Expressibility in the Lambda Calculus with mu

TL;DR

This paper characterizes infinite lambda-terms that are lambda_{mu}-expressible, introducing concepts like regularity and binding-capturing chains, and shows their equivalence to expressibility in lambda_{mu}.

Contribution

It provides the first characterization of lambda_{mu}-expressible infinite lambda-terms using regularity and binding-capturing chains.

Findings

Lambda_{mu}-expressible terms are exactly the strongly regular ones.

Regularity is characterized via finitely many subterms under specific rewrite relations.

Binding-capturing chains determine the expressibility in lambda_{mu}.

Abstract

We address a problem connected to the unfolding semantics of functional programming languages: give a useful characterization of those infinite lambda-terms that are lambda_{letrec}-expressible in the sense that they arise as infinite unfoldings of terms in lambda_{letrec}, the lambda-calculus with letrec. We provide two characterizations, using concepts we introduce for infinite lambda-terms: regularity, strong regularity, and binding-capturing chains. It turns out that lambda_{letrec}-expressible infinite lambda-terms form a proper subclass of the regular infinite lambda-terms. In this paper we establish these characterizations only for expressibility in lambda_{mu}, the lambda-calculus with explicit mu-recursion. We show that for all infinite lambda-terms T the following are equivalent: (i): T is lambda_{mu}-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 lambda-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 relations for decomposing lambda-terms. These rewrite relations act on infinite lambda-terms furnished with a marked prefix of abstractions for collecting decomposed lambda-abstractions and keeping the terms closed under decomposition. They differ in how vacuous abstractions in the prefix are removed. This report accompanies the article with the same title for the proceedings of the conference RTA 2013, and mainly differs from that by providing the proof of the characterization of lambda_{mu}-expressibility with binding-capturing chains.