Solution to the Range Problem for Combinatory Logic

Authors:
Benedetto Intrigila;Richard Statman
Affiliations:
(Correspd.) Dipartimento di Matematica, Università di Roma “Tor Vergata”, Rome, Italy. intrigil@mat.uniroma2.it;Department of Mathematical Sciences, Carnegie-Mellon University, Pittsburgh, PA, USA. rs31@andrew.cmu.edu
Venue:
Fundamenta Informaticae
Year:
2011

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Abstract

The λ-theory ℋ is obtained from β-conversion by identifying all closed unsolvable terms (or, equivalently, terms without head normal form). The range problem for the theory ℋ asks whether a closed term has always (up to equality in ℋ) either an infinite range or a singleton range (that is, it is a constant function). Here we give a solution to a natural version of this problem, giving a positive answer for the theory ℋ restricted to Combinatory Logic. The method of proof applies also to the Lazy λ-Calculus.