First-Order Universality for Real Programs

J. Raymundo Marcial–Romero and M. H. Escard´o described onal 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 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, first-order partial functions in several arguments a...

• Type: Conference
• Year: 2009
• Where: CIE
• Authors: Thomas Anberrée
• Keywords: Applied Computing, CIE 2009, Data Type Real, Non-deterministic Operator Rtest, Real Number