Real number computability and domain theory

Abstract

We present the different constructive definitions of real number that can be found in the literature. Using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them. This definition of computability turns out to be equivalent to other definitions given in the literature using different methods.Domain theory is a useful tool to study higher order computability on real numbers. An interesting connection between Scott-topology and the standard topologies on the real line and on the space of continuous functions on reals is stated. An important result in this paper is the proof that every computable functional on real numbers is continuous w.r.t. the compact open topology on the function space.

Continuity ∑—Algebras (Extended Abstract)

“…In [3] is shown that no set of sequential primitives are sufficient to define the reals as an abstract data type but is left open the use of non-sequential primitives. [11] restricts to algebraic cpo case (and as a result the reals are obtained indirectly via retracts). [18] uses quasiuniformities and do obtain the real line as the subspace of total elements but no notion of continuous algebras is introduced.…”

“…One of the most widely used representation of the reals in both practical and theoretical computer science is the signed digit representation (q. v. [3,11,15,21,13]). The reason comes from its simplicity, computability of the arithmetic operations, and easy implementation in lazy functional languages.…”

Applications of the Kleene–Kreisel Density Theorem to Theoretical Computer Science

“…This also suggests that based on this model, we may use this representation, its higher type semantics, and a P CF -like language to give a semantics to typed computations involving reals. This idea is discussed in more detail in DiGianantonio [6,7,8] and Simpson [32].…”

The appearance of big integers in exact real arithmetic based on Linear Fractional Transformations

“…There are many ways to represent real numbers as infinite objects [3,2,4,5]. Here, we are only concerned with representations as infinite streams of "digits".…”

“…There are several different stream representations which can be grouped into two large families: variations of the familiar decimal representation [1,3,2,5,7,11,10], and continued fraction expansions [8,16,9].…”