arXiv subjects

Satyadev Nandakumar

Publications and source records attributed to Satyadev Nandakumar.

At least 19 recordsLinked to original sources

arXiv2026

Resource bounded Ku\v{c}era-G\'{a}cs Theorems

The Ku\v{c}era--G\'{a}cs theorem is a fundamental result in algorithmic randomness. It states that every infinite sequence $X$ is Turing reducible to a Martin-L\"of random $R$. This paper studies resource-bounded analogues of the Ku\v{c}era-G\'acs Theorem, at the resource bounds of polynomial-time and finite-state computation. We prove a {quasi-polynomial-time}{ Ku\v{c}era-G\'acs Theorem}, showing that every infinite sequence $X$ is quasi-polynomial-time reducible to a \emph{polynomial-time random} sequence $R$. We also show that for any $X$, the oracle use of $R$ is $n+o(n)$ bits for obtaining the first $n$ bits of $X$. We then study the relationship between compressibility and Turing reductions, in the polynomial-time setting. We establish that $\rho^-_{\mathsf{poly}}(X) = K_{poly}(X)$, demonstrating that the lower polynomial-time Turing decompression ratio is precisely characterized by the polynomial-time Kolmogorov complexity rate. We note that this characterization fails for the polynomial-time dimension if one-way functions exist, resolving an open problem from Doty's work. We use these results to strengthen the {quasi-polynomial-time}{ Ku\v{c}era-G\'acs Theorem}. We show that every infinite sequence $X$ is quasi-polynomial-time reducible to a {polynomial-time random} sequence $R$, where the lower oracle use rate of the reduction is less than ${K}_{poly}(X)$. We also show that any sequence extracted from the (even larger) set of \emph{normal sequences} by a finite-state reduction must have a convergent asymptotic frequency for its symbols. Since sequences lacking this invariant property exist, they cannot be finite-state reduced from any normal sequence. Hence we show that the Ku\v{c}era-G\'acs theorem \emph{fails} for finite-state reductions.

cs.CC↗

arXiv2026

On the Constructive Dimension Spectrum of Polynomials

Recently, Stull [18], [17] resolved a long-standing open problem posed by Lutz, on whether the set of effective Hausdorff dimensions of points on a straight line in $\mathbb{R}^2$ -- the effective dimension spectrum of the line -- contains a unit interval. This question is related to problems in classical fractal geometry like the Kakeya conjecture and Furstenberg sets. Stull posed an open question on the dimension spectra of polynomial curves. For the first result, with new techniques which adapt the theory of classical real root-finding of polynomials to the current setting, we show that the dimension spectra of every polynomial curve contains at least two points. This answers an open question posed by Stull [18], [17]. We use the main result to construct a class of polynomials which have width strictly greater than 1, answering a second problem stated in [18],[17]. Stull [18] resolved the dimension spectrum conjecture for planar lines, showing that it contains a unit interval. For the second result, we resolve the conjecture for a subfamily of polynomials whose coefficients form a "low" dimension point in $\mathbb{R}^{d+1}$.

math.GM↗

arXiv2025

On Effective Banach-Mazur Games and an application to the Poincar\'e Recurrence Theorem for Category

The classical Banach-Mazur game characterizes sets of first category in a topological space. In this work, we show that an effectivized version of the game yields a characterization of sets of effective first category. Using this, we give a proof for the effective Banach Category Theorem. Further, we provide a game-theoretic proof of an effective theorem in dynamical systems, namely the category version of Poincar\'e Recurrence. The Poincar\'e Recurrence Theorem for category states that for a homeomorphism without open wandering sets, the set of non recurrent points forms a first category (meager) set. As an application of the effectivization of the Banach-Mazur game, we show that such a result holds true in effective settings as well.

math.LO↗

arXiv2024

One-Way Functions and Polynomial Time Dimension

This paper demonstrates a duality between the non-robustness of polynomial time dimension and the existence of one-way functions. Polynomial-time dimension (denoted $\mathrm{cdim}_\mathrm{P}$) quantifies the density of information of infinite sequences using polynomial time betting algorithms called $s$-gales. An alternate quantification of the notion of polynomial time density of information is using polynomial-time Kolmogorov complexity rate (denoted $\mathcal{K}_\text{poly}$). Hitchcock and Vinodchandran (CCC 2004) showed that $\mathrm{cdim}_\mathrm{P}$ is always greater than or equal to $\mathcal{K}_\text{poly}$. We first show that if one-way functions exist then there exists a polynomial-time samplable distribution with respect to which $\mathrm{cdim}_\mathrm{P}$ and $\mathcal{K}_\text{poly}$ are separated by a uniform gap with probability $1$. Conversely, we show that if there exists such a polynomial-time samplable distribution, then (infinitely-often) one-way functions exist. Using our main results, we solve a long standing open problem posed by Hitchcock and Vinodchandran (CCC 2004) and Stull under the assumption that one-way functions exist. We demonstrate that if one-way functions exist, then there are individual sequences $X$ whose poly-time dimension strictly exceeds $\mathcal{K}_\text{poly}(X)$, that is $\mathrm{cdim}_\mathrm{P}(X) > \mathcal{K}_\text{poly}(X)$. Further, we show that the gap between these quantities can be made as large as possible (i.e. close to 1). We also establish similar bounds for strong poly-time dimension versus asymptotic upper Kolmogorov complexity rates.

cs.CC↗

arXiv2024

Point-to-set Principle and Constructive Dimension Faithfulness

Hausdorff $\Phi$-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce an effective version of Hausdorff $\Phi$-dimension, which we call constructive $\Phi$-dimension. We prove a point-to-set principle for $\Phi$-dimension. We also provide a characterization of constructive $\Phi$-dimension using Kolmogorov complexity and $s$-gales. Finally, we apply these tools to study faithfulness of coverings $\Phi$. A family of coverings $\Phi$ is said to be faithful to Hausdorff dimension if the $\Phi$-dimension and Hausdorff dimension coincide for every set. Similarly, $\Phi$ is said to be faithful to constructive dimension if the constructive $\Phi$-dimension and constructive dimension coincide for every set. We derive the necessary and sufficient conditions for the constructive dimension faithfulness of the coverings generated by the Cantor series expansion, based on the terms of the expansion. Using the point-to-set principle for Cantor coverings, we show that the same condition characterises Hausdorff dimension faithfulness of Cantor coverings, thereby giving an information theoretic proof of the result by Albeverio, Ivanenko, Lebid, and Torbin. We investigate the question of weather the notions of faithfulness at Hausdorff and constructive levels are equivalent. Using a new technique for the construction of sequences satisfying a certain Kolmogorov complexity condition, we show that the notions of ``faithfulness'' of Cantor coverings at the Hausdorff and constructive levels are equivalent, independent of the log-limit condition.

cs.IT↗

arXiv2023

Analysis of Linux-PRNG (Pseudo Random Number Generator)

The Linux pseudorandom number generator (PRNG) is a PRNG with entropy inputs and is widely used in many security-related applications and protocols. This PRNG is written as an open-source code which is subject to regular changes. It has been analysed in the works of Gutterman et al., Lacharme et al., while in the meantime, several changes have been applied to the code, to counter the attacks presented since then. Our work describes the Linux PRNG of kernel versions 5.3 and upwards. We discuss the PRNG architecture briefly and in detail about the entropy mixing function. Our goal is to study the entropy mixing function and analyse it over two properties, namely, injectivity and length of the longest chain. For this purpose, we will be using SAT solving and model counting over targetted formulas involving multiple states of the Linux entropy store.

cs.PL↗

arXiv2023

Effective Continued Fraction Dimension versus Effective Hausdorff Dimension of Reals

We establish that constructive continued fraction dimension originally defined using $s$-gales is robust, but surprisingly, that the effective continued fraction dimension and effective (base-$b$) Hausdorff dimension of the same real can be unequal in general. We initially provide an equivalent characterization of continued fraction dimension using Kolmogorov complexity. In the process, we construct an optimal lower semi-computable $s$-gale for continued fractions. We also prove new bounds on the Lebesgue measure of continued fraction cylinders, which may be of independent interest. We apply these bounds to reveal an unexpected behavior of continued fraction dimension. It is known that feasible dimension is invariant with respect to base conversion. We also know that Martin-Löf randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation. In contrast, for any $0 < \varepsilon < 0.5$, we prove the existence of a real whose effective Hausdorff dimension is less than $\varepsilon$, but whose effective continued fraction dimension is greater than or equal to $0.5$. This phenomenon is related to the ``non-faithfulness'' of certain families of covers, investigated by Peres and Torbin and by Albeverio, Ivanenko, Lebid and Torbin. We also establish that for any real, the constructive Hausdorff dimension is at most its effective continued fraction dimension.

cs.IT↗

arXiv2023

Finite-State Relative Dimension, dimensions of A. P. subsequences and a Finite-State van Lambalgen's theorem

Finite-state dimension (Dai, Lathrop, Lutz, and Mayordomo (2004)) quantifies the information rate in an infinite sequence as measured by finite-state automata. In this paper, we define a relative version of finite-state dimension. The finite-state relative dimension $dim_{FS}^Y(X)$ of a sequence $X$ relative to $Y$ is the finite-state dimension of $X$ measured using the class of finite-state gamblers with an oracle access to $Y$. We show its mathematical robustness by equivalently characterizing this notion using the relative block entropy rate of $X$ conditioned on $Y$. We derive inequalities relating the dimension of a sequence to the relative dimension of its subsequences along any arithmetic progression (A.P.). These enable us to obtain a strengthening of Wall's Theorem on the normality of A.P. subsequences of a normal number, in terms of relative dimension. In contrast to the original theorem, this stronger version has an exact converse yielding a new characterization of normality. We also obtain finite-state analogues of van Lambalgen's theorem on the symmetry of relative normality.

cs.IT↗

arXiv2023

A Weyl Criterion for Finite-State Dimension and Applications

Finite-state dimension, introduced early in this century as a finite-state version of classical Hausdorff dimension, is a quantitative measure of the lower asymptotic density of information in an infinite sequence over a finite alphabet, as perceived by finite automata. Finite-state dimension is a robust concept that now has equivalent formulations in terms of finite-state gambling, lossless finite-state data compression, finite-state prediction, entropy rates, and automatic Kolmogorov complexity. The Schnorr-Stimm dichotomy theorem gave the first automata-theoretic characterization of normal sequences, which had been studied in analytic number theory since Borel defined them. This theorem implies that a sequence (or a real number having this sequence as its base-b expansion) is normal if and only if it has finite-state dimension 1. One of the most powerful classical tools for investigating normal numbers is the Weyl criterion, which characterizes normality in terms of exponential sums. Such sums are well studied objects with many connections to other aspects of analytic number theory, and this has made use of Weyl criterion especially fruitful. This raises the question whether Weyl criterion can be generalized from finite-state dimension 1 to arbitrary finite-state dimensions, thereby making it a quantitative tool for studying data compression, prediction, etc. This paper does exactly this. We extend the Weyl criterion from a characterization of sequences with finite-state dimension 1 to a criterion that characterizes every finite-state dimension. This turns out not to be a routine generalization of the original Weyl criterion. Even though exponential sums may diverge for non-normal numbers, finite-state dimension can be characterized in terms of the dimensions of the subsequence limits of the exponential sums. We demonstrate the utility of our criterion though examples.

cs.IT↗

arXiv2022

Real numbers equally compressible in every base

This work solves an open question in finite-state compressibility posed by Lutz and Mayordomo about compressibility of real numbers in different bases. Finite-state compressibility, or equivalently, finite-state dimension, quantifies the asymptotic lower density of information in an infinite sequence. Absolutely normal numbers, being finite-state incompressible in every base of expansion, are precisely those numbers which have finite-state dimension equal to $1$ in every base. At the other extreme, for example, every rational number has finite-state dimension equal to $0$ in every base. Generalizing this, Lutz and Mayordomo (2021) posed the question: are there numbers which have absolute positive finite-state dimension strictly between 0 and 1 - equivalently, is there a real number $ξ$ and a compressibility ratio $s \in (0,1)$ such that for every base $b$, the compressibility ratio of the base-$b$ expansion of $ξ$ is precisely $s$? It is conceivable that there is no such number. Indeed, some works explore ``zero-one'' laws for other feasible dimensions - i.e. sequences with certain properties either have feasible dimension 0 or 1, taking no value strictly in between. However, we answer the question of Lutz and Mayordomo affirmatively by proving a more general result. We show that given any sequence of rational numbers $\langle q_b \rangle$, we can explicitly construct a single number $ξ$ such that for any base $b$, the finite-state dimension/compression ratio of $ξ$ in base-$b$ is $q_b$. As a special case, this result implies the existence of absolutely dimensioned numbers for any given rational dimension between $0$ and $1$, as posed by Lutz and Mayordomo. In our construction, we combine ideas from Wolfgang Schmidt's construction of absolutely normal numbers (1962), results regarding low discrepancy sequences and several new estimates related to exponential sums.

cs.IT↗

arXiv2021

Ergodic Theorems for PSPACE functions and their converses

We initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow. In contrast, we show that for a class of PSPACE L1 functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists for all PSPACE randoms and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x.

cs.CC↗

arXiv2019

An analogue of Pillai's theorem for continued fraction normality and an application to subsequences

We show that two notions of continued fraction normality, one where overlapping occurrences of finite patterns are counted as distinct occurrences, and another where only disjoint occurrences are counted as distinct, are identical. This equivalence involves an analogue of a theorem due to S. S. Pillai in 1940 for base-$b$ expansions. The proof requires techniques which are fundamentally different, since the continued fraction expansion utilizes a countably infinite alphabet, leading to a non-compact space. Utilizing the equivalence of these two notions, we provide a new proof of Heersink and Vandehey's recent result that selection of subsequences along arithmetic progressions does not preserve continued fraction normality.

math.DS↗

arXiv2017

Normality of the Ehrenfeucht-Mycielski Sequence

We study the binary Ehrenfeucht Mycielski sequence seeking a balance between the number of occurrences of different binary strings. There have been numerous attempts to prove the balance conjecture of the sequence, which roughly states that 1 and 0 occur equally often in it. Our contribution is twofold. First, we study weaker forms of the conjecture proved in the past and lay out detailed proofs for many lemmas which were stated without proofs. Secondly, we extend the claim of balance to that of normality and prove a weaker form of simple normality to word length 2.

cs.DM↗

arXiv2017

On Resource-bounded versions of the van Lambalgen theorem

The van Lambalgen theorem is a surprising result in algorithmic information theory concerning the symmetry of relative randomness. It establishes that for any pair of infinite sequences $A$ and $B$, $B$ is Martin-Löf random and $A$ is Martin-Löf random relative to $B$ if and only if the interleaved sequence $A \uplus B$ is Martin-Löf random. This implies that $A$ is relative random to $B$ if and only if $B$ is random relative to $A$ \cite{vanLambalgen}, \cite{Nies09}, \cite{HirschfeldtBook}. This paper studies the validity of this phenomenon for different notions of time-bounded relative randomness. We prove the classical van Lambalgen theorem using martingales and Kolmogorov compressibility. We establish the failure of relative randomness in these settings, for both time-bounded martingales and time-bounded Kolmogorov complexity. We adapt our classical proofs when applicable to the time-bounded setting, and construct counterexamples when they fail. The mode of failure of the theorem may depend on the notion of time-bounded randomness.

cs.CC↗

arXiv2016

Multiple Recurrence and Algorithmic Randomness

This work contributes to the programme of studying effective versions of "almost everywhere" theorems in analysis and ergodic theory via algorithmic randomness. We determine the level of randomness needed for a point in a Cantor space $ \{0,1\}^{\NN}$ with the uniform measure and the usual shift so that effective versions of the multiple recurrence theorem of Furstenberg holds for iterations starting at the point. We consider recurrence into closed sets that possess various degrees of effectiveness: clopen, $\PPI$ with computable measure, and $\PPI$. The notions of Kurtz, Schnorr, and \ML\ randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the $k$ commuting shift operators on $\{0,1\}^{\NN^{\normalsize k}}$.

math.LO↗

arXiv2016

Ornstein Isomorphism and Algorithmic Randomness

In 1970, Donald Ornstein proved a landmark result in dynamical systems, viz., two Bernoulli systems with the same entropy are isomorphic except for a measure 0 set. Keane and Smorodinsky gave a finitary proof of this result. They also indicated how one can generalize the result to mixing Markov Shifts. We adapt their construction to show that if two computable mixing Markov systems have the same entropy, then there is a layerwise computable isomorphism defined on all Martin-Lof random points in the system. Since the set of Martin-Lof random points forms a measure 1 set, it implies the classical result for such systems. This result uses several recent developments in computable analysis and algorithmic randomness. Following the work by Braverman, Nandakumar, and Hoyrup and Rojas introduced discontinuous functions into the study of algorithmic randomness. We utilize Hoyrup and Rojas' elegant notion of layerwise computable functions to produce the test of randomness in our result. Further, we use the recent result of the effective Shannon-McMillan-Breiman theorem, independently established by Hochman and Hoyrup to prove the properties of our construction. We show that the result cannot be improved to include all points in the systems - only trivial computable isomorphisms exist between systems with the same entropy.

cs.IT↗

arXiv2016

Predictive Complexity and Generalized Entropy Rate of Stationary Ergodic Processes

In the online prediction framework, we use generalized entropy of to study the loss rate of predictors when outcomes are drawn according to stationary ergodic distributions over the binary alphabet. We show that the notion of generalized entropy of a regular game \cite{KVV04} is well-defined for stationary ergodic distributions. In proving this, we obtain new game-theoretic proofs of some classical information theoretic inequalities. Using Birkhoff's ergodic theorem and convergence properties of conditional distributions, we prove that a classical Shannon-McMillan-Breiman theorem holds for a restricted class of regular games, when no computational constraints are imposed on the prediction strategies. If a game is mixable, then there is an optimal aggregating strategy which loses at most an additive constant when compared to any other lower semicomputable strategy. The loss incurred by this algorithm on an infinite sequence of outcomes is called its predictive complexity. We use our version of Shannon-McMillan-Breiman theorem to prove that when a restriced regular game has a predictive complexity, the predictive complexity converges to the generalized entropy of the game almost everywhere with respect to the stationary ergodic distribution.

cs.IT↗

arXiv2014

Normality and Finite-state Dimension of Liouville numbers

Liouville numbers were the first class of real numbers which were proven to be transcendental. It is easy to construct non-normal Liouville numbers. Kano and Bugeaud have proved, using analytic techniques, that there are normal Liouville numbers. Here, for a given base k >= 2, we give two simple constructions of a Liouville number which is normal to the base k. The first construction is combinatorial, and is based on de Bruijn sequences. A real number in the unit interval is normal if and only if its finite-state dimension is 1. We generalize our construction to prove that for any rational r in the closed unit interval, there is a Liouville number with finite state dimension r. This refines Staiger's result that the set of Liouville numbers has constructive Hausdorff dimension zero, showing a new quantitative classification of Liouville numbers can be attained using finite-state dimension. In the second number-theoretic construction, we use an arithmetic property of numbers - the existence of primitive roots - to construct Liouville numbers normal in finitely many bases, assuming a Generalized Artin's conjecture on primitive roots.

cs.IT↗