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George Barmpalias

Publications and source records attributed to George Barmpalias.

At least 37 records · Page 2Linked to original sources
arXiv2018

Aspects of Chaitin's Omega

The halting probability of a Turing machine,also known as Chaitin's Omega, is an algorithmically random number with many interesting properties. Since Chaitin's seminal work, many popular expositions have appeared, mainly focusing on the metamathematical or philosophical significance of Omega (or debating against it). At the same time, a rich mathematical theory exploring the properties of Chaitin's Omega has been brewing in various technical papers, which quietly reveals the significance of this number to many aspects of contemporary algorithmic information theory. The purpose of this survey is to expose these developments and tell a story about Omega, which outlines its multifaceted mathematical properties and roles in algorithmic randomness.

math.LO↗
arXiv2018

Equivalences between learning of data and probability distributions, and their applications

Algorithmic learning theory traditionally studies the learnability of effective infinite binary sequences (reals), while recent work by [Vitanyi and Chater, 2017] and [Bienvenu et al., 2014] has adapted this framework to the study of learnability of effective probability distributions from random data. We prove that for certain families of probability measures that are parametrized by reals, learnability of a subclass of probability measures is equivalent to learnability of the class of the corresponding real parameters. This equivalence allows to transfer results from classical algorithmic theory to learning theory of probability measures. We present a number of such applications, providing many new results regarding EX and BC learnability of classes of measures, thus drawing parallels between the two learning theories.

math.LO↗
arXiv2018

Algorithmic learning of probability distributions from random data in the limit

We study the problem of identifying a probability distribution for some given randomly sampled data in the limit, in the context of algorithmic learning theory as proposed recently by Vinanyi and Chater. We show that there exists a computable partial learner for the computable probability measures, while by Bienvenu, Monin and Shen it is known that there is no computable learner for the computable probability measures. Our main result is the characterization of the oracles that compute explanatory learners for the computable (continuous) probability measures as the high oracles. This provides an analogue of a well-known result of Adleman and Blum in the context of learning computable probability distributions. We also discuss related learning notions such as behaviorally correct learning and orther variations of explanatory learning, in the context of learning probability distributions from data.

cs.LG↗
arXiv2017

A note on the differences of computably enumerable reals

We show that given any non-computable left-c.e. real $α$ there exists a left-c.e. real $β$ such that $α\neq β+γ$ for all left-c.e. reals and all right-c.e. reals $γ$. The proof is non-uniform, the dichotomy being whether the given real $α$ is Martin-Loef random or not. It follows that given any universal machine $U$, there is another universal machine $V$ such that the halting probability of U is not a translation of the halting probability of V by a left-c.e. real. We do not know if there is a uniform proof of this fact.

math.LO↗
arXiv2017

Random numbers as probabilities of machine behaviour

A fruitful way of obtaining meaningful, possibly concrete, algorithmically random numbers is to consider a potential behaviour of a Turing machine and its probability with respect to a measure (or semi-measure) on the input space of binary codes. For example, Chaitin's Omega is a well known Martin-Loef random number that is obtained by considering the halting probability of a universal prefix-free machine. In the last decade, similar examples have been obtained for higher forms of randomness, i.e. randomness relative to strong oracles. In this work we obtain characterizations of the algorithmically random reals in higher randomness classes, as probabilities of certain events that can happen when an oracle universal machine runs probabilistically on a random oracle. Moreover we apply our analysis to different machine models, including oracle Turing machines, prefix-free machines, and models for infinite online computation. We find that in many cases the arithmetical complexity of a property is directly reflected in the strength of the algorithmic randomness of the probability with which it occurs, on any given universal machine. On the other hand, we point to many examples where this does not happen and the probability is a number whose algorithmic randomness is not the maximum possible (with respect to its arithmetical complexity). Finally we find that, unlike the halting probability of a universal machine, the probabilities of more complex properties like totality, cofinality, computability or completeness do not necessarily have the same Turing degree when they are defined with respect to different universal machines.

cs.CC↗
arXiv2017

Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers

The Kučera-Gács theorem is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Löf random real. If the computation of the first $n$ bits of a sequence requires $n+h(n)$ bits of the random oracle, then $h$ is the redundancy of the computation. Kučera implicitly achieved redundancy $n\log n$ while Gács used a more elaborate coding procedure which achieves redundancy $\sqrt{n}\log n$. A similar upper bound is implicit in the later proof by Merkle and Mihailović. In this paper we obtain strict optimal lower bounds on the redundancy in computations from Martin-Löf random oracles. We show that any nondecreasing computable function $g$ such that $\sum_n 2^{-g(n)}=\infty$ is not a general upper bound on the redundancy in computations from Martin-Löf random oracles. In fact, there exists a real $X$ such that the redundancy $g$ of any computation of $X$ from a Martin-Löf random oracle satisfies $\sum_n 2^{-g(n)}<\infty$. Moreover, the class of such reals is comeager and includes a $Δ^0_2$ real as well as all weakly 2-generic reals. This excludes many slow growing functions such as $\log n$ from bounding the redundancy in computations from random oracles for a large class of reals. On the other hand it was recently shown that if $\sum_n 2^{-g(n)}<\infty$ then $g$ is a general upper bound for the redundancy in computations of any real from some Martin-Löf random oracle. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop.

cs.CC↗
arXiv2017

Optimal redundancy in computations from random oracles

A classic result in algorithmic information theory is that every infinite binary sequence is computable from a Martin-Loef random infinite binary sequence. Proved independently by Kucera and Gacs, this result answered a question by Charles Bennett and has seen numerous applications in the last 30 years. The optimal redundancy in such a coding process has, however, remained unknown. If the computation of the first n bits of a sequence requires n + g(n) bits of the random oracle, then g is the redundancy of the computation. Kucera implicitly achieved redundancy n log n while Gacs used a more elaborate block-coding procedure which achieved redundancy sqrt(n) log n. Different approaches to coding such as the one by Merkle and Mihailovic have not improved this redundancy bound. In this paper we devise a new coding method that achieves optimal logarithmic redundancy. Our redundancy bound is exponentially smaller than the previously best known bound and is known to be the best possible. It follows that redundancy r log n in computation from a random oracle is possible for every stream, if and only if r > 1.

cs.CC↗
arXiv2017

Differences of halting probabilities

The halting probabilities of universal prefix-free machines are universal for the class of reals with computably enumerable left cut (also known as left-c.e. reals), and coincide with the Martin-Loef random elements of this class. We study the differences of Martin-Loef random left-c.e. reals and show that for each pair of such reals a, b there exists a unique number r > 0 such that qa - b is a 1-random left-c.e. real for each positive rational q > r and a 1-random right-c.e. real for each positive rational q < r. Based on this result we develop a theory of differences of halting probabilities, which answers a number of questions about Martin-Loef random left-c.e. reals, including one of the few remaining open problems from the list of open questions in algorithmic randomness by Miller and Nies in 2006. The halting probability of a prefix-free machine M restricted to a set X is the probability that the machine halts and outputs an element of X. These numbers Omega_M(X) were studied by a number of authors in the last decade as a way to obtain concrete highly random numbers. When X is the complement of a computably enumerable set, the number Omega_M(X) is the difference of two halting probabilities. Becher, Figueira, Grigorieff, and Miller asked whether Omega_U(X) is Martin-Loef random when U is universal and X is the complement of a computably enumerable set. This problem has resisted numerous attempts in the last decade. We apply our theory of differences of halting probabilities to give a positive answer, and show that Omega_U(X) is a Martin-Loef random left-c.e. real whenever X is nonempty and the complement of a computably enumerable set.

cs.CC↗
arXiv2017

The probability of a computable output from a random oracle

Consider a universal Turing machine that produces a partial or total function (or a binary stream), based on the answers to the binary queries that it makes during the computation. We study the probability that the machine will produce a computable function when it is given a random stream of bits as the answers to its queries. Surprisingly, we find that these probabilities are the entire class of real numbers in (0, 1) that can be written as the difference of two halting probabilities relative to the halting problem. In particular, there are universal Turing machines which produce a computable output with probability exactly 1/2. Our results contrast a large array of facts (the most well-known being the randomness of Chaitin's halting probability) which witness maximal initial segment complexity of probabilities associated with universal machines. Our proof uses recent advances in algorithmic randomness.

cs.CC↗
arXiv2017

Limits of the Kucera-Gacs coding method

Every real is computable from a Martin-Loef random real. This well known result in algorithmic randomness was proved by Kucera and Gacs. In this survey article we discuss various approaches to the problem of coding an arbitrary real into a Martin-Loef random real,and also describe new results concerning optimal methods of coding. We start with a simple presentation of the original methods of Kucera and Gacs and then rigorously demonstrate their limitations in terms of the size of the redundancy in the codes that they produce. Armed with a deeper understanding of these methods, we then proceed to motivate and illustrate aspects of the new coding method that was recently introduced by Barmpalias and Lewis-Pye and which achieves optimal logarithmic redundancy, an exponential improvement over the original redundancy bounds.

math.LO↗
arXiv2017

Pointed computations and Martin-Löf randomness

Schnorr showed that a real is Martin-Loef random if and only if all of its initial segments are incompressible with respect to prefix-free complexity. Fortnow and independently Nies, Stephan and Terwijn noticed that this statement remains true if we can merely require that the initial segments of the real corresponding to a computable increasing sequence of lengths are incompressible. The purpose of this note is to establish the following generalization of this fact. We show that a real is X Martin-Loef random if and only if its initial segments corresponding to a pointedly X-computable sequence (r_n) (where r_n is computable from X in a self-delimiting way, so that at most the first r_n bits of X are queried in the computation) of lengths are incompressible. On the other hand we also show that there are reals which are very far from being Martin-Loef random, yet they compute an increasing sequence of lengths at which their initial segments are incompressible.

cs.CC↗
arXiv2017

Kobayashi compressibility

Kobayashi introduced a uniform notion of compressibility of infinite binary sequences in terms of relative Turing computations with sub-identity use of the oracle. Kobayashi compressibility has remained a relatively obscure notion, with the exception of some work on resource bounded Kolmogorov complexity. The main goal of this note is to show that it is relevant to a number of topics in current research on algorithmic randomness. We prove that Kobayashi compressibility can be used in order to define Martin-Loef randomness, a strong version of finite randomness and Kurtz randomness, strictly in terms of Turing reductions. Moreover these randomness notions naturally correspond to Turing reducibility, weak truth-table reducibility and truth-table reducibility respectively. Finally we discuss Kobayashi's main result from his 1981 technical report regarding the compressibility of computably enumerable sets, and provide additional related original results.

cs.CC↗
arXiv2016

Computing halting probabilities from other halting probabilities

The halting probability of a Turing machine is the probability that the machine will halt if it starts with a random stream written on its one-way input tape. When the machine is universal, this probability is referred to as Chaitin's omega number, and is the most well known example of a real which is random in the sense of Martin-Löf. Although omega numbers depend on the underlying universal Turing machine, they are robust in the sense that they all have the same Turing degree, namely the degree of the halting problem. In this paper we give precise bounds on the redundancy growth rate that is generally required for the computation of an omega number from another omega number. We show that for each $ε>1$, any pair of omega numbers compute each other with redundancy $ε\log n$. On the other hand, this is not true for $ε=1$. In fact, we show that for each omega number there exists another omega number which is not computable from the first one with redundancy $\log n$. This latter result improves an older result of Frank Stephan.

cs.CC↗
arXiv2016

Unperturbed Schelling segregation in two or three dimensions

Schelling's model of segregation, first described in 1969, has become one of the best known models of self-organising behaviour. While Schelling's explicit concern was to understand the mechanisms underlying racial segregation in large cities from a game theoretic perspective, the model should be seen as one of a family, arising in fields as diverse as statistical mechanics, neural networks and the social sciences, and which are concerned with interacting populations situated on network structures. Despite extensive study, however, the (unperturbed) Schelling model has largely resisted rigorous analysis, prior results in the literature generally pertaining to variants of the model in which noise is introduced into the dynamics of the system, the resulting model then being amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory. A series of recent papers (one by Brandt, Immorlica, Kamath, and Kleinberg, and two by the authors), has seen the first rigorous analysis of the one dimensional version of the unperturbed model. Here we provide the first rigorous analysis of the two and three dimensional unperturbed models, establishing most of the phase diagram, and answering a challenge from a recent paper by Brandt, Immorlica, Kamath, and Kleinberg.

cs.MA↗
arXiv2016

Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega

Chaitin's number Omega is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every computably enumerable (c.e.) real, there exists a Turing functional via which Omega computes it, and such that the number of bits of omega that are needed for the computation of the first n bits of the given number (i.e. the use on argument n) is bounded above by a computable function h(n) = n+o(n). We characterise the asymptotic upper bounds on the use of Chaitin's omega in oracle computations of halting probabilities (i.e. c.e. reals). We show that the following two conditions are equivalent for any computable function h such that h(n)-n is non-decreasing: (1) h(n)-n is an information content measure, (2) for every c.e. real there exists a Turing functional via which omega computes the real with use bounded by h. We also give a similar characterisation with respect to computations of c.e. sets from Omega, by showing that the following are equivalent for any computable non-decreasing function g: (1) g is an information-content measure, (2) for every c.e. set A, Omega computes A with use bounded by g. Further results and some connections with Solovay functions are given.

math.LO↗
arXiv2015

Minority population in the one-dimensional Schelling model of segregation

The Schelling model of segregation looks to explain the way in which a population of agents or particles of two types may come to organise itself into large homogeneous clusters, and can be seen as a variant of the Ising model in which the system is subjected to rapid cooling. While the model has been very extensively studied, the unperturbed (noiseless) version has largely resisted rigorous analysis, with most results in the literature pertaining to versions of the model in which noise is introduced into the dynamics so as to make it amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory. We rigorously analyse the one-dimensional version of the model in which one of the two types is in the minority, and establish various forms of threshold behaviour. Our results are in sharp contrast with the case when the distribution of the two types is uniform (i.e. each agent has equal chance of being of each type in the initial configuration), which was studied by Brandt, Immorlica, Kamath, and Kleinberg.

cs.DM↗
arXiv2015

Digital morphogenesis via Schelling segregation

Schelling's model of segregation looks to explain the way in which particles or agents of two types may come to arrange themselves spatially into configurations consisting of large homogeneous clusters, i.e.\ connected regions consisting of only one type. As one of the earliest agent based models studied by economists and perhaps the most famous model of self-organising behaviour, it also has direct links to areas at the interface between computer science and statistical mechanics, such as the Ising model and the study of contagion and cascading phenomena in networks. While the model has been extensively studied it has largely resisted rigorous analysis, prior results from the literature generally pertaining to variants of the model which are tweaked so as to be amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory. In \cite{BK}, Brandt, Immorlica, Kamath and Kleinberg provided the first rigorous analysis of the unperturbed model, for a specific set of input parameters. Here we provide a rigorous analysis of the model's behaviour much more generally and establish some surprising forms of threshold behaviour, notably the existence of situations where an \emph{increased} level of intolerance for neighbouring agents of opposite type leads almost certainly to \emph{decreased} segregation.

cs.GT↗