# Infinitary Rewriting - Theory and Applications

## Abstract

Infinitary rewriting generalises usual finitary rewriting by providing infinite reduction sequences with a notion of convergence. The idea of – at least conceptually – assigning a meaning to infinite derivations is well-known, for example, from lazy functional programming or from process calculi. Infinitary rewriting makes it possible to apply rewriting in order to obtain a formal model for such infinite derivations. The goal of this thesis is to comprehensively survey the field of infinitary term rewriting, to point out its shortcomings, and to try to overcome some of these shortcomings. The most significant problem that arises in infinitary rewriting is the inherent difficulty to finitely represent and, hence, to implement it. To this end, we consider term graph rewriting, which is able to finitely represent restricted forms of infinitary term rewriting. Moreover, we study different models that are used to formalise infinite reduction sequences: The well-established metric approach as well as an alternative approach using partial orders. Both methods together with the consequent infinitary versions of confluence and termination properties are analysed on an abstract level. Based on this, we argue that the partial order model has more advantageous properties and represents the intuition of convergence in a more natural way. This assessment is also backed up by the results that we obtain for infinitary term rewriting: Unlike the metric approach, the partial order approach admits to generalise some results known from finitary orthogonal term rewriting – most importantly, confluence. It is also shown that so-called Böhm trees, usually constructed rather intricately, naturally arise as normal forms in the partial order model. Finally, we devise a complete ultrametric and a complete semilattice on term graphs both of which are used to introduce infinitary term graph rewriting. This is supposed to serve as a tool in order to investigate the limitations of term graph rewriting for implementing infinitary term rewriting.

Category: Rewriting