First-Order Universality for Real Programs

Abstract. J.Raymundo Marcial–Romero and M.H.Escardo ́ described a functional programming language with an abstract data type Real for the real numbers and a non-deterministic operator rtest: Real → Bool. We show that this language is universal at first order, as conjectured by these authors: all computable, first-order total functions on the real num-bers are definable. To be precise, we show that each computable function f: R → R we consider is the extension of the denotation JMf K of some program Mf: Real → Real, in a model based on powerdomains, de-scribed in previous work. Whereas this semantics is only an approximate one, in the sense that programs may have a denotation strictly below their true outputs, our result shows that, to compute a given function, it is in fact always possible to find a program with a faithful denotation. We briefly indicate how our proof extends to show that functions taken from a large class of computable, first-order partial functions in several arguments are definable. Key words: computability, real number computation, simply typed lambda-calculus, denotational semantics.