PCF extended with real numbers: a domain-theoretic approach to higher-order exact real number computation

Martín Hötzel Escardó


We develop a theory of higher-order exact real number computation based on Scott domain theory. Our main object of investigation is a higher-order functional programming language, Real PCF, which is an extension of PCF with a data type for real numbers and constants for primitive real functions. Real PCF has both operational and denotational semantics, related by a computational adequacy property.

In the standard interpretation of Real PCF, types are interpreted as continuous Scott domains. We refer to the domains in the universe of discourse of Real PCF induced by the standard interpretation of types as the real numbers type hierarchy. Sequences are functions defined on natural numbers, and predicates are truth-valued functions. Thus, in the real numbers types hierarchy we have real numbers, functions between real numbers, predicates defined on real numbers, sequences of real numbers, sequences of sequences of real numbers, sequences of functions, functionals mapping sequences to numbers (such as limiting operators), functionals mapping functions to numbers (such as integration and supremum operators), functionals mapping predicates to truth-values (such as existential and universal quantification operators), and so on.

As it is well-known, the notion of computability on a domain depends on the choice of an effective presentation. We say that an effective presentation of the real numbers type hierarchy is sound if all Real PCF definable elements and functions are computable with respect to it. The idea is that Real PCF has an effective operational semantics, and therefore the definable elements and functions should be regarded as concretely computable. We then show that there is a unique sound effective presentation of the real numbers type hierarchy, up to equivalence with respect to the induced notion of computability. We can thus say that there is an absolute notion of computability for the real numbers type hierarchy.

All computable elements and all computable first-order functions in the real numbers type hierarchy are Real PCF definable. However, as it is the case for PCF, some higher-order computable functions, including an existential quantifier, fail to be definable. If a constant for the existential quantifier (or, equivalently, a computable supremum operator) is added, the computational adequacy property remains true, and Real PCF becomes a computationally complete programming language, in the sense that all computable functions of all orders become definable.

We introduce induction principles and recursion schemes for the real numbers domain, which are formally similar to the so-called Peano axioms for natural numbers. These principles and schemes abstractly characterize the real numbers domain up to isomorphism, in the same way as the so-called Peano axioms for natural numbers characterize the natural numbers. On the practical side, they allow us to derive recursive definitions of real functions, which immediately give rise to correct Real PCF programs (by an application of computational adequacy). Also, these principles form the core of the proof of absoluteness of the standard effective presentation of the real numbers type hierarchy, and of the proof of computational completeness of Real PCF.

Finally, results on integration in Real PCF consisting of joint work with Abbas Edalat are included.


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