Proving Infinitary Normalization

Abstract

We investigate the notion of `infinitary strong normalization' (SN *** ), introduced in [6], the analogue of termination when rewriting infinite terms. A (possibly infinite) term is SN *** if along every rewrite sequence each fixed position is rewritten only finitely often. In [9], SN *** has been investigated as a system-wide property, i.e. SN *** for all terms of a given rewrite system. This global property frequently fails for trivial reasons. For example, in the presence of the collapsing rule tail(x:*** )****** , the infinite term t =tail(0:t) rewrites to itself only. Moreover, in practice one usually is interested in SN *** of a certain set of initial terms. We give a complete characterization of this (more general) `local version' of SN *** using interpretations into weakly monotone algebras (as employed in [9]). Actually, we strengthen this notion to continuous weakly monotone algebras (somewhat akin to [5]). We show that tree automata can be used as an automatable instance of our framework; an actual implementation is made available along with this paper.