arXiv subjects

Christopher P. Porter

Publications and source records attributed to Christopher P. Porter.

At least 19 recordsLinked to original sources

arXiv2024

Length Functions and the Dimension of Points in Self-Similar Fractal Trees

In this paper, we study the effective dimension of points in infinite fractal trees generated recursively by a finite tree over some alphabet. Using unequal costs coding, we associate a length function with each such fractal tree and show that the channel capacity of the length function is equal to the similarity dimension of the fractal tree (up to a multiplicative constant determined by the size of the alphabet over which our tree is defined). Using this result, we derive formulas for calculating the effective dimension and strong effective dimension of points in fractal trees, establishing analogues of several results due to Lutz and Mayordomo, who studied the effective dimension of points in self-similar fractals in Euclidean space. Lastly, we explore the connections between the channel capacity of a length function derived from a finite tree and the measure of maximum entropy on a related directed multigraph that encodes the structure of our tree, drawing on work by Abram and Lagarias on path sets, where a path set is a generalization of the notion of a sofic shift.

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arXiv2024

Bridging Computational Notions of Depth

In this article, we study the relationship between notions of depth for sequences, namely, Bennett's notions of strong and weak depth, and deep $Π^0_1$ classes, introduced by the authors and motivated by previous work of Levin. For the first main result of the study, we show that every member of a $Π^0_1$ class is order-deep, a property that implies strong depth. From this result, we obtain new examples of strongly deep sequences based on properties studied in computability theory and algorithmic randomness. We further show that not every strongly deep sequence is a member of a deep $Π^0_1$ class. For the second main result, we show that the collection of strongly deep sequences is negligible, which is equivalent to the statement that the probability of computing a strongly deep sequence with some random oracle is 0, a property also shared by every deep $Π^0_1$ class. Finally, we show that variants of strong depth, given in terms of a priori complexity and monotone complexity, are equivalent to weak depth.

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arXiv2024

Continuous Randomness via Transformations of 2-Random Sequences

Reimann and Slaman initiated the study of sequences that are Martin-Löf random with respect to a continuous measure, establishing fundamental facts about NCR, the collection of sequences that are not Martin-Löf random with respect to any continuous measure. In the case of sequences that are random with respect to a computable, continuous measure, the picture is fairly well-understood: such sequences are truth-table equivalent to a Martin-Löf random sequence. However, given a sequence that is random with respect to a continuous measure but not with respect to any computable measure, we can ask: how close to effective is the measure with respect to which it is continuously random? In this study, we take up this question by examining various transformations of 2-random sequences (sequences that are Martin-Löf random relative to the halting set $\emptyset'$) to establish several results on sequences that are continuously random with respect to a measure that is computable in $\emptyset'$. In particular, we show that (i) every noncomputable sequence that is computable from a 2-random sequence is Martin-Löf random with respect to a continuous, $\emptyset'$-computable measure and (ii) the Turing jump of every 2-random sequence is Martin-Löf random with respect to a continuous, $\emptyset'$-computable measure. From these results, we obtain examples of sequences that are not proper, i.e., not random with respect to any computable measure, but are random with respect to a continuous, $\emptyset'$-computable measure. Lastly, we consider the behavior of 2-randomness under a wider class of effective operators (c.e. operators, pseudojump operators, and operators defined in terms of pseudojump inversion), showing that these too yield sequences that are Martin-Löf random with respect to a continuous, $\emptyset'$-computable measure.

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arXiv2021

Degrees of Randomized Computability

In this survey we discuss work of Levin and V'yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V'yugin introduced an ordering on collections of sequences that are closed under Turing equivalence. Roughly speaking, given two such collections $\mathcal{A}$ and $\mathcal{B}$, $\mathcal{A}$ is below $\mathcal{B}$ in this ordering if $\mathcal{A}\setminus\mathcal{B}$ is negligible. The degree structure associated with this ordering, the Levin-V'yugin degrees (or LV-degrees), can be shown to be a Boolean algebra, and in fact a measure algebra. We demonstrate the interactions of this work with recent results in computability theory and algorithmic randomness: First, we recall the definition of the Levin-V'yugin algebra and identify connections between its properties and classical properties from computability theory. In particular, we apply results on the interactions between notions of randomness and Turing reducibility to establish new facts about specific LV-degrees, such as the LV-degree of the collection of 1-generic sequences, that of the collection of sequences of hyperimmune degree, and those collections corresponding to various notions of effective randomness. Next, we provide a detailed explanation of a complex technique developed by V'yugin that allows the construction of semi-measures into which computability-theoretic properties can be encoded. We provide two examples of the use of this technique by explicating a result of V'yugin's about the LV-degree of the collection of Martin-Löf random sequences and extending the result to the LV-degree of the collection of sequences of DNC degree.

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arXiv2021

The intersection of algorithmically random closed sets and effective dimension

In this article, we study several aspects of the intersections of algorithmically random closed sets. First, we answer a question of Cenzer and Weber, showing that the operation of intersecting relatively random closed sets (with respect to certain underlying measures induced by Bernoulli measures on the space of codes of closed sets), which preserves randomness, can be inverted: a random closed set of the appropriate type can be obtained as the intersection of two relatively random closed sets. We then extend the Cenzer/Weber analysis to the intersection of multiple random closed sets, identifying the Bernoulli measures with respect to which the intersection of relatively random closed sets can be non-empty. We lastly apply our analysis to provide a characterization of the effective Hausdorff dimension of sequences in terms of the degree of intersectability of random closed sets that contain them.

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arXiv2021

Randomness extraction in computability theory

In this article, we study a notion of the extraction rate of Turing functionals that translate between notions of randomness with respect to different underlying probability measures. We analyze several classes of extraction procedures: a first class that generalizes von Neumann's trick for extracting unbiased randomness from the tosses of a biased coin, a second class based on work of generating biased randomness from unbiased randomness by Knuth and Yao, and a third class independently developed by Levin and Kautz that generalizes the data compression technique of arithmetic coding. For the first two classes of extraction procedures, we identify a level of algorithmic randomness for an input that guarantees that we attain the extraction rate along that input, while for the third class, we calculate the rate attained along sufficiently random input sequences.

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arXiv2020

Key developments in algorithmic randomness

The goal of this introductory survey is to present the major developments of algorithmic randomness with an eye toward its historical development. While two highly comprehensive books and one thorough survey article have been written on the subject, our goal is to provide an introduction to algorithmic randomness that will be both useful for newcomers who want to develop a sense of the field quickly and interesting for researchers already in the field who would like to see these results presented in chronological order.

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arXiv2019

Rank and randomness

We show that for each computable ordinal $α>0$ it is possible to find in each Martin-Löf random $Δ^0_2$ degree a sequence $R$ of Cantor-Bendixson rank $α$, while ensuring that the sequences that inductively witness $R$'s rank are all Martin-Löf random with respect to a single countably supported and computable measure. This is a strengthening for random degrees of a recent result of Downey, Wu, and Yang, and can be understood as a randomized version of it.

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arXiv2019

Kolmogorov complexity and generalized length functions

Kolmogorov complexity measures the algorithmic complexity of a finite binary string $σ$ in terms of the length of the shortest description $σ^*$ of $σ$. Traditionally, the length of a string is taken to measure the amount of information contained in the string. However, we may also view the length of $σ$ as a measure of the cost of producing $σ$, which permits one to generalize the notion of length, wherein the cost of producing a 0 or a 1 can vary in some prescribed manner. In this article, we initiate the study of this generalization of length based on the above information cost interpretation. We also modify the definition of Kolmogorov complexity to use such generalized length functions instead of standard length. We further investigate conditions under which the notion of complexity defined in terms of a given generalized length function preserves some essential properties of Kolmogorov complexity. We focus on a specific class of generalized length functions that are intimately related to a specific subcollection of Bernoulli $p$-measures, namely those corresponding to the unique computable real $p\in(0,1)$ such that $p^k=1-p$, for integers $k\geq 1$. We then study randomness with respect to such measures, by proving a generalization version of the classic Levin-Schnorr theorem that involves $k$-length functions and then proving subsequent results that involve effective dimension and entropy.

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arXiv2019

Effective Aspects of Bernoulli Randomness

In this paper, we study Bernoulli random sequences, i.e., sequences that are Martin-Löf random with respect to a Bernoulli measure $μ_p$ for some $p\in[0,1]$, where we allow for the possibility that $p$ is noncomputable. We focus in particular on the case in which the underlying Bernoulli parameter $p$ is proper (that is, Martin-Löf random with respect to some computable measure). We show for every Bernoulli parameter $p$, if there is a sequence that is both proper and Martin-Löf random with respect to $μ_p$, then $p$ itself must be proper, and explore further consequences of this result. We also study the Turing degrees of Bernoulli random sequences, showing, for instance, that the Turing degrees containing a Bernoulli random sequence do not coincide with the Turing degrees containing a Martin-Löf random sequence. Lastly, we consider several possible approaches to characterizing blind Bernoulli randomness, where the corresponding Martin-Löf tests do not have access to the Bernoulli parameter $p$, and show that these fail to characterize blind Bernoulli randomness.

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arXiv2018

On the interplay between effective notions of randomness and genericity

In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence (which answers an open question posed by Barmpalias, Day, and Lewis) and that every Demuth random sequence forms a minimal pair with every pb-generic sequence (where pb-genericity is an effective notion of genericity that is strictly between 1-genericity and 2-genericity). Moreover, we prove that for every comeager $\mathcal{G}\subseteq 2^ω$, there is some weakly 2-random sequence $X$ that computes some $Y\in\mathcal{G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity.

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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.

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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.

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arXiv2017

Deep $Π^0_1$ Classes

A set of infinite binary sequences $\mathcal{C}\subseteq2^ω$ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of $Π^0_1$ classes. In this paper, we introduce the notion of depth for $Π^0_1$ classes, which is a stronger form of negligibility. Whereas a negligible $Π^0_1$ class $\mathcal{C}$ has the property that one cannot probabilistically compute a member of $\mathcal{C}$ with positive probability, a deep $Π^0_1$ class $\mathcal{C}$ has the property that one cannot probabilistically compute an initial segment of a member of $\mathcal{C}$ with high probability. That is, the probability of computing a length $n$ initial segment of a deep $Π^0_1$ class converges to 0 effectively in $n$. We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call $\mathit{tt}$-negligibility. We also provide a number of examples of deep $Π^0_1$ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, we examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin's $Ω$.

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arXiv2016

The random members of a $Π^0_1$ class

We examine several notions of randomness for elements in a given $Π^0_1$ class $\mathcal{P}$. Such an effectively closed subset $\mathcal{P}$ of $2^ω$ may be viewed as the set of infinite paths through the tree $T_{\mathcal{P}}$ of extendible nodes of $\mathcal{P}$, i.e., those finite strings that extend to a member of $\mathcal{P}$, so one approach to defining a random member of $\mathcal{P}$ is to randomly produce a path through $T_{\mathcal{P}}$ using a sufficiently random oracle for advice. In addition, this notion of randomness for elements of $\mathcal{P}$ may be induced by a map from $2^ω$ onto $\mathcal{P}$ that is computable relative to $T_{\mathcal{P}}$, and the notion even has a characterization in term of Kolmogorov complexity. Another approach is to define a relative measure on $\mathcal{P}$ by conditionalizing the Lebesgue measure on $\mathcal{P}$, which becomes interesting if $\mathcal{P}$ has Lebesgue measure 0. Lastly, one can alternatively define a notion of incompressibility for members of $\mathcal{P}$ in terms of the amount of branching at levels of $T_{\mathcal{P}}$. We explore some notions of homogeneity for $Π^0_1$ classes, inspired by work of van Lambalgen. A key finding is that in a specific class of sufficiently homogeneous $Π^0_1$ classes $\mathcal{P}$, each of these approaches coincides. We conclude with a discussion of random members of $Π^0_1$ classes of positive measure.

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arXiv2016

Randomness for computable measures and initial segment complexity

We study the possible growth rates of the Kolmogorov complexity of initial segments of sequences that are random with respect to some computable measure on $2^ω$, the so-called proper sequences. Our main results are as follows: (1) We show that the initial segment complexity of a proper sequence $X$ is bounded from below by a computable function (that is, $X$ is complex) if and only if $X$ is random with respect to some computable, continuous measure. (2) We prove that a uniform version of the previous result fails to hold: there is a family of complex sequences that are random with respect to a single computable measure such that for every computable, continuous measure $μ$, some sequence in this family fails to be random with respect to $μ$. (3) We show that there are proper sequences with extremely slow-growing initial segment complexity, that is, there is a proper sequence the initial segment complexity of which is infinitely often below every computable function, and even a proper sequence the initial segment complexity of which is dominated by all computable functions. (4) We prove various facts about the Turing degrees of such sequences and show that they are useful in the study of certain classes of pathological measures on $2^ω$, namely diminutive measures and trivial measures.

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arXiv2015

The interplay of classes of algorithmically random objects

We study algorithmically random closed subsets of $2^ω$, algorithmically random continuous functions from $2^ω$ to $2^ω$, and algorithmically random Borel probability measures on $2^ω$, especially the interplay between these three classes of objects. Our main tools are preservation of randomness and its converse, the no randomness ex nihilo principle, which say together that given an almost-everywhere defined computable map between an effectively compact probability space and an effective Polish space, a real is Martin-Löf random for the pushforward measure if and only if its preimage is random with respect to the measure on the domain. These tools allow us to prove new facts, some of which answer previously open questions, and reprove some known results more simply. Our main results are the following. First we answer an open question of Barmapalias, Brodhead, Cenzer, Remmel, and Weber by showing that $\mathcal{X}\subseteq2^ω$ is a random closed set if and only if it is the set of zeros of a random continuous function on $2^ω$. As a corollary we obtain the result that the collection of random continuous functions on $2^ω$ is not closed under composition. Next, we construct a computable measure $Q$ on the space of measures on $2^ω$ such that $\mathcal{X}\subseteq2^ω$ is a random closed set if and only if $\mathcal{X}$ is the support of a $Q$-random measure. We also establish a correspondence between random closed sets and the random measures studied by Culver in previous work. Lastly, we study the ranges of random continuous functions, showing that the Lebesgue measure of the range of a random continuous function is always contained in $(0,1)$.

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arXiv2015

Algorithmically random functions and effective capacities

We continue the investigation of algorithmically random functions and closed sets, and in particular the connection with the notion of capacity. We study notions of random continuous functions given in terms of a family of computable measures called symmetric Bernoulli measures. We isolate one particular class of random functions that we refer to as online random functions $F$, where the value of $y(n)$ for $y = F(x)$ may be computed from the values of $x(0),\dots,x(n)$. We show that random online functions are neither onto nor one-to-one. We give a necessary condition on the members of the ranges of online random functions in terms of initial segment complexity and the associated computable capacity. Lastly, we introduce the notion of online \emph{partial} Martin-Löf random function on $2^ω$ and give a family of online partial random functions the ranges of which are precisely the random closed sets introduced by Barmpalias, Brodhead, Cenzer, Dashti, and Weber.

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