Acyclic Term Graphs are able to represent terms with sharing, and the relationship between Term Graph Rewriting (TGR) and Term Rewrtiting (TR) is now well understood [BvEG + 87, HP91]. During the last years, some researchers considered the extension of TGR to possibly cyclic term graphs, which can represent possibly infinite, rational terms. In [KKSdV94] the authors formalize the classical relationship between TGR and TR as an "adequate mapping" between rewriting systems, and extend it by proving that unraveling is an adequate mapping from cyclic TGR to rational, infinitary term rewriting: In fact, a single graph reduction may correspond to an infinite sequence of term reductions. Using the same notions, we propose a different adequacy result, showing that unraveling is an adequate mapping from cyclic TGR to rational parallel term rewriting, where at each reduction infinitely many rules can be applied in parallel. We also argue that our adequacy result is more natural...

. Rational terms (possibly infinite terms with finitely many subterms) can be represented in a finite way via -terms, that is, terms over a signature extended with self-instantiation operators. For example, f ! = f(f(f(: : :))) can be represented as x :f(x) (or also as x :f(f(x)), f(x :f(x)), . . . ). Now, if we reduce a -term t to s via a rewriting rule using standard notions of the theory of Term Rewriting Systems, how are the rational terms corresponding to t and to s related? We answer to this question in a satisfactory way, resorting to the definition of infinite parallel rewriting proposed in [7]. We also provide a simple, algebraic description of -term rewriting through a variation of Meseguer's Rewriting Logic formalism. 1 Introduction Rational terms are possibly infinite terms with a finite set of subterms. They show up in a natural way in Theoretical Computer Science whenever some finite cyclic structures are of concern (for example data flow diagrams, cyclic te...

. We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2-theories. We show that this presentation is equivalent to the well-accepted operational definition proposed by Barendregt et alii---but for the case of circular redexes, for which we propose (and justify formally) a different treatment. The categorical framework allows us to model in a concise way also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures, and to relate term graph rewriting to other rewriting formalisms. R'esum'e. Nous pr'esentons une formulation cat'egorique de la r'e'ecriture des graphes cycliques des termes, bas'ee sur une variante de 2-theorie alg'ebrique. Nous prouvons que cette pr'esentation est 'equivalente `a la d'efinition op'erationnelle propos'ee par Barendregt et d'autres auteurs, mais pas dons le cas des radicaux circulaires, pour lesquels nous proposons (et justifions formellem...

. Dealing properly with sharing is important for expressing some of the common compiler optimizations, such as common subexpressions elimination, lifting of free expressions and removal of invariants from a loop, as source-to-source transformations. Graph rewriting is a suitable vehicle to accommodate these concerns. In [4] we have presented a term model for graph rewriting systems (GRSs) without interfering rules, and shown the partial correctness of the aforementioned optimizations. In this paper we define a different model for GRSs, which allows us to prove total correctness of those optimizations. Differently from [4] we will discard sharing from our observations and introduce more restrictions on the rules. We will introduce the notion of Bohm tree for GRSs, and show that in a system without interfering and non-left linear rules (orthogonal GRSs), Bohm tree equivalence defines a congruence. Total correctness then follows in a straightforward way from showing that if a program M co...

.We study properties of rewrite systems that are not necessarily terminating, but allow instead for trans#nite derivations that have a limit. In particular, we give conditions for the existence of a limit and for its uniqueness and relate the operational and algebraic semantics of in#nitary theories. We also consider su#cient completeness of hierarchical systems. Is there no limit? ---Job 16:3 1. Introduction Rewrite systems are sets of directed equations used to compute by repeatedly replacing equal terms in a given formula, as long as possible. For one approach to their use in computing, see #23#. The theory of rewriting is an outgrowth of the study of the lambda calculus and combinatory logic, and # Preliminary versions #6, 7# of ideas in this paper were presented at the Sixteenth ACM Symposium on Principles of Programming Languages, Austin, TX #January 1989# and at the Sixteenth EATCS International Colloquium on Automata, Languages and Programming, Stresa, Italy #July 1989#. ...

We define the notion of transfinite term rewriting: rewriting in which terms may be infinitely large and rewrite sequences may be of any ordinal length. For orthogonal rewrite systems, some fundamental properties known in the finite case are extended to the transfinite case. Among these are the Parallel Moves lemma and the Unique Normal Form property. The transfinite Church-Rosser property (CR 1 ) fails in general, even for orthogonal systems, including such well-known systems as Combinatory Logic. Syntactic characterisations are given of some classes of orthogonal TRSs which do satisfy CR 1 . We also prove a weakening of CR 1 for all orthogonal systems, in which the property is only required to hold up to a certain equivalence relation on terms. Finally, we extend the theory of needed reduction from the finite to the transfinite case. The reduction strategy of needed reduction is normalising in the finite case, but not in the transfinite case. To obtain a normalising str...

A methodology for polymorphic type inference for general term graph rewriting systems is presented. This requires modified notions of type and of type inference due to the absence of structural induction over graphs. Induction over terms is replaced by dataflow analysis. 1 Introduction Term graphs are objects that locally look like terms, but globally have a general directed graph structure. Since their introduction in Barendregt et al. (1987), they have served the purpose of defining a rigorous framework for graph reduction implementations of functional languages (Peyton-Jones (1987)). This was the original intention. However the rewriting of term graphs defined in the operational semantics of the model, makes term graph rewriting systems (TGRSs) interesting models of computation in their own right. One can thus study all sorts of issues in the specific TGRS context. Typically one might be interested in how close TGRSs are to TRSs and this problem is examined in Barendregt et al. (19...

Infinitary rewriting allows infinitely large terms and infinitely long reduction sequences. There are two computational motivations for studying these: the infinite data structures implicit in lazy functional programming, and the use of rewriting of possibly cyclic graphs as an implementation technique for functional languages. We survey the fundamental properties of infinitary rewriting in orthogonal term rewrite systems, and its relation to cyclic graph rewriting. 1 Introduction Our interest in term and graph rewriting arises from functional languages and their implementation. Functional programs can be seen as term rewrite systems. 2 Terms can be thought of as trees. Representing these trees as graphs allows repeated subterms to be replaced by multiple pointers to the same subgraph. This optimisation has a dramatic effect when rewrite steps are performed. Whenever a variable appears more than once on the right-hand side of a rule, when that rule is applied to a graph multiple poi...

We will present a syntactical proof of correctness and completeness of shared reduction. This work is an application of type theory extended with infinite objects and co-induction.

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