Search

A major open problem in computational complexity is the existence of a one-way function, namely a function from strings to strings which is computationally easy to compute but hard to invert. Levin (2023) formulated the notion of one-way functions from reals (infinite bit-sequences) to reals in terms of computability, and asked whether partial computable one-way functions exist. We give a strong positive answer using the hardness of the halting problem and exhibiting a total computable one-way function.

Arranging the bits of a random string or real into k columns of a two-dimensional array or higher dimensional structure is typically accompanied with loss in the Kolmogorov complexity of the columns, which depends on k. We quantify and characterize this phenomenon for arrays and trees and its relationship to negligible classes.

We study the compressibility of enumerations, and its role in the relative Kolmogorov complexity of computably enumerable sets, with respect to density. With respect to a strong and a weak form of compression, we examine the gain: the amount of auxiliary information embedded in the compressed enumeration. Strong compression and weak gainless compression is shown for any computably enumerable set, and a positional game is studied toward understanding strong gainless compression.

We study effective randomness-preserving transformations of path-incompressible trees. Some path-incompressible trees with infinitely many paths do not compute perfect path-random trees with computable oracle-use. Sparse perfect path-incompressible trees can be effectively densified, almost surely. We characterize the branching density of path-random trees., Comment: Revised

We show that degrees containing a complete extensions of arithmetic have the random join property: they are the supremum of any random real they compute, with another random real. The same is true for the truth-table and weak truth-table reducibilities.

Muchnik's paradox says that enumerable betting strategies are not always reducible to enumerable strategies whose bets are restricted to either even rounds or odd rounds. In other words, there are outcome sequences x where an effectively enumerable strategy succeeds, but no such parity-restricted effectively enumerable strategy does. We characterize the effective Hausdorff dimension of such $x$, showing that it can be as low as 1/2 but not less. We also show that such reals that are random with respect to parity-restricted effectively enumerable strategies with packing dimension as low as $\log\sqrt3$. Finally we exhibit Muchnik's paradox in the case of computable integer-valued strategies.

We study the problem of whether a betting-strategy can be decomposed into an equivalent set of simpler betting-strategies, such as betting-strategies that bet on a restricted set of stages or bet on a restricted of favorable outcomes. We show that the class of effectively enumerable betting-strategies has irreducible members which cannot be decomposed into an equivalent set of simpler betting-strategies. We answer questions of Kastermans and Hitchcock by constructing a real on which no kastergale (which is a left-c.e. supermartingale whose favorable outcomes are effectively determined) succeeds, but some unrestricted left-c.e. supermartingale succeeds on it. We generalize a result of Muchnick by showing that there is a non-1-random real such that no muchgale (which is a left-c.e. supermartingale who does not bet on certain stages) succeeds on it. Our methodology is then used to obtain further irreducibility results, strongly supporting a conjecture that if a natural class of left-c.e. supermartingales defines 1-randomness, then a single member of that class can do so. In another word, the class of left-c.e. supermartingales cannot be reduced to a simpler, natural subclass of it. For example, we show that there is a non-1-random real such that no kastergale or muchgale succeeds on it.

Leonid Levin (arxiv.org/abs/cs/0503039v14, p.7) published a new (and very nice) proof of G\'acs-Ku\v{c}era's theorem that occupies only a few lines when presented in his style. We try to explain more details and discuss the connection of this proof with image randomness theorems, making explicit some result (see Proposition 4) that is implicit in Levin's exposition. Then we review the previous work about the oracle use when reducing a given sequence to another one, and its connection with algorithmic dimension theory., Comment: published version

A tree is pathwise-random if all of its paths are Martin-Lof random. We show that (a) no weakly 2-random real computes a perfect pathwise-random tree; it follows that the class of perfect pathwise-random trees is null, with respect to any computable measure; (b) there exists a positive-measure pathwise-random tree which does not compute any complete extension of Peano arithmetic; and (c) there exists a perfect pathwise-random tree which does not compute any tree of positive measure and finite randomness deficiency. We then obtain models of second-order arithmetic that separate compactness principles below weak Konigs lemma, answering questions by Chong et al.(2019)., Comment: Revision

Suppose that the outcomes of a roulette table are not entirely random, in the sense that there exists a successful betting strategy. Is there a successful `separable' strategy, in the sense that it does not use the winnings from betting on red in order to bet on black, and vice-versa? We study this question from an algorithmic point of view and observe that every strategy $M$ can be replaced by a separable strategy which is computable from $M$ and successful on any outcome-sequence where $M$ is successful. We then consider the case of mixtures and show: (a) there exists an effective mixture of separable strategies which succeeds on every casino sequence with effective Hausdorff dimension less than 1/2; (b) there exists a casino sequence of effective Hausdorff dimension 1/2 on which no effective mixture of separable strategies succeeds. Finally we extend (b) to a more general class of strategies.

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.

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.

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.

We show that there is a strong minimal pair in the computably enumerable Turing degrees.

We show that given any non-computable left-c.e. real $\alpha$ there exists a left-c.e. real $\beta$ such that $\alpha\neq \beta+\gamma$ for all left-c.e. reals and all right-c.e. reals $\gamma$. The proof is non-uniform, the dichotomy being whether the given real $\alpha$ 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.

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.

We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefix-free complexity. A computably enumerable set $A$ is $K$-trivial if and only if the family of sets with complexity bounded by the complexity of $A$ is uniformly computable from the halting problem.

The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov's 0-1 law that for any property which may or may not be satisfied by any given Turing degree, the satisfying class will either be of Lebesgue measure 0 or 1, so long as it is measurable. So either the \emph{typical} degree satisfies the property, or else the typical degree satisfies its negation. Further, there is then some level of randomness sufficient to ensure typicality in this regard. A similar analysis can be made in terms of Baire category, where a standard form of genericity now plays the role that randomness plays in the context of measure. We describe and prove a number of results in a programme of research which aims to establish the properties of the typical Turing degree, where typicality is gauged either in terms of Lebesgue measure or Baire category., Comment: 42 pages

We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set $A$ with non-trivial prefix-free initial segment complexity, there exists a Turing complete computably enumerable set $B$ with complexity strictly less than the complexity of $A$. On the other hand it is known that sets with trivial initial segment prefix-free complexity are not Turing complete. Moreover we give a generalization of this result for any finite collection of computably enumerable sets $A_i, i