Expressibility in the Lambda Calculus with mu

Abstract

We address a problem connected to the unfolding semantics of functional programming languages: give a useful characterization of those infinite λ-terms that are λ letrec -expressible in the sense that they arise as infinite unfoldings of terms in λ letrec , the λ-calculus with letrec. We provide two characterizations, using concepts we introduce for infinite λ-terms: regularity, strong regularity, and binding-capturing chains. It turns out that λ letrec -expressible infinite λ-terms form a proper subclass of the regular infinite λ-terms. In this paper we establish these characterizations only for expressibility in λ µ , the λ-calculus with explicit µ-recursion. We show that for all infinite λ-terms T the following are equivalent: (i): T is λ µ -expressible; (ii): T is strongly regular; (iii): T is regular, and it only has finite binding-capturing chains.We define regularity and strong regularity for infinite λ-terms as two different generalizations of regularity for infinite first-order terms: as the existence of only finitely many subterms that are defined as the reducts of two rewrite systems for decomposing λ-terms. These rewrite systems act on infinite λ-terms furnished with a bracketed prefix of abstractions for collecting decomposed λ-abstractions and keeping the terms closed under decomposition. They differ in which vacuous abstractions in the prefix are removed.This report accompanies the article [9], and mainly differs from that by providing the proof of the characterization of λ µ -expressibility with binding-capturing chains.

Unfolding Semantics of the Untyped λ-Calculus with letrec

“…The presented methods deal with sharing as expressed by letrec that is horizontal, vertical, or twisted (see remark 3.3.4). By contrast, the µ-construct [8,20] expresses only vertical, and the non-recursive let only horizontal, sharing (see remark 3.3.4). By restricting bisimulation (either artificially or by adding special backlinks), our methods can be adapted to the λ-calculus with µ [20], or with let.…”