Infinitary rewriting: meta-theory and convergence

Abstract

When infinitary rewriting was introduced by Kaplan et. al. [9] at the beginning of the 1990s, its term universe was explained as the metric completion of a metric on finite terms. The motivation for this connection to topology was that it allowed to import other well-studied notions from metric spaces, in particular the notion of convergence as a replacement for normalisation.This paper generalises the approach by parameterising it with a term metric, and applying the process of metric completion not only to terms but also to operations on and relations between terms. The resulting meta-theory is studied, leading to a revised notion of infinitary rewrite system. For these systems a method is devised to prove their convergence.

Modes of Convergence for Term Graph Rewriting

“…However, in contrast to the majority of the literature on infinitary term rewriting, which is concerned with strong convergence [24,27], we will only consider weak notions of convergence in this paper; cf. [14,20,33]. This weak form of convergence, also called Cauchy convergence, is entirely based on the sequence of objects produced by rewriting without considering how the rewrite rules are applied.…”

Modularity of Convergence and Strong Convergence in Infinitary Rewriting

“…A topological abstract reduction system (short: TARS) is a structure (S, O, →) such that (S, O) is a topological space and (S, →) an abstract reduction system, i.e. → is a binary relation on S. In the world of infinitary terms the underlying topological spaces are typically ultra-metric spaces, see [9,Proposition 3] and [20,Section 4.3]. The reason for the added generality of this section is that (i) the notion of convergence is a topological concept, (ii) certain convergence results about abstract reduction systems can be proven at this more general level, and (iii) not all forms of infinitary rewriting derive from metric spaces, e.g.…”

“…, d m (t n , u n )). Some fundamental results about term metrics from [9] are: d m is an ultra-metric; the topology induced by d m is discrete. Moreover, the set of infinitary terms over Σ and m is called Ter m (Σ) and defined as the metric completion of the metric space (Ter (Σ ), d m ).…”

“…As the definition only depends on a finite part of t it is even defined for infinite terms t which are not in Ter m (Σ). The notions of contexts, substitutions, positions and the operations on them lift canonically from Ter (Σ ) to Ter m (Σ), at least for continuous term metrics, see [9].…”

“…As a consequence it is impossible to construct an LTree with an infinite right spine. The same effect can be achieved by modifying the distance function [9]; in this case, we could define d m (Bin(a, b, c), Bin(x, y, z)) = max(d m (c, z), d m (b, y),…”