In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption).
The theory of algorithmic randomness begins with the definition of individual random infinite sequence introduced in 1966 by Martin-Löf [1]. Since then, many efforts have contributed to the development of this theory which is now well established and intensively studied, yet restricted to the Cantor space. In order to carry out an extension of this theory to more general infinite objects as encountered in most mathematical models of physical random phenomena, a necessary step is to understand what means for a probability measure on a general space to be computable (this is very simple expressed on the Cantor Space). Only then algorithmic randomness can be extended.
The problem of computability of (Borel) probability measures over more general spaces has been investigated by several authors: by Edalat for compact spaces using domain-theory ( [2]); by Weihrauch for the unit interval ( [3]) and by Schröder for sequential topological spaces ( [4]) both using representations; and by Gács for computable metric spaces ( [5]). Probability measures can be seen from different points of view and those works develop, each in its own framework, the corresponding computability notions. Mainly, Borel probability measures can be regarded as points of a metric space, as valuations on open sets or as integration operators. We express the computability counterparts of these different views in a unified framework, and show them to be equivalent.
Extensions of the algorithmic theory of randomness to general spaces have previously been proposed: on effective topological spaces by Hertling and Weihrauch (see [6], [7]) and on computable metric spaces by Gács (see [5]), both of them generalizing the notion of randomness tests and investigating the problem of the existence of a universal test. In [7], to prove the existence of such a test, ad hoc computability conditions on the measure are required, which a posteriori turn out to be incompatible with the notion of computable measure. The second one ( [5]), carrying the extension of Levin’s theory of randomness, considers uniform tests which are tests parametrized by measures. A computability condition on the basis of ideal balls (namely, recognizable Boolean inclusions) is needed to prove the existence of a universal uniform test.
In this article, working in computable metric spaces with any probability measure, we consider both uniform and non-uniform tests and prove the following points:
• uniformity and non-uniformity do not essentially differ,
• the existence of a universal test is assured without any further condition.
Another issue addressed in [5] is the characterization of randomness in terms of Kolmogorov Complexity (a central result in Cantor Space). There, this characterization is proved to hold (for a compact computable metric space X with a computable measure) under the assumption that there exists a computable injective encoding of a full-measure subset of X into binary sequences. In the real line for example, the base-two numeral system (or binary expansion) constitutes such encoding for the Lebesgue measure. This fact was already been (implicitly) used in the definition of random reals (reals with a random binary expansion, w.r.t the uniform measure).
We introduce, for computable metric spaces with a computable measure, a notion of binary representation generalizing the base-two numeral system of the reals, and prove that:
• such a binary representation always exists, • a point is random if and only if it has a unique binary expansion, which is random.
Moreover, our notion of binary representation allows to identify any computable probability space with the Cantor space (in a computable-measuretheoretic sense). It provides a tool to directly transfer elements of algorithmic randomness theory from the Cantor space to any computable probability space. In particular, the characterization of randomness in terms of Kolmogorov complexity, even in a non-compact space, is a direct consequence of this.
The way we handle computability on continuous spaces is largely inspired by representation theory. However, the main goal of that theory is to study, in general topological spaces, the way computability notions depend on the chosen representation. Since we focus only on Computable Metric Spaces (see [8] for instance) and Enumerative Lattices (introduced in setion 2.2) we shall consider only one canonical representation for each set, so we do not use representation theory in its general setting.
Our study of measures and randomness, although restricted to computable metric spaces, involves computability notions on various sets which do not have natural metric structures. Fortunately, all these sets become enumerative lattices in a very natural way and the canonical representation provides in each case the right computability notions.
In section 2, we develop a language intended to express computability concepts, statements and proofs in a rigorous but still (we hope)
