arXiv subjects

Subin Pulari

Publications and source records attributed to Subin Pulari.

11 recordsLinked to original sources

arXiv2026

Efficient Constructions of Finite-State Independent Normal Pairs

Finite-state independence is a robust notion of algorithmic independence for infinite words. It was introduced for general infinite words by Becher, Carton, and Heiber via deterministic asynchronous two-tape finite automata. \'Alvarez, Becher, and Carton then studied the normal case and characterized finite-state independence in terms of deterministic finite-state shufflers. A shuffler is a finite automaton that reads from two input tapes $x,y\in\Sigma^\infty$ and, at each step, chooses one tape to read next, outputs the symbol read, and updates its state based only on that output symbol. In terms of this characterization, two normal sources are finite-state independent if every deterministic finite-state way of shuffling (interleaving) them still produces a normal sequence. \'Alvarez, Becher, and Carton posed the following questions: (1) can one compute finite-state independent normal pairs efficiently, improving their doubly-exponential procedure; and (2) given a normal word $x$, can one effectively construct a normal word $y$ that is finite-state independent from $x$? We answer both questions by explicit deterministic constructions. First, we give a deterministic polynomial-time algorithm that, on input $N$, outputs the first $N$ symbols of two normal words $x$ and $y$ such that for every shuffler $S$, the shuffled output $S(x,y)$ is normal; hence $(x,y)$ is finite-state independent. Second, we solve the one-sided companion problem effectively. Given any computable normal word $x\in\Sigma^\infty$, we give an explicit deterministic construction of a computable normal word $y\in\Sigma^\infty$ such that for every shuffler $S$, the shuffled output $S(x,y)$ is normal. In particular, $x$ and $y$ are finite-state independent by the shuffler characterization theorem.

cs.FL↗

arXiv2026

On Normality and Equidistribution for Separator Enumerators

A separator is a countable dense subset of $[0,1)$, and a separator enumerator is a naming scheme that assigns a real number in $[0,1)$ to each finite word so that the set of all named values is a separator. Mayordomo introduced separator enumerators to define $f$-normality and a relativized finite-state dimension $\dim^{f}_{\mathrm{FS}}(x)$, where finite-state dimension measures the asymptotic lower rate of finite-state information needed to approximate $x$ through its $f$-names. This framework extends classical base-$k$ normality, and Mayordomo showed that it supports a point-to-set principle for finite-state dimension. This representation-based viewpoint has since been developed further in follow-up work, including by Calvert et al., yielding strengthened randomness notions such as supernormal and highly normal numbers. Mayordomo posed the following open question: can $f$-normality be characterized via equidistribution properties of the sequence $\left(|\Sigma|^{n} a^{f}_{n}(x)\right)_{n=0}^{\infty}$, where $a^{f}_{n}(x)$ is the sequence of best approximations to $x$ from below induced by $f$? We give a strong negative answer: we construct computable separator enumerators $f_0,f_1$ and a point $x$ such that $a^{f_0}_{n}(x)=a^{f_1}_{n}(x)$ for all $n$, yet $\dim^{f_0}_{\mathrm{FS}}(x)=0$ while $\dim^{f_1}_{\mathrm{FS}}(x)=1$. Consequently, no criterion depending only on the sequence $\left(|\Sigma|^{n} a^{f}_{n}(x)\right)_{n=0}^{\infty}$ - in particular, no equidistribution property of this sequence - can characterize $f$-normality uniformly over all separator enumerators. On the other hand, for a natural finite-state coherent class of separator enumerators we recover a complete equidistribution characterization of $f$-normality. We also show that beyond finite-state coherence, this characterization can fail even for a separator enumerator computable in nearly linear time.

cs.FL↗

arXiv2025

A Markov-Chain Characterization of Finite-State Dimension and a Generalization of Agafonov's Theorem

Finite-state dimension quantifies the asymptotic rate of information in an infinite sequence as perceived by finite automata. For a fixed alphabet, the infinite sequences that have maximal finite-state dimension are exactly those that are Borel normal, i.e., in which all words of any given length appear with the same frequency. A theorem of Schnorr and Stimm (1972) shows that a real number is Borel normal if and only if, for every finite-state irreducible Markov chain with fair transitions, when the chain is simulated using the binary expansion of the given number, the empirical distribution of states converges to its stationary distribution. In this paper we extend this correspondence beyond normal numbers. We show that the finite-state dimension of a sequence can be characterized in terms of the conditional Kullback-Leibler divergence between the limiting distributions arising from the simulation of Markov chains using the given sequence and their stationary distributions. This provides a new information-theoretic characterization of finite-state dimension which generalizes the Schnorr-Stimm result. As an application, we prove a generalization of Agafonov's theorem for normal numbers. Agafonov's theorem states that a sequence is normal if and only if every subsequence selected by a finite automaton is also normal. We extend this to arbitrary sequences by establishing a tight quantitative relationship between the finite-state dimension of a sequence and the finite-state dimensions of its automatic subsequences.

cs.IT↗

arXiv2025

The Agafonov and Schnorr-Stimm theorems for probabilistic automata

For a fixed alphabet $A$, an infinite sequence $X$ is said to be normal if every word $w$ over $A$ appears in $X$ with the same frequency as any other word of the same length. A classical result of Agafonov (1966) relates normality to finite automata as follows: a sequence $X$ is normal if and only if any subsequence of $X$ selected by a finite automaton is itself normal. Another theorem of Schnorr and Stimm (1972) gives an alternative characterization: a sequence $X$ is normal if and only if no gambler can win large amounts of money by betting on the sequence $X$ using a strategy that can be described by a finite automaton. Both of these theorems are established in the setting of deterministic finite automata. This raises the question as to whether they can be extended to the setting of probabilistic finite automata. In the case of the Agafonov theorem, this question was positively answered by L\'echine et al. (2024) in a restricted case of probabilistic automata with rational transition probabilities. In this paper, we settle the full conjecture by proving that both the Agafonov and the Schnorr-Stimm theorems hold true for arbitrary probabilistic automata. Specifically, we show that a sequence $X$ is normal if and only if any probabilistic automaton selects a normal subsequence of $X$ with probability $1$. We also show that a sequence $X$ is normal if and only if a probabilistic finite-state gambler fails to win on $X$ with probability $1$.

cs.FL↗

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

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↗

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 $\xi$ and a compressibility ratio $s \in (0,1)$ such that for every base $b$, the compressibility ratio of the base-$b$ expansion of $\xi$ 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 $\xi$ such that for any base $b$, the finite-state dimension/compression ratio of $\xi$ 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

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↗

arXiv2020

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↗