arXiv subjects

Jack H. Lutz

Publications and source records attributed to Jack H. Lutz.

At least 19 recordsLinked to original sources

arXiv2026

Algorithmic Randomness in Continuous-Time Markov Chains

In this paper we develop the elements of the theory of algorithmic randomness in continuous-time Markov chains (CTMCs). Our main contribution is a rigorous, useful notion of what it means for an individual trajectory of a CTMC to be random. CTMCs have discrete state spaces and operate in continuous time. This, together with the fact that trajectories may or may not halt, presents challenges not encountered in more conventional developments of algorithmic randomness. Although we formulate algorithmic randomness in the general context of CTMCs, we are primarily interested in the computational power of stochastic chemical reaction networks, which are special cases of CTMCs. This leads us to embrace situations in which the long-term behavior of a network depends essentially on its initial state and hence to eschew assumptions that are frequently made in Markov chain theory to avoid such dependencies. After defining the randomness of trajectories in terms of martingales (algorithmic betting strategies), we prove equivalent characterizations in terms of algorithmic measure theory and Kolmogorov complexity. As a preliminary application we prove that, in any stochastic chemical reaction network, \emph{every} random trajectory with bounded molecular counts has the non-Zeno property that infinitely many reactions do not occur in any finite interval of time.

cs.IT↗

arXiv2026

Multi-Head Finite-State Dimension

We introduce multi-head finite-state dimension, a generalization of finite-state dimension in which a group of finite-state agents (the heads) with oblivious, one-way movement rules, each reporting only one symbol at a time, enable their leader to bet on subsequent symbols in an infinite data stream. In aggregate, such a scheme constitutes an $h$-head finite state gambler whose maximum achievable growth rate of capital in this task, quantified using betting strategies called gales, determines the multi-head finite-state dimension of the sequence. The 1-head case is equivalent to finite-state dimension as defined by Dai, Lathrop, Lutz and Mayordomo (2004). In our main theorem, we prove a strict hierarchy as the number of heads increases, giving an explicit sequence family that separates, for each positive integer $h$, the earning power of $h$-head finite-state gamblers from that of $(h+1)$-head finite-state gamblers. We prove that multi-head finite-state dimension is stable under finite unions but that the corresponding quantity for any fixed number $h>1$ of heads--the $h$-head finite-state predimension--lacks this stability property.

cs.IT↗

arXiv2025

Algorithmically Optimal Outer Measures

We investigate the relationship between algorithmic fractal dimensions and the classical local fractal dimensions of outer measures in Euclidean spaces. We introduce global and local optimality conditions for lower semicomputable outer measures. We prove that globally optimal outer measures exist. Our main theorem states that the classical local fractal dimensions of any locally optimal outer measure coincide exactly with the algorithmic fractal dimensions. Our proof uses an especially convenient locally optimal outer measure $\boldsymbolκ$ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.

cs.CC↗

arXiv2024

Algorithmic Dimensions via Learning Functions

We characterize the algorithmic dimensions (i.e., the lower and upper asymptotic densities of information) of infinite binary sequences in terms of the inability of learning functions having an algorithmic constraint to detect patterns in them. Our pattern detection criterion is a quantitative extension of the criterion that Zaffora Blando used to characterize the algorithmically random (i.e., Martin-Löf random) sequences. Our proof uses Lutz's and Mayordomo's respective characterizations of algorithmic dimension in terms of gales and Kolmogorov complexity.

cs.IT↗

arXiv2023

The Point-to-Set Principle and the Dimensions of Hamel Bases

We prove that every real number in [0,1] is the Hausdorff dimension of a Hamel basis of the vector space of reals over the field of rationals. The logic of our proof is of particular interest. The statement of our theorem is classical; it does not involve the theory of computing. However, our proof makes essential use of algorithmic fractal dimension--a computability-theoretic construct--and the point-to-set principle of J. Lutz and N. Lutz (2018).

cs.LO↗

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↗

arXiv2021

Finite-State Mutual Dimension

In 2004, Dai, Lathrop, Lutz, and Mayordomo defined and investigated the finite-state dimension (a finite-state version of algorithmic dimension) of a sequence $S \in Σ^\infty$ and, in 2018, Case and Lutz defined and investigated the mutual (algorithmic) dimension between two sequences $S \in Σ^\infty$ and $T \in Σ^\infty$. In this paper, we propose a definition for the lower and upper finite-state mutual dimensions $mdim_{FS}(S:T)$ and $Mdim_{FS}(S:T)$ between two sequences $S \in Σ^\infty$ and $T \in Σ^\infty$ over an alphabet $Σ$. Intuitively, the finite-state dimension of a sequence $S \in Σ^\infty$ represents the density of finite-state information contained within $S$, while the finite-state mutual dimension between two sequences $S \in Σ^\infty$ and $T \in Σ^\infty$ represents the density of finite-state information shared by $S$ and $T$. Thus ``finite-state mutual dimension'' can be viewed as a ``finite-state'' version of mutual dimension and as a ``mutual'' version of finite-state dimension. The main results of this investigation are as follows. First, we show that finite-state mutual dimension, defined using information-lossless finite-state compressors, has all of the properties expected of a measure of mutual information. Next, we prove that finite-state mutual dimension may be characterized in terms of block mutual information rates. Finally, we provide necessary and sufficient conditions for two normal sequences to achieve $mdim_{FS}(S:T) = Mdim_{FS}(S:T) = 0$.

cs.IT↗

arXiv2021

Dimension and the Structure of Complexity Classes

We prove three results on the dimension structure of complexity classes. 1. The Point-to-Set Principle, which has recently been used to prove several new theorems in fractal geometry, has resource-bounded instances. These instances characterize the resource-bounded dimension of a set $X$ of languages in terms of the relativized resource-bounded dimensions of the individual elements of $X$, provided that the former resource bound is large enough to parameterize the latter. Thus for example, the dimension of a class $X$ of languages in EXP is characterized in terms of the relativized p-dimensions of the individual elements of $X$. 2. Every language that is $\leq^P_m$-reducible to a p-selective set has p-dimension 0, and this fact holds relative to arbitrary oracles. Combined with a resource-bounded instance of the Point-to-Set Principle, this implies that if NP has positive dimension in EXP, then no quasipolynomial time selective language is $\leq^P_m$-hard for NP. 3. If the set of all disjoint pairs of NP languages has dimension 1 in the set of all disjoint pairs of EXP languages, then NP has positive dimension in EXP.

cs.CC↗

arXiv2021

Extending the Reach of the Point-to-Set Principle

The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that their statements do not involve computation or related aspects of logic. In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces $X$. We first extend two fractal dimensions--computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions $\dim(x)$ and $\textrm{Dim}(x)$ to individual points $x\in X$--to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families. We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces. (For a concrete computational example, the stages $E_0, E_1, E_2, \ldots$ used to construct a self-similar fractal $E$ in the plane are elements of the hyperspace of the plane, and they converge to $E$ in the hyperspace.) Our third main result, proven via our extended point-to-set principle, states that, under a wide variety of gauge families, the classical packing dimension agrees with the classical upper Minkowski dimension on all hyperspaces of compact sets. We use this theorem to give, for all sets $E$ that are analytic, i.e., $\mathbfΣ^1_1$, a tight bound on the packing dimension of the hyperspace of $E$ in terms of the packing dimension of $E$ itself.

cs.CC↗

arXiv2020

Algorithmic Fractal Dimensions in Geometric Measure Theory

The development of algorithmic fractal dimensions in this century has had many fruitful interactions with geometric measure theory, especially fractal geometry in Euclidean spaces. We survey these developments, with emphasis on connections with computable functions on the reals, recent uses of algorithmic dimensions in proving new theorems in classical (non-algorithmic) fractal geometry, and directions for future research.

cs.CC↗

arXiv2020

Computing Absolutely Normal Numbers in Nearly Linear Time

A real number $x$ is absolutely normal if, for every base $b\ge 2$, every two equally long strings of digits appear with equal asymptotic frequency in the base-$b$ expansion of $x$. This paper presents an explicit algorithm that generates the binary expansion of an absolutely normal number $x$, with the $n$th bit of $x$ appearing after $n$polylog$(n)$ computation steps. This speed is achieved by simultaneously computing and diagonalizing against a martingale that incorporates Lempel-Ziv parsing algorithms in all bases.

cs.DS↗

arXiv2020

Population-Induced Phase Transitions and the Verification of Chemical Reaction Networks

We show that very simple molecular systems, modeled as chemical reaction networks, can have behaviors that exhibit dramatic phase transitions at certain population thresholds. Moreover, the magnitudes of these thresholds can thwart attempts to use simulation, model checking, or approximation by differential equations to formally verify the behaviors of such systems at realistic populations. We show how formal theorem provers can successfully verify some such systems at populations where other verification methods fail.

cs.ET↗

arXiv2020

Nonregularity via Ordinal Extensions

We present a simple new method for proving that languages are not regular. We prove the correctness of the method, illustrate the ease of using the method on well-known examples of nonregular languages, and prove two additional theorems on the power and limitations of the method.

cs.FL↗

arXiv2019

Who Asked Us? How the Theory of Computing Answers Questions about Analysis

Algorithmic fractal dimensions -- constructs of computability theory -- have recently been used to answer open questions in classical geometric measure theory, questions of mathematical analysis whose statements do not involve computability theory or logic. We survey these developments and the prospects for future such results.

cs.CC↗

arXiv2019

Asymptotic Divergences and Strong Dichotomy

The Schnorr-Stimm dichotomy theorem concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet $Σ$. In this paper we use the Kullback-Leibler divergence to formulate the $\textit{lower asymptotic divergence}$ $\text{div}(S||α)$ of a probability measure $α$ on $Σ$ from a sequence $S$ over $Σ$ and the $\textit{upper asymptotic divergence}$ $\text{Div}(S||α)$ of $α$ from $S$ in such a way that a sequence $S$ is $α$-normal (meaning that every string $w$ has asymptotic frequency $α(w)$ in $S$) if and only if $\text{Div}(S||α)=0$. We also use the Kullback-Leibler divergence to quantify the $\textit{total risk }$ $\text{Risk}_G(w)$ that a finite-state gambler $G$ takes when betting along a prefix $w$ of $S$. Our main theorem is a $\textit{strong dichotomy theorem}$ that uses the above notions to $\textit{quantify}$ the exponential rates of winning and losing on the two sides of the Schnorr-Stimm dichotomy theorem (with the latter routinely extended from normality to $α$-normality). Modulo asymptotic caveats in the paper, our strong dichotomy theorem says that the following two things hold for prefixes $w$ of $S$. (1) The infinitely-often exponential rate of winning is $2^{\text{Div}(S||α)|w|}$. (2) The exponential rate of loss is $2^{-\text{Risk}_G(w)}$. We also use (1) to show that $1-\text{Div}(S||α)/c$, where $c= \log(1/ \min_{a\inΣ}α(a))$, is an upper bound on the finite-state $α$-dimension of $S$ and prove the dual fact that $1-\text{div}(S||α)/c$ is an upper bound on the finite-state strong $α$-dimension of $S$.

cs.IT↗

arXiv2019

Robustness and Games Against Nature in Molecular Programming

Matter, especially DNA, is now programmed to carry out useful processes at the nanoscale. As these programs and processes become more complex and their envisioned safety-critical applications approach deployment, it is essential to develop methods for engineering trustworthiness into molecular programs. Some of this can be achieved by adapting existing software engineering methods, but molecular programming also presents new challenges that will require new methods. This paper presents a method for dealing with one such challenge, namely, the difficulty of ascertaining how robust a molecular program is to perturbations of the relative "clock speeds" of its various reactions. The method proposed here is game-theoretic. The robustness of a molecular program is quantified in terms of its ability to win (achieve its original objective) in games against other molecular programs that manipulate its relative clock speeds. This game-theoretic approach is general enough to quantify the security of a molecular program against malicious manipulations of its relative clock speeds. However, this preliminary report focuses on games against nature, games in which the molecular program's opponent perturbs clock speeds randomly (indifferently) according to the probabilities inherent in chemical kinetics.

cs.GT↗

arXiv2018

Robust Biomolecular Finite Automata

We present a uniform method for translating an arbitrary nondeterministic finite automaton (NFA) into a deterministic mass action input/output chemical reaction network (I/O CRN) that simulates it. The I/O CRN receives its input as a continuous time signal consisting of concentrations of chemical species that vary to represent the NFA's input string in a natural way. The I/O CRN exploits the inherent parallelism of chemical kinetics to simulate the NFA in real time with a number of chemical species that is linear in the size of the NFA. We prove that the simulation is correct and that it is robust with respect to perturbations of the input signal, the initial concentrations of species, the output (decision), and the rate constants of the reactions of the I/O CRN.

cs.CC↗

arXiv2018

Runtime Fault Detection in Programmed Molecular Systems

Watchdog timers are devices that are commonly used to monitor the health of safety-critical hardware and software systems. Their primary function is to raise an alarm if the monitored systems fail to emit periodic "heartbeats" that signal their well-being. In this paper we design and verify a molecular watchdog timer for monitoring the health of programmed molecular nanosystems. This raises new challenges because our molecular watchdog timer and the system that it monitors both operate in the probabilistic environment of chemical kinetics, where many failures are certain to occur and it is especially hard to detect the absence of a signal. Our molecular watchdog timer is the result of an incremental design process that uses goal-oriented requirements engineering, simulation, stochastic analysis, and software verification tools. We demonstrate the molecular watchdog's functionality by having it monitor a molecular oscillator. Both the molecular watchdog timer and the oscillator are implemented as chemical reaction networks, which are the current programming language of choice for many molecular programming applications.

cs.ET↗