## Computability of probability measures and Martin-Löf randomness over metric spaces

Venue: | Information and Computation |

Citations: | 11 - 5 self |

### BibTeX

@ARTICLE{Rojas_computabilityof,

author = {Cristóbal Rojas},

title = {Computability of probability measures and Martin-Löf randomness over metric spaces},

journal = {Information and Computation},

year = {}

}

### OpenURL

### Abstract

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). 1

### Citations

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Citation Context ...or which µn → µ if and only if � fdµn → � fdµ for all continuous bounded function f : X → R. This topology is metrizable and when X is separable and complete, M(X) is also separable and complete (see =-=[Bil68]-=-). Moreover, a computable metric structure on X induces in a canonical way a computable metric structure on M(X). Let D ⊂ M(X) be the set of those probability measures that are concentrated in finitel... |

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Citation Context ...s some i such that E = Ei := {ϕ(〈i,n〉) : n ∈ N}. Moreover, we can take ϕ such that whenever Ei �= ∅ the function ϕ(〈i,.〉) : N → N is total (this is a classical construction from recursion theory, see =-=[HR87]-=-). Then consider the associated algorithm Aϕ = νP ◦ ϕ: for every constructive element x there is some i such that Aϕ(〈i,.〉) : N → P enumerates x (∅ is an enumeration of ⊥). � Remark 2.2.1 Observe that... |

481 | Domain theory
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Citation Context ...hen it is defined, runs forever otherwise. In the same vein, a robust notion of (partial) recursive function F : N N → N N can be characterized by different formal definitions: Via domain theory (see =-=[AJ94]-=-). This approach takes the notion of recursive function as primitive, which avoids the definition of a new computation model. A partial function F : N N → N N is recursive if there is a recursive func... |

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Citation Context ...rm randomness test (without further assumption). 1 Introduction The theory of algorithmic randomness begins with the definition of individual random infinite sequence introduced in 1966 by Martin-Löf =-=[ML66]-=-. Since then, many efforts have contributed to the development of this theory which is now well etablished and intensively studied, yet restricted to the Cantor space. In order to carry out an extensi... |

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Citation Context ...le and is allowed to read elements σn of the oracle sequence. On an input n ∈ N, it may stop and output a natural number, interpreted as F(σ)n. Via type-two Turing machines (defined by Weihrauch, see =-=[Wei00]-=-). Expressed differently, it is essentially the same computation model (it works on symbols instead of integers). Again, to show that a function F : N N → N N is recursive, we will exhibit an algorith... |

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Giuseppe Gradient flows in metric spaces and in the space of probability measures
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Citation Context ...enotes � fdµ. Definition 4.1.3 The Wasserstein metric on M(X) is defined by: W(µ,ν) = sup f∈1−Lip(X) (|µf − νf|) (2) where 1 − Lip(X) is the space of 1-Lipschitz functions from X to R. We recall (see =-=[LNG05]-=-) that W has the following properties: Proposition 4.1.2 1. W is a distance and if X is separable and complete then M(X) with this distance is a separable and complete metric space. 10s2. The topology... |

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Complexity Theory of Real Functions
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Citation Context ...be embedded into the set of finite and infinite sequences of integers ordered by the prefix relation, which is an ω-algebraic domain). Via oracle Turing machines (used by Ko and Friedman, see [KF82], =-=[Ko91]-=-). An oracle Turing machine M [σ] is a Turing machine which works with a sequence σ ∈ N N provided as oracle and is allowed to read elements σn of the oracle sequence. On an input n ∈ N, it may stop a... |

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Complexity of Real Functions
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Citation Context ...ace can be embedded into the set of finite and infinite sequences of integers ordered by the prefix relation, which is an ω-algebraic domain). Via oracle Turing machines (used by Ko and Friedman, see =-=[KF82]-=-, [Ko91]). An oracle Turing machine M [σ] is a Turing machine which works with a sequence σ ∈ N N provided as oracle and is allowed to read elements σn of the oracle sequence. On an input n ∈ N, it ma... |

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Citation Context ...tructure: a partial constructive function from some represented space to another cannot in general be extended to a total constructive one. 3.2 The Open Subsets of a computable metric space Following =-=[BW99]-=-, [BP03], we define constructivity notions on the open subsets of a computable metric space. The topology τ induced by the metric has the numbered set B of ideal balls as a countable basis: any open s... |

41 | Uniform test of algorithmic randomness over a general space
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Citation Context ...domain-theory ([Eda96]); by Weihrauch for the unit interval ([Wei99]) and by Schröder for sequential topological spaces ([Sch07]) both using representations; and by Gács for computable metric spaces (=-=[Gác05]-=-). 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... |

24 |
Computability on subsets of metric spaces
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Citation Context ...: a partial constructive function from some represented space to another cannot in general be extended to a total constructive one. 3.2 The Open Subsets of a computable metric space Following [BW99], =-=[BP03]-=-, we define constructivity notions on the open subsets of a computable metric space. The topology τ induced by the metric has the numbered set B of ideal balls as a countable basis: any open set can t... |

22 | Randomness Spaces
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(Show Context)
Citation Context ...ork, 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 =-=[HW98]-=-,[HW03]) and on computable metric spaces by Gács (see [Gác05]), both of them generalizing the notion of randomness tests and investigating the problem of the existence of a universal test. In [HW03], ... |

14 |
Random elements in effective topological spaces with measure
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Citation Context ...d 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 [HW98],=-=[HW03]-=-) and on computable metric spaces by Gács (see [Gác05]), both of them generalizing the notion of randomness tests and investigating the problem of the existence of a universal test. In [HW03], to prov... |

13 |
On the empirical validity of the Bayesian method
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Citation Context ...sure, in an effective way: it is then called an effective null set. Equivalently, a µ-randomness test can be defined as a positive lower semi-computable function t : 2 ω → R satisfying � tdµ ≤ 1 (see =-=[VV93]-=- for instance). The associated effective null set is {x : t(x) = +∞} = � n {x : t(x) > 2n }. Actually, every effective null set can be put in this form for some t. A point is then called µ-random if i... |

13 | Computability on the probability measures on the Borel sets of the unit interval, Theoret
- Weihrauch
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(Show Context)
Citation Context ...lity of (Borel) probability measures over more general spaces has been investigated by several authors: by Edalat for compact spaces using domain-theory ([Eda96]); by Weihrauch for the unit interval (=-=[Wei99]-=-) and by Schröder for sequential topological spaces ([Sch07]) both using representations; and by Gács for computable metric spaces ([Gác05]). Probability measures can be seen from different points of ... |

11 |
The Scott topology induces the weak topology
- Edalat
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(Show Context)
Citation Context ...ness 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 (=-=[Eda96]-=-); by Weihrauch for the unit interval ([Wei99]) and by Schröder for sequential topological spaces ([Sch07]) both using representations; and by Gács for computable metric spaces ([Gác05]). Probability ... |

11 | Admissible representations of probability measures
- Schröder
(Show Context)
Citation Context ...s has been investigated by several authors: by Edalat for compact spaces using domain-theory ([Eda96]); by Weihrauch for the unit interval ([Wei99]) and by Schröder for sequential topological spaces (=-=[Sch07]-=-) both using representations; and by Gács for computable metric spaces ([Gác05]). Probability measures can be seen from different points of view and those works develop, each in its own framework, the... |

10 | Effective convergence in probability and an ergodic theorem for individual random sequences - V’yugin - 1997 |

8 | An effective Borel-Cantelli lemma. Constructing orbits with required statistical properties
- Galatolo, Hoyrup, et al.
(Show Context)
Citation Context ...ntruct a fast sequence of ideal measures converging to µ in the W metric and vice-versa. � This equiavalence offers an alternative method to prove computability of measures. It is used for example in =-=[GHRb]-=- to show the computability of the physical measures for some classes of dynamical systems. 4.2 Measures as valuations We now investigate the first problem: can the measure of sets be computed from the... |

7 | Effective metric spaces and representations of the reals
- Hemmerling
(Show Context)
Citation Context ...owever, the main goal of that theory is to study, in general topological spaces, the way computablity notions depend on the chosen representation. Since we focus only on Computable Metric Spaces (see =-=[Hem02]-=- 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 ... |

7 |
Yoshiki Tsujii. Effective properties of sets and functions in metric spaces with computability stqucture. Theoretical Computer Science
- Yasugi, Mori
- 1999
(Show Context)
Citation Context ...d µ is finite, so the set of r for which µ(Sr) ≥ 1/k is finite. Define V 〈i,j〉 = R + \ {d(si,sj)}: this is a dense r.e open set, uniformly in 〈i,j〉. Then by the computable Baire Category Theorem (see =-=[YMT99]-=-, [Bra01]), the dense Π 0 2 -set � 〈i,k〉 U 〈i,k〉 ∩ � 〈i,j〉 V 〈i,j〉 contains a sequence (rn)n∈N of uniformly computable real numbers which is dense in R + . In other words, all rn are computable, unifo... |

5 | Computable versions of baire’s category theorem
- Brattka
- 2001
(Show Context)
Citation Context ...nite, so the set of r for which µ(Sr) ≥ 1/k is finite. Define V 〈i,j〉 = R + \ {d(si,sj)}: this is a dense r.e open set, uniformly in 〈i,j〉. Then by the computable Baire Category Theorem (see [YMT99], =-=[Bra01]-=-), the dense Π 0 2 -set � 〈i,k〉 U 〈i,k〉 ∩ � 〈i,j〉 V 〈i,j〉 contains a sequence (rn)n∈N of uniformly computable real numbers which is dense in R + . In other words, all rn are computable, uniformly in n... |

3 |
Algorithmically random points in measure preserving systems, statistical behaviour, complexity and entropy
- Galatolo, Hoyrup, et al.
(Show Context)
Citation Context .... Then x is µ-random if and only if there is c such that for all n: Hn(x) ≥ − log µ(Γn(x)) − c All this allows to treat algorithmic randomness within probability theory over general metric spaces. In =-=[GHRa]-=- for instance, it is applied to show that in ergodic systems over metric spaces, algoritmically random points are well-behavied: they are generic with respect to any computable measure preserving tran... |

1 | HR87] [HW98] [HW03] [KF82] [Ko91] Jr. Hartley Rogers. Theory of Recursive Functions and Effective Computability - Sci - 2002 |