@MISC{Kuper_usability:formalising, author = {Jan Kuper}, title = {Usability: Formalising (un)definedness in Typed Lambda Calculus}, year = {} }

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Abstract

. In this paper we discuss usability, and propose to take that notion as a formalisation of (un)definedness in typed lambda calculus, especially in calculi based on PCF. We discuss some important properties that make usability attractive as a formalisation of (un)definedness. There is a remarkable difference between usability and solvability: in the untyped lambda calculus the solvable terms are precisely the terms with a head normal form, whereas in typed lambda calculus the usable terms are "between" the terms with a normal form and the terms with a (weak) head normal form. 1 Introduction The elementary form of undefinedness arises on the level of natural numbers, when the evaluation of a (closed) term M of type Nat does not terminate, i.e., when M does not have a normal form. Such a term is also often called meaningless. However, for higher types it is not so evident which terms should be called meaningless. Analogous to the situation for ground types, it is often felt to be attrac...