arXiv subjects

Neil Lutz

Publications and source records attributed to Neil Lutz.

At least 19 recordsLinked to original sources

arXiv2026

One Adaptive Trailing Head Can Outperform Many Oblivious Trailing Heads

In the setting of multi-head finite-state dimensions, trailing heads lag behind a leading head, accessing past data to aid a finite-state gambler placing bets on successive bits read by the leading head. Cruz, Glashausser, Li, and Lutz (2026) proved that, for any fixed number of trailing heads, adaptive (data-dependent) movement rules can strictly outperform oblivious (data-independent) movement schedules. In this paper we strengthen that separation by proving that a single trailing head with adaptive movements can outperform, by a large and uniform margin, arbitrarily many trailing heads with oblivious movements. Formally, our main theorem states that there is a binary sequence whose adaptive two-head finite-state strong dimension is less than its oblivious multi-head finite-state dimension, and that the gap is greater than 0.3.

cs.FL↗

arXiv2026

Adaptive Multi-Head Finite-State Gamblers

Multi-head finite-state dimensions and predimensions quantify the predictability of a sequence by a gambler with trailing heads acting as "probes to the past." These additional heads allow the gambler to exploit patterns that are simple but non-local, such as in a sequence $S$ with $S[n]=S[2n]$ for all $n$. In the original definitions of Huang, Li, Lutz, and Lutz (2025), the head movements were required to be oblivious (i.e., data-independent). Here, we introduce a model in which head movements are adaptive (i.e., data-dependent) and compare it to the oblivious model. We establish that for each $h\geq 2$, adaptivity enhances the predictive power of $h$-head finite-state gamblers, in the sense that there are sequences whose oblivious $h$-head finite-state predimensions strictly exceed their adaptive $h$-head finite-state predimensions. We further prove that adaptive finite-state predimensions admit a strict hierarchy as the number of heads increases, and in fact that for all $h\geq 1$ there is a sequence whose adaptive $(h+1)$-head finite-state predimension is strictly less than its adaptive $h$-head predimension.

cs.IT↗

arXiv2025

Multihead Finite-State Compression

This paper develops multihead finite-state compression, a generalization of finite-state compression, complementary to the multihead finite-state dimensions of Huang, Li, Lutz, and Lutz (2025). In this model, an infinite sequence of symbols is compressed by a compressor that produces outputs according to finite-state rules, based on the symbols read by a constant number of finite-state read heads moving forward obliviously through the sequence. The main theorem of this work establishes that for every sequence and every positive integer $h$, the infimum of the compression ratios achieved by $h$-head finite-state information-lossless compressors equals the $h$-head finite-state predimension of the sequence. As an immediate corollary, the infimum of these ratios over all $h$ is the multihead finite-state dimension of the sequence.

cs.IT↗

arXiv2025

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

Algorithmic Information Bounds for Distances and Orthogonal Projections

We introduce a new technique for proving bounds on the Kolmogorov complexity of geometric objects in Euclidean space, such as points and lines. We apply this technique to prove two theorems on algorithmic information theory, both of which have consequences for well-known problems in geometric measure theory. First, we show that for any point $x$ in the plane and any other point $y$ sufficiently independent of $x$, the distance between $x$ and $y$ retains at least half the complexity of the original point $x$. By the point-to-set principle of J. Lutz and N. Lutz, this yields an improved lower bound on the Hausdorff dimension of pinned distance sets, a topic closely related to Falconer's distance set conjecture. Second, we prove an analogous result for orthogonal projections: for any point $x$ in the plane and any line through the origin which is sufficiently independent of $x$, the projection of $x$ onto that line retains at least half the complexity of $x$. As a consequence, we obtain a generalization of a theorem of Bourgain on exceptional sets for orthogonal projections.

cs.CC↗

arXiv2025

Lines in Every Direction with No ee-Random Points

We prove that in every direction in the Euclidean plane, there exists a line containing no double exponential time random (ee-random) points. This means each point on these lines has an algorithmically predictable location, to the extent that a gambler in an environment with fair payouts can, using double exponential time computing resources, amass unbounded capital placing bets on increasingly precise estimates of the point's location. Our proof relies on effectivizing the construction of the lineal extension of a Kakeya set. This resolves an open question of Lutz and Lutz (2015).

cs.CC↗

arXiv2024

Bounding the dimension of exceptional sets for orthogonal projections

It is well known that if $A \subseteq \mathbb{R}^n$ is an analytic set of Hausdorff dimension $a$, then $\dim_H(\pi_VA)=\min\{a,k\}$ for a.e.\ $V\in G(n,k)$, where $G(n,k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{R}^n$ and $\pi_V$ is the orthogonal projection of $A$ onto $V$. In this paper we study how large the exceptional set \begin{equation*} \{V\in G(n,k) \mid \dim_H(\pi_V A) < s\} \end{equation*} can be for a given $s\le\min\{a,k\}.$ We improve previously known estimates on the dimension of the exceptional set, and we show that our estimates are sharp for $k=1$ and for $k=n-1$. Hence we completely resolve this question for $n=3$.

math.CA↗

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↗

arXiv2020

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 $\boldsymbol{\kappa}$ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.

cs.CC↗

arXiv2020

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., $\mathbf{\Sigma}^1_1$, a tight bound on the packing dimension of the hyperspace of $E$ in terms of the packing dimension of $E$ itself.

cs.CC↗

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

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↗

arXiv2019

A Center in Your Neighborhood: Fairness in Facility Location

When selecting locations for a set of facilities, standard clustering algorithms may place unfair burden on some individuals and neighborhoods. We formulate a fairness concept that takes local population densities into account. In particular, given $k$ facilities to locate and a population of size $n$, we define the "neighborhood radius" of an individual $i$ as the minimum radius of a ball centered at $i$ that contains at least $n/k$ individuals. Our objective is to ensure that each individual has a facility within at most a small constant factor of her neighborhood radius. We present several theoretical results: We show that optimizing this factor is NP-hard; we give an approximation algorithm that guarantees a factor of at most 2 in all metric spaces; and we prove matching lower bounds in some metric spaces. We apply a variant of this algorithm to real-world address data, showing that it is quite different from standard clustering algorithms and outperforms them on our objective function and balances the load between facilities more evenly.

cs.DS↗

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

Quantifying the Burden of Exploration and the Unfairness of Free Riding

We consider the multi-armed bandit setting with a twist. Rather than having just one decision maker deciding which arm to pull in each round, we have $n$ different decision makers (agents). In the simple stochastic setting, we show that a "free-riding" agent observing another "self-reliant" agent can achieve just $O(1)$ regret, as opposed to the regret lower bound of $\Omega (\log t)$ when one decision maker is playing in isolation. This result holds whenever the self-reliant agent's strategy satisfies either one of two assumptions: (1) each arm is pulled at least $\gamma \ln t$ times in expectation for a constant $\gamma$ that we compute, or (2) the self-reliant agent achieves $o(t)$ realized regret with high probability. Both of these assumptions are satisfied by standard zero-regret algorithms. Under the second assumption, we further show that the free rider only needs to observe the number of times each arm is pulled by the self-reliant agent, and not the rewards realized. In the linear contextual setting, each arm has a distribution over parameter vectors, each agent has a context vector, and the reward realized when an agent pulls an arm is the inner product of that agent's context vector with a parameter vector sampled from the pulled arm's distribution. We show that the free rider can achieve $O(1)$ regret in this setting whenever the free rider's context is a small (in $L_2$-norm) linear combination of other agents' contexts and all other agents pull each arm $\Omega (\log t)$ times with high probability. Again, this condition on the self-reliant players is satisfied by standard zero-regret algorithms like UCB. We also prove a number of lower bounds.

cs.LG↗

arXiv2017

Projection Theorems Using Effective Dimension

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand's projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal. We use Kolmogorov complexity to give two new results on the Hausdorff and packing dimensions of orthogonal projections onto lines. The first shows that the conclusion of Marstrand's theorem holds whenever the Hausdorff and packing dimensions agree on the set E, even if E is not analytic. Our second result gives a lower bound on the packing dimension of projections of arbitrary sets. Finally, we give a new proof of Marstrand's theorem using the theory of computing.

cs.CC↗

arXiv2017

Dimension Spectra of Lines

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp(L) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim(a, b) is equal to the effective packing dimension Dim(a, b), then sp(L) contains a unit interval. We also show that, if the dimension dim(a, b) is at least one, then sp(L) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

cs.CC↗

arXiv2016

A Note on Pointwise Dimensions

This short note describes a connection between algorithmic dimensions of individual points and classical pointwise dimensions of measures.

cs.CC↗