Explicit Substitution On the Edge of Strong Normalization

We use the recursive path ordering (RPO) technique of semantic labelling to show the preservation of strong normalization (PSN) property for several calculi of explicit substitution. PSN states that if a term M is strongly normalizing under ordinary p-reduction (using ‘global’ substitutions), then it is strongly normalizing if the substitution is made explicit (‘local’). There are different ways of making global substitution explicit and PSN is a quite natural and desirable property for the explicit substitution calculus. Our method for proving PSN is very general and applies to several known systems of explicit substitutions, both with named variables and with De Bruijn indices: lv of Lescanne et al., ILr of Kamareddine and Rios and Ix of Rose and Bloo. We also look at two small extensions of the explicit substitution calculus that allow to permute substitutions. For one of these extensions PSN fails (using the counterexample in Mellits 1995). For the other we can prove PSN using our method, thus showing the subtlety of the subject and the generality of our method. One of the key ideas behind our proof is that, for Lx the set of terms of the explicit substitution calculus, we look at the set Ixcos, consisting of the terms A such that the substitution-normalform of each subterm of A is /I-SN. This is a kind of ‘induction loading’: if we prove that Ax-reduction is SN on the set kxCm, then we have proved PSN for 2.x. To prove Ix-SN on the set kxcM, we define the /&size; of a term A E 3bxrpo on labelled terms, such that any infinite Ix-reduction path starting from an A E ,Ix<~ translates to an infinite >rpO -descending sequence. The well-founded order > rp0 is defined by using a technique similar to semantic labelling. @ 1999 Elsevier Science B.V. All rights reserved Keyw0rd.x Lambda-calculus; Explicit substitution; Recursive path order * Corresponding author. E-mail: [email protected]. ’ Supported by the Netherlands Computer Science Research Foundation (SION) with financial support from the Netherlands Organization for Scientific Research (NWO). 0304-3975/99/$ see front matter @ 1999-Elsevier Science B.V. All rights reserved PII: so304-3975(97)00183-7 376 R. Bloo, H. Geuversl Theoretical Computer Science 211 (1999) 375-395