Summary: First-Order Universality for Real Programs
Division of Computer Science, University of Nottingham,
199 Taikang East Road, Ningbo, 315100, China
Abstract. J. Raymundo MarcialRomero and M. H. Escard´o 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 Mf 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.