 
Summary: FirstOrder Universality for Real Programs
Thomas Anberr´ee
Division of Computer Science, University of Nottingham
199 Taikang East Road, Ningbo, 315100, China
thomas.anberree@nottingham.edu.cn
Abstract
J. Raymundo MarcialRomero and M. H. Escard´o described a func
tional programming language with an abstract data type real for the
real numbers and a nondeterministic operator rtest: real bool. We
show that this language is universal at first order, as conjectured by these
authors: all computable, firstorder total functions on the real numbers
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, described
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, firstorder partial functions in several arguments are
