Publications on FANG Nan

Speedability of Computably Approximable Reals and Their Approximations

Fri, 10 Apr 2026 00:00:00 +0000

Abstract

An approximation of a real is a sequence of rational numbers that converges to the real. An approximation is left-c.e. if it is computable and nondecreasing and is d.c.e. if it is computable and has bounded variation. A real is computably approximable if it has some computable approximation, and left-c.e. and d.c.e. reals are defined accordingly. An approximation {as}s∈ω is speedable if there exists a nondecreasing computable function f such that the approximation {af(s)}s∈ω converges in a certain formal sense faster than {as}s∈ω. This leads to various notions of speedability for reals, e.g., one may require for a computably approximable real that either all or some of its approximations of a specific type are speedable. Merkle and Titov established the equivalence of several speedability notions for left-c.e. reals that are defined in terms of left-c.e. approximations. We extend these results to d.c.e. reals and d.c.e. approximations, and we prove that in this setting, being speedable is equivalent to not being Martin-Löf random. Finally, we demonstrate that every computably approximable real has a computable approximation that is speedable.

Integer-Valued Martingales and Cl-Turing Reductions

Fri, 28 Mar 2025 00:00:00 +0000

Abstract

We prove that a sequence X is integer-valued random (IVR) if and only if for every cl-tt reduction Φ, there are only finitely many oracles that compute X via Φ. A well-known property of integer-valued supermartingales is that under any cone, there is a subcone where the values are constant. We extend this property and prove that for a special integer-valued martingale M, which starts with initial capital 1 and always bets 1 on bit “1” if possible, and for any integervalued supermartingale f , under any cone there is a subcone such that f actually simulates M. Using this property, we prove that there exists a non-IVR sequence that is cl-computable by only countably many oracles, as a complement to the first result.

Extending CL-reducibility on Array Noncomputable Degrees

Mon, 18 Nov 2024 00:00:00 +0000

Abstract

Given a function f , f -bounded-Turing ( f -bT-) reducibility is the Turing reducibility with use function bounded by f . In the special case where f = id + c (with id being the identity function and c a constant), this is referred to as cl-reducibility. In a work by Barmpalias, Fang, and Lewis-Pye, it was proven that there exist two left-c.e. reals such that no leftc.e. real (id+g)-bT-computes both of them whenever g is computable, nondecreasing, and satisfies ∑ n 2−g(n) = ∞. Moreover, such maximal pairs exist precisely within every array noncomputable degree. This result generalizes a prior result on cl-reducibility, which states that there exist two left-c.e. reals such that no left-c.e. real cl-computes both of them. An open question remained as to whether a similar extension could apply to another result on cl-reducibility, which asserts that there exists a left-c.e. real not cl-reducible to any random left-c.e. real. We answer this question affirmatively, providing a simpler proof compared to previous works. Additionally, we streamline the proof of the initial extension.

Granularity of Wagers in Games and the Possibility of Saving

Tue, 30 Jun 2020 00:00:00 +0000

Abstract

In a casino where arbitrarily small bets are admissible, any betting strategy M can be modified into a saving strategy that, not only is successful on each casino sequence where M is (thus accumulating unbounded wealth inside the casino) but also saves an unbounded capital, by permanently and gradually withdrawing it from the game. Teutsch showed that this is no longer the case when a fixed minimum wager is imposed by the casino, thus exemplifying a savings paradox where a player can win unbounded wealth inside the casino, but upon withdrawing a sufficiently large amount out of the game, he is forced into bankruptcy. We study the potential for saving under a shrinking minimum wager rule (granularity) and its dependence on the rate of decrease (inflation) as well as timid versus bold play.

Monotonous Betting Strategies in Warped Casinos

Wed, 16 Oct 2019 00:00:00 +0000

Abstract

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.

Equivalences between Learning of Data and Probability Distributions, and Their Applications

Wed, 01 Aug 2018 00:00:00 +0000

Abstract

Algorithmic learning theory traditionally studies the learnability of effective infinite binary sequences (reals), while recent work by Vitányi and Chater 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.

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

Sat, 11 Jun 2016 00:00:00 +0000

Abstract

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, i.e. the series ∑n2n−h(n) converges, (2) for every c.e. real α there exists a Turing functional via which Ω computes α with use bounded by h. We also give a similar characterisation with respect to computations of c.e. sets from Ω, 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, Ω computes A with use bounded by g. Further results and some connections with Solovay functions (studied by a number of authors [38], [3], [26], [11]) are given.