George Davie - Academia.edu (original) (raw)

Papers by George Davie

Journal of Logic and Analysis

In this paper we look at the convergence rates for the ergodic averages in the pointwise ergodic ... more In this paper we look at the convergence rates for the ergodic averages in the pointwise ergodic theorem for computable ergodic transformations on the unit interval. While these rates are layerwise computable for Martin-Löf random points and effectively open sets with Lebesgue measure a computable real, they are also layerwise computable for an arbitrary interval. There are however, effectively open sets for which there are \emph{no} effective rates, in particular, not layerwise computable ones. We also show that, when the measure of the effectively open set is any real alpha\alphaalpha, the convergence rates are computable in alpha\alphaalpha and the layers relative to alpha\alphaalpha.

The Annals of Probability, 2001

We formulate effective versions of the Borel-Cantelli lemmas using a coefficient from Kolmogorov ... more We formulate effective versions of the Borel-Cantelli lemmas using a coefficient from Kolmogorov complexity. We then use these effective versions to lift the effective content of the law of large numbers and the law of the iterated logarithm.

arXiv (Cornell University), Jul 31, 2022

We are interested in the computability between left c.e. reals α and their initial segments. We s... more We are interested in the computability between left c.e. reals α and their initial segments. We show that the quantity C(C(α n)|α n) plays a crucial role in this and in their completeness. We look in particular at Chaitin's theorem and its relativisation due to Frank Stephan.

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) s... more Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth-table reducibility to RK (the set of Kolmogorov-random strings) that lies between BPP and PSPACE [4, 3]. In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆C⊆PSPACE ∩ P/poly. We conjecture that C is equal to P, and discuss the possibility this might be an avenue for trying to prove the equality of BPP and P.

Statistics & Probability Letters, 2012

In a previous paper, we showed how the compressibility of an (algorithmically) random sequence se... more In a previous paper, we showed how the compressibility of an (algorithmically) random sequence sets upper bounds for events taking place in such sequences. In this paper, we show that the compressibility also determines allowed and forbidden regions for such events to occur.

Characterising the Martin-L�f random sequences using computably enumerable sets of measure one

Ipl, 2004

Kolmogorov complexity and recursive events

Logical Methods in Computer Science, 2014

In this paper we study the behaviour at infinity of the Fourier transform of Radon measures suppo... more In this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these sets, the Fourier transform of the associated measures have, relative to the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity. The argument relies heavily on a direct characterisation, due to Asarin and Pokrovskii, of algorithmically random Brownian motion in terms of the prefix-free Kolmogorov complexity of finite binary sequences. The study also necessitates a closer look at the potential theory over fractals from a computable point of view.

Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie, 2010

There is a general consensus that it is not possible to gamble successfully against a random se-q... more There is a general consensus that it is not possible to gamble successfully against a random se-quence. This consensus is based on results from probability theory that all gambling systems arein some sense futile and the idea that at any stage of the sequence, the next outcome is entirelyunpredictable.

Decidable lim sup and Borel–Cantelli-like lemmas for random sequences

Statistics & Probability Letters, 2013

We prove computable versions of limsup events and Borel–Cantelli-like results for algorithmically... more We prove computable versions of limsup events and Borel–Cantelli-like results for algorithmically random sequences using a coefficient from Kolmogorov complexity. In particular we show that under suitable conditions on events, limsup is layerwise decidable.

Mathematical Structures in Computer Science, 2013

We examine a construction due to Fouché in which a Brownian motion is constructed from an algorit... more We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

Kolmogorov Complexity and Noncomputability

MLQ, 2002

Characterising the Martin-Löf random sequences using computably enumerable sets of measure one

Information Processing Letters, 2004

A sequence @w is Martin-Lof random if and only if it appears early in every Lebesgue measure one ... more A sequence @w is Martin-Lof random if and only if it appears early in every Lebesgue measure one set of computably enumerable intervals.

Archive for Mathematical Logic, 2001

Let ω be a Kolmogorov-Chaitin random sequence with ω 1:n denoting the first n digits of ω. Let P ... more Let ω be a Kolmogorov-Chaitin random sequence with ω 1:n denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP (ω 1:n)} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any such P and α finds an n such that P holds within the first n digits of ω or not in ω at all. We apply the result to the halting probability and show that various generalizations of the result fail.

We introduce the notion of being Weihrauch-complete for layerwise computability and provide sever... more We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoff's theorem. We also consider hitting time operators, which share the Weihrauch degree of the former examples but fail to be layerwise computable.

Logical Methods in Computer Science, 2014

In this paper we study the behaviour at infinity of the Fourier transform of Radon measures suppo... more In this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these sets, the Fourier transform of the associated measures have, relative to the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity. The argument relies heavily on a direct characterisation, due to Asarin and Pokrovskii, of algorithmically random Brownian motion in terms of the prefix-free Kolmogorov complexity of finite binary sequences. The study also necessitates a closer look at the potential theory over fractals from a computable point of view.

Mathematical Structures in Computer Science, 2013

We examine a construction due to Fouché in which a Brownian motion is constructed from an algorit... more We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) s... more Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth-table reducibility to R K (the set of Kolmogorov-random strings) that lies between BPP and PSPACE . In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆ C ⊆ PSPACE ∩ P/poly.

We introduce the notion of being Weihrauch-complete for layerwise computability and provide sever... more We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoff's theorem. We also consider the hitting time operators, which share the Weihrauch degree of the former examples, but fail to be layerwise computable.

Journal of Logic and Analysis

In this paper we look at the convergence rates for the ergodic averages in the pointwise ergodic ... more In this paper we look at the convergence rates for the ergodic averages in the pointwise ergodic theorem for computable ergodic transformations on the unit interval. While these rates are layerwise computable for Martin-Löf random points and effectively open sets with Lebesgue measure a computable real, they are also layerwise computable for an arbitrary interval. There are however, effectively open sets for which there are \emph{no} effective rates, in particular, not layerwise computable ones. We also show that, when the measure of the effectively open set is any real alpha\alphaalpha, the convergence rates are computable in alpha\alphaalpha and the layers relative to alpha\alphaalpha.

The Annals of Probability, 2001

We formulate effective versions of the Borel-Cantelli lemmas using a coefficient from Kolmogorov ... more We formulate effective versions of the Borel-Cantelli lemmas using a coefficient from Kolmogorov complexity. We then use these effective versions to lift the effective content of the law of large numbers and the law of the iterated logarithm.

arXiv (Cornell University), Jul 31, 2022

We are interested in the computability between left c.e. reals α and their initial segments. We s... more We are interested in the computability between left c.e. reals α and their initial segments. We show that the quantity C(C(α n)|α n) plays a crucial role in this and in their completeness. We look in particular at Chaitin's theorem and its relativisation due to Frank Stephan.

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) s... more Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth-table reducibility to RK (the set of Kolmogorov-random strings) that lies between BPP and PSPACE [4, 3]. In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆C⊆PSPACE ∩ P/poly. We conjecture that C is equal to P, and discuss the possibility this might be an avenue for trying to prove the equality of BPP and P.

Statistics & Probability Letters, 2012

In a previous paper, we showed how the compressibility of an (algorithmically) random sequence se... more In a previous paper, we showed how the compressibility of an (algorithmically) random sequence sets upper bounds for events taking place in such sequences. In this paper, we show that the compressibility also determines allowed and forbidden regions for such events to occur.

Characterising the Martin-L�f random sequences using computably enumerable sets of measure one

Ipl, 2004

Kolmogorov complexity and recursive events

Logical Methods in Computer Science, 2014

In this paper we study the behaviour at infinity of the Fourier transform of Radon measures suppo... more In this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these sets, the Fourier transform of the associated measures have, relative to the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity. The argument relies heavily on a direct characterisation, due to Asarin and Pokrovskii, of algorithmically random Brownian motion in terms of the prefix-free Kolmogorov complexity of finite binary sequences. The study also necessitates a closer look at the potential theory over fractals from a computable point of view.

Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie, 2010

There is a general consensus that it is not possible to gamble successfully against a random se-q... more There is a general consensus that it is not possible to gamble successfully against a random se-quence. This consensus is based on results from probability theory that all gambling systems arein some sense futile and the idea that at any stage of the sequence, the next outcome is entirelyunpredictable.

Decidable lim sup and Borel–Cantelli-like lemmas for random sequences

Statistics & Probability Letters, 2013

We prove computable versions of limsup events and Borel–Cantelli-like results for algorithmically... more We prove computable versions of limsup events and Borel–Cantelli-like results for algorithmically random sequences using a coefficient from Kolmogorov complexity. In particular we show that under suitable conditions on events, limsup is layerwise decidable.

Mathematical Structures in Computer Science, 2013

We examine a construction due to Fouché in which a Brownian motion is constructed from an algorit... more We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

Kolmogorov Complexity and Noncomputability

MLQ, 2002

Characterising the Martin-Löf random sequences using computably enumerable sets of measure one

Information Processing Letters, 2004

A sequence @w is Martin-Lof random if and only if it appears early in every Lebesgue measure one ... more A sequence @w is Martin-Lof random if and only if it appears early in every Lebesgue measure one set of computably enumerable intervals.

Archive for Mathematical Logic, 2001

Let ω be a Kolmogorov-Chaitin random sequence with ω 1:n denoting the first n digits of ω. Let P ... more Let ω be a Kolmogorov-Chaitin random sequence with ω 1:n denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP (ω 1:n)} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any such P and α finds an n such that P holds within the first n digits of ω or not in ω at all. We apply the result to the halting probability and show that various generalizations of the result fail.

We introduce the notion of being Weihrauch-complete for layerwise computability and provide sever... more We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoff's theorem. We also consider hitting time operators, which share the Weihrauch degree of the former examples but fail to be layerwise computable.

Logical Methods in Computer Science, 2014

In this paper we study the behaviour at infinity of the Fourier transform of Radon measures suppo... more In this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these sets, the Fourier transform of the associated measures have, relative to the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity. The argument relies heavily on a direct characterisation, due to Asarin and Pokrovskii, of algorithmically random Brownian motion in terms of the prefix-free Kolmogorov complexity of finite binary sequences. The study also necessitates a closer look at the potential theory over fractals from a computable point of view.

Mathematical Structures in Computer Science, 2013

We examine a construction due to Fouché in which a Brownian motion is constructed from an algorit... more We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) s... more Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth-table reducibility to R K (the set of Kolmogorov-random strings) that lies between BPP and PSPACE . In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆ C ⊆ PSPACE ∩ P/poly.

We introduce the notion of being Weihrauch-complete for layerwise computability and provide sever... more We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoff's theorem. We also consider the hitting time operators, which share the Weihrauch degree of the former examples, but fail to be layerwise computable.