Transfinite reductions in orthogonal term rewriting systems

Abstract

Abstract. Strongly convergent reduction is the fundamental notion of reduction in infinitary orthogonal term rewriting systems (OTRSs). For these we prove the Transfinite Parallel Moves Lemma and the Compressing Lemma. Strongness is necessary as shown by counterexamples. Normal forms, ·which we allow to be infinite, are unique, in contrast to oo-normal forms. Strongly converging fair reductions result in normal forms.In general OTRSs the infinite Church-Rosser Property fails for strongly converging reductions. However for BOhm reduction (as in Lambda Calculus, subterms without head normal forms may be replaced by J.) the infinite Church-Rosser property does hold. The infinite Church-Rosser Property for non-unifiable OTRSs follows. The top-terminating OTRSs of Dershowitz c.s. are examples of nonunifiable OTRSs.

The infinitary lambda calculus of the infinite eta Böhm trees

“…Since each term N i coincides with M up to depth i, the limit N ω of this sequence is exactly M . (Kennaway et al, 1995b;Kennaway and de Vries, 2003) it is sufficient to prove that a sequence of length ω + 1 can be compressed into one of length ω. Without loss of generality, we may suppose that we have a strongly convergent η!-reduction sequence of length ω + 1 as follows:…”

Infinitary Combinatory Reduction Systems: Confluence

“…To start, recall that confluence in general does not hold for iTRSs, even under assumption of orthogonality [8]. As every iTRS can be seen as a fully-extended iCRS, it follows that fully-extended, orthogonal iCRSs are in general not confluent either.…”

“…In case of iTRSs two approaches are known for restoring confluence [8], namely (1) identifying all subterms that disrupt confluence, and (2) restricting the rewrite rules that are allowed. Identifying all subterms that disrupt confluence leads to the definition of socalled hypercollapsing subterms and yields the result that orthogonal iTRSs are confluent modulo these subterms.…”

“…We would like to have a characterisation of confluence that appeals only to the syntax of iCRSs without any need to consider equality modulo some relation. The first correct, fundamental confluence result for iTRSs [8] stated that an orthogonal iTRSs is confluent iff it has the property of being 'almost non-collapsing': There is at most one rule that is collapsing and the variable at the root of the right-hand side of that rule is the only variable occurring in the left-hand side of that rule. Unfortunately, this concept does not carry over trivially to iCRSs, when replacing the variables from iTRSs by meta-variables: We currently do not know how to give a precise characterisation of the class of confluent iCRSs.…”

“…The properties extend their usual finitary counterparts to infinitary rewriting. Ample motivation for the formulation of the properties can be found in [8]. In the definition, (և·։) * denotes the symmetric, transitive, reflexive closure of ։.…”