Intersection and Singleton Type Assignment Characterizing Finite Böhm-Trees

Toshihiko Kurata
2002 Information and Computation  
Intersection types and type constants representing unsolvability and singleton sets of λ-terms are incorporated into the Curry version of a simple type assignment system. Two restricted forms of typability in the system turn out to be equivalent to finiteness of Böhm-trees. C 2002 Elsevier Science (USA) 1 0890-5401/02 $35.00 C 2002 Elsevier Science (USA) All rights reserved. TOSHIHIKO KURATA The finiteness of Böhm-trees can be characterized, in terms of reduction theory, as the weak
more » ... ity with respect to the following restricted β-reduction. This reduction is not effective, but would be reasonable in a theoretical sense. Indeed, it can be proved by the genericity lemma [3, Proposition 12.3.24] that if a λ-term has a normal form then one can obtain the normal form by the restricted β-reduction. It has been an important topic in the theory of types to investigate typability of λ-terms in relation to various notions of normalizability, such as solvability, weak normalizability and strong normalizability. As classical results concerning this, it is well known that in simply typed λ-calculus and second order typed λ-calculus typable terms are all strongly normalizable [7] . Furthermore, in intersection type assignment systems [2, 6, 12] , each of the three notions of normalizability above can be neatly characterized by typability under a certain limited use of the (ω)-axiom. In this paper, we introduce a type theory that allows us to characterize the finite Böhm-trees. This kind of attempt has already been made in [10] , where an intersection type assignment system with a refinement of the universal type ω is introduced to characterize the property. In our system, instead of the refinement of ω, we use intersection types, type constants representing unsolvability, and singleton types. We prove a characterization theorem for this system, which states that a restricted form of typability in this system is equivalent to finiteness of Böhm-trees.
doi:10.1006/inco.2002.2907 fatcat:gw24cmftlfawzfbznz2rdmtvme