First-Order Universality for Real Programs

Abstract

J. Raymundo Marcial–Romero and M. H. Escardó described a functional programming language with an abstract data type Real for the real numbers and a non-deterministic operator $f\colon\mathbb{R}\to\mathbb{R}$ we consider is the extension of the denotation of some program, 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, first-order partial functions in several arguments are definable.