpler and more general. The latter is demonstrated by the fact that our notions of convergence give rise to two independent canonical and exhaustive constructions of infinite term graphs from finite term graphs via metric and ideal completion. In addition, we show that our notions of convergence on term graphs are sound w.r.t. the ones employed in infinitary term rewriting in the sense that convergence is preserved by unravelling term graphs to terms. Moreover, the resulting infinitary term graph calculi provide a unified framework for both infinitary term rewriting and term graph rewriting, which makes it possible to study the correspondences between these two worlds more closely.

Termination of Graph and Term Graph Rewriting

Termination of Graph and Term Graph Rewriting

Kruskal's Tree Theorem for Acyclic Term Graphs

In this paper we study termination of term graph rewriting, where we restrict our attention to acyclic term graphs. Motivated by earlier work by Plump we aim at a definition of the notion of simplification order for acyclic term graphs. For this we adapt the homeomorphic embedding relation to term graphs. In contrast to earlier extensions, our notion is inspired by morphisms. Based on this, we establish a variant of Kruskal's Tree Theorem formulated for acyclic term graphs. In proof, we rely on the new notion of embedding and follow Nash-Williams' minimal bad sequence argument. Finally, we propose a variant of the lexicographic path order for acyclic term graphs.

Convergence in Infinitary Term Graph Rewriting Systems is Simple (Extended Abstract)

In this extended abstract, we present a simple approach to convergence on term graphs that allows us to unify term graph rewriting and infinitary term rewriting. This approach is based on a partial order and a metric on term graphs. These structures arise as straightforward generalisations of the corresponding structures used in infinitary term rewriting. We compare our simple approach to a more complicated approach that we developed earlier and show that this new approach is superior in many ways. The only unfavourable property that we were able to identify, viz. failure of full correspondence between weak metric and partial order convergence, is rectified by adopting a strong convergence discipline.

Rewriting and narrowing for constructor systems with call-time choice semantics

AbstractNon-confluent and non-terminating {constructor-based term rewriting systems are useful for the purpose of specification and programming. In particular, existing functional logic languages use such kinds of rewrite systems to define possibly non-strict non-deterministic functions. The semantics adopted for non-determinism iscall-time choice, whose combination with non-strictness is a non-trivial issue, addressed years ago from a semantic point of view with the Constructor-based Rewriting Logic (CRWL), a well-known semantic framework commonly accepted as suitable semantic basis of modern functional logic languages. A drawback of CRWL is that it does not come with a proper notion of one-step reduction, which would be very useful to understand and reason about how computations proceed. In this paper, we develop thoroughly the theory for the first-order version of let-rewriting, a simple reduction notion close to that of classical term rewriting, but extended with a let-binding construction to adequately express the combination of call-time choice with non-strict semantics. Let-rewriting can be seen as a particular textual presentation of term graph rewriting. We investigate the properties of let-rewriting, most remarkably their equivalence with respect to a conservative extension of the CRWL-semantics coping with let-bindings, and we show by some case studies that having two interchangeable formal views (reduction/semantics) of the same language is a powerful reasoning tool. After that, we provide a notion of let-narrowing, which is adequate for call-time choice as proved by soundness and completeness results of let-narrowing with respect to let-rewriting. Moreover, we relate those let-rewriting and let-narrowing relations (and hence CRWL) with ordinary term rewriting and narrowing, providing in particular soundness and completeness of let-rewriting with respect to term rewriting for a class of programs which are deterministic in a semantic sense.

Modes of Convergence for Term Graph Rewriting

Term graph rewriting provides a simple mechanism to finitely represent restricted forms of infinitary term rewriting. The correspondence between infinitary term rewriting and term graph rewriting has been studied to some extent. However, this endeavour is impaired by the lack of an appropriate counterpart of infinitary rewriting on the side of term graphs. We aim to fill this gap by devising two modes of convergence based on a partial order respectively a metric on term graphs. The thus obtained structures generalise corresponding modes of convergence that are usually studied in infinitary term rewriting. We argue that this yields a common framework in which both term rewriting and term graph rewriting can be studied. In order to substantiate our claim, we compare convergence on term graphs and on terms. In particular, we show that the modes of convergence on term graphs are conservative extensions of the corresponding modes of convergence on terms and are preserved under unravelling term graphs to terms. Moreover, we show that many of the properties known from infinitary term rewriting are preserved. This includes the intrinsic completeness of both modes of convergence and the fact that convergence via the partial order is a conservative extension of the metric convergence.

Term Graph Rewriting and Parallel Term Rewriting

The relationship between Term Graph Rewriting and Term Rewriting is well understood: a single term graph reduction may correspond to several term reductions, due to sharing. It is also known that if term graphs are allowed to contain cycles, then one term graph reduction may correspond to infinitely many term reductions. We stress that this fact can be interpreted in two ways. According to the "sequential interpretation", a term graph reduction corresponds to an infinite sequence of term reductions, as formalized by Kennaway et.al. using strongly converging derivations over the complete metric space of infinite terms. Instead according to the "parallel interpretation" a term graph reduction corresponds to the parallel reduction of an infinite set of redexes in a rational term. We formalize the latter notion by exploiting the complete partial order of infinite and possibly partial terms, and we stress that this interpretation allows to explain the result of reducing circular redexes in several approaches to term graph rewriting.

From Infinitary Term Rewriting to Cyclic Term Graph Rewriting and back

From Infinitary Term Rewriting to Cyclic Term Graph Rewriting and back

On Term-Graph Rewrite Strategies

We tackle the problem of cyclic term-graph rewriting. We first revisit the classical algorithmic approach to term-graph rewriting by providing a definition of rewrite rules of the form lhs→rhs where the left-hand sides are term-graphs and the right-hand sides are sequences of actions. Such actions, which specify how to rewrite a term-graph in a stepwise manner, contribute to simplify substantially the definition of cyclic term-graph rewriting. Then we define a new class of term-graph rewrite systems which are confluent over the so-called admissible term-graphs. Finally, we provide an efficient rewrite strategy which contracts only needed redexes and give pointers to other results regarding optimal rewrite strategies of admissible term-graphs.

Term-graph Rewriting in Tom Using Relative Positions

In this paper, we present the implementation in Tom of a de Bruijn indices generalization allowing the representation of term-graphs over an algebraic signature. By adding pattern matching and traversal controls to Java, Tom is a well-suited environment for defining program transformations or analyses. As some analyses, e.g. based on control flow, require graph-like structures, the use of this formalism is a natural way of expressing them by graph rewriting.

Deduction Graphs with Universal Quantification

Deduction Graphs are meant to generalise both Gentzen-Prawitz style natural deductions and Fitch style flag deductions. They have the structure of acyclic directed graphs with boxes. In [Herman Geuvers and Iris Loeb. Natural Deduction via Graphs: Formal Definition and Computation Rules. Mathematical Structures in Computer Science (Special Issue on Theory and Applications of Term Graph Rewriting), Volume 17(03):485–526, 2007.] we have investigated the deduction graphs for minimal proposition logic. This paper studies the extension with first-order universal quantification, showing the robustness of the concept of deduction graphs.

Lifting Infinite Normal Form Definitions From Term Rewriting to Term Graph Rewriting

nfinite normal forms are a way of giving semantics to non-terminating rewrite systems. The notion is a generalization of the Böhm tree in the lambda calculus. It was first introduced in [Ariola, Z. M. and S. Blom, Cyclic lambda calculi, in: Abadi and Ito [Abadi, M. and T. Ito, editors, “Theoretical Aspects of Computer Software,” Lecture Notes in Computer Science 1281, Springer Verlag, 1997], pp. 77–106] to provide semantics for a lambda calculus on terms with letrec. In that paper infinite normal forms were defined directly on the graph rewrite system. In [Blom, S., “Term Graph Rewriting - syntax and semantics,” Ph.D. thesis, Vrije Universiteit Amsterdam (2001)] the framework was improved by defining the infinite normal form of a term graph using the infinite normal form on terms. This approach of lifting the definition makes the non-confluence problems introduced into term graph rewriting by substitution rules much easier to deal with. In this paper, we give a simplified presentation of the latter approach.

Conditional Term Graph Rewriting with Indirect Sharing

For reasons of efficiency, term rewriting is usually implemented by term graph rewriting. In term rewriting, expressions are represented as terms, whereas in term graph rewriting these are represented as directed graphs. Unlike terms, graphs allow a sharing of common subexpressions. In previous work, we have shown that conditional term graph rewriting is a sound and complete implementation for a certain class of CTRSs with strict equality, provided that a minimal structure sharing scheme is used. In this paper, we will show that this is also true for two different extensions of normal CTRSs. In contrast to the previous work, however, a non-minimal structure sharing scheme can be used. That is, the amount of sharing is increased.

Inequational Deduction as Term Graph Rewriting

Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results.We propose a simple inequational deduction system, based on term graphs, for inferring inclusions of derived relations in a multi-algebra, and we show that term graph rewriting provides a sound and complete implementation of it.

Using Term-Graph Rewriting Models to Analyse Relative Space Efficiency

Space leaks are a common operational problem in programming languages with automated memory man-agement. Graph rewriting is a natural model of operational behaviour. This paper summarises a PhD thesis which gives a graph-rewriting framework suitable for modelling language implementations and proof techniques for determining the presence or absence of leaks. The approach is to model implementations as graph evaluators with garbage collectors. An evaluator may leak relative to another evaluator, with respect to a translation between their states. Leaks are classified according to their cause and the behaviour which exposes them.Graphs naturally model state size, but we argue that this is too concrete. Accurate evaluators are introduced which allow for a more abstract model in which initial program size is ignored. Evaluators are compared by defining a translation between graphs. Space-safe translations, and non-standard garbage collectors, are defined as another kind of term-graph rewrite system. Leaky evaluators are detected by a proof method which searches for graphs whose evaluation trace is a self-feeding rule sequence.

Theory and applications of term graph rewriting: introduction

Term graph rewriting is concerned with the representation of functional expressions as graphs and the evaluation of these expressions by rule-based graph transformation. The advantage of computing with graphs rather than terms is that common subexpressions can be shared, improving the efficiency of computations in space and time. Sharing is ubiquitous in implementations of programming languages: many functional, logic, object-oriented and concurrent calculi are implemented using term graphs.

A rewriting calculus for cyclic higher-order term graphs

The Rewriting Calculus (ρ-calculus, for short) was introduced at the end of the 1990s and fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the structured results obtained are first class objects of the calculus. The evaluation mechanism, which is a generalisation of beta-reduction, relies strongly on term matching in various theories.In this paper we propose an extension of the ρ-calculus, called ρg-calculus, that handles structures with cycles and sharing rather than simple terms. This is obtained by using recursion constraints in addition to the standard ρ-calculus matching constraints, which leads to a term-graph representation in an equational style. Like in the ρ-calculus, the transformations are performed by explicit application of rewrite rules as first-class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.We show that the ρg-calculus, under suitable linearity conditions, is confluent. The proof of this result is quite elaborate, due to the non-termination of the system and the fact that ρg-calculus-terms are considered modulo an equational theory. We also show that the ρg-calculus is expressive enough to simulate first-order (equational) left-linear term-graph rewriting and α-calculus with explicit recursion (modelled using a letrec-like construct).

Modeling Pointer Redirection as Cyclic Term-graph Rewriting

We tackle the problem of data-structure rewriting including global and local pointer redirections. Each basic rewrite step may perform three kinds of actions: (i) Local redirection, the aim of which is to redirect specific pointers determined by means of a pattern; (ii) Replacement, that may add new information to data-structures; (iii) Global redirection, which is aimed at redirecting all pointers targeting a node towards another one. We define a new framework, following the double-pushout approach, where graph rewrite rules may mix these three kinds of actions in a row. We define first the category of graphs we consider and then we define rewrite rules as pairs of graph homomorphisms of the form L←K→R. In our setting, graph K is not arbitrary, it is used to encode pointer redirection. Furthermore, pushouts do not always exist and complement pushouts, when they exist, are not unique. Despite these concerns, our definition of rewriting steps is such that a rewrite rule can always be fired, once a matching is found.

Committees and Sponsors

This volume contains the Proceedings of SEGRAGRA'95, the Joint COMPUGRAPH/SEMAGRAPH Workshop on Graph Rewriting and Computation. The Workshop was held in the “Centro Studi Santa Maria Maddalena” of Volterra, near Pisa, Italy, from August 28 to September 1, 1995.SEGRAGRA'95 addressed the area of Graph Rewriting and Term Graph Rewriting. It was the final workshop of two Esprit Basic Research Working Groups, COMPUGRAPH II and SEMAGRAPH II. Both working groups have been active in the three year period 1992/95. Unicode entity – undefined95. They have decided to organize their final workshop as a common initiative, with the aim of presenting the achieved results and their view of the state of the art in the field.Out of 39 submitted papers, the Program Committee selected 23 for presentation at the Workshop. These were grouped into sessions on Graph Rewriting I and II, Term Graph Rewriting I and II, Computing on Graphs, Concurrency & Parallelism, Modules, Software Engineering, Types and Proofs. The program included also invited talks by Eike Best (Universität Hildesheim), Thomas Johnsson (Chalmers University), Detlef Seese (Universität Karlsruhe) and Chris Wadsworth (Rutherford Appleton Laboratory); and tutorials by Henk Barendregt (Katholieke Universiteit Nijmegen), Andrea Corradini (Università di Pisa), Bruno Courcelle (Université Bordeaux I), Hartmut Ehrig (Technische Universität Berlin), Richard Kennaway (University of East Anglia, Norwich), Hans-Jörg Kreowski (Universität Bremen), Rinus Plasmeijer (Katholieke Universiteit Nijmegen) and Ronan Sleep (University of East Anglia, Norwich).The Conference site was made available at no cost to SEGRAGRA'95 by the Cassa di Risparmio di Volterra. The Workshop was supported by Gruppo Nazionale Informatica Matematica (CNR) and Istituto di Elaborazione dell'Informazione (CNR).We thank the anonymous referees, the Program Committee members: the Organizing Committee members: Fabio Gadducci and Marco Pistore from the University of Pisa gave us a valuable help in the local organization during the Workshop. We also thank the Managing Editors of the Electronic Notes in Computer Science series, Michael Mislove, Maurice Nivat, and Christos Papadimitriou, for giving us the opportunity of publishing the proceedings in this new electronic series, and Mike Mislove and Joost W.M. Kok (from Elsevier) for the constant support and the valuable suggestions they provided us during the preparation of this electronic volume.Andrea Corradini and Ugo Montanari, Guest Editors