arXiv subjects

Neil Lutz

Publications and source records attributed to Neil Lutz.

25 records · Page 2Linked to original sources

arXiv2017

Dynamics at the Boundary of Game Theory and Distributed Computing

We use ideas from distributed computing and game theory to study dynamic and decentralized environments in which computational nodes, or decision makers, interact strategically and with limited information. In such environments, which arise in many real-world settings, the participants act as both economic and computational entities. We exhibit a general non-convergence result for a broad class of dynamics in asynchronous settings. We consider implications of our result across a wide variety of interesting and timely applications: game dynamics, circuit design, social networks, Internet routing, and congestion control. We also study the computational and communication complexity of testing the convergence of asynchronous dynamics. Our work opens a new avenue for research at the intersection of distributed computing and game theory.

cs.GT↗

arXiv2017

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↗

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

Algorithmic information, plane Kakeya sets, and conditional dimension

We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways. 1. We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set E in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of E. We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya conjecture. This theorem of geometric measure theory, proved by Davies in 1971, says that every plane set containing a unit line segment in every direction has Hausdorff dimension 2. 2. We use conditional Kolmogorov complexity in Euclidean spaces to develop the lower and upper conditional dimensions dim(x|y) and Dim(x|y) of x given y, where x and y are points in Euclidean spaces. Intuitively these are the lower and upper asymptotic algorithmic information densities of x conditioned on the information in y. We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the well-studied dimensions dim(x) and Dim(x) and the mutual dimensions mdim(x:y) and Mdim(x:y).

cs.CC↗

arXiv2016

Stateless Computation

We present and explore a model of stateless and self-stabilizing distributed computation, inspired by real-world applications such as routing on today's Internet. Processors in our model do not have an internal state, but rather interact by repeatedly mapping incoming messages ("labels") to outgoing messages and output values. While seemingly too restrictive to be of interest, stateless computation encompasses both classical game-theoretic notions of strategic interaction and a broad range of practical applications (e.g., Internet protocols, circuits, diffusion of technologies in social networks). We embark on a holistic exploration of stateless computation. We tackle two important questions: (1) Under what conditions is self-stabilization, i.e., guaranteed "convergence" to a "legitimate" global configuration, achievable for stateless computation? and (2) What is the computational power of stateless computation? Our results for self-stabilization include a general necessary condition for self-stabilization and hardness results for verifying that a stateless protocol is self-stabilizing. Our main results for the power of stateless computation show that labels of logarithmic length in the number of processors yield substantial computational power even on ring topologies. We present a separation between unidirectional and bidirectional rings (L/poly vs. P/poly), reflecting the sequential nature of computation on a unidirectional ring, as opposed to the parallelism afforded by the bidirectional ring. We leave the reader with many exciting directions for future research.

cs.DC↗

arXiv2014

Lines Missing Every Random Point

We prove that there is, in every direction in Euclidean space, a line that misses every computably random point. We also prove that there exist, in every direction in Euclidean space, arbitrarily long line segments missing every double exponential time random point.

cs.CC↗

arXiv2014

Self-stabilizing uncoupled dynamics

Dynamics in a distributed system are self-stabilizing if they are guaranteed to reach a stable state regardless of how the system is initialized. Game dynamics are uncoupled if each player's behavior is independent of the other players' preferences. Recognizing an equilibrium in this setting is a distributed computational task. Self-stabilizing uncoupled dynamics, then, have both resilience to arbitrary initial states and distribution of knowledge. We study these dynamics by analyzing their behavior in a bounded-recall synchronous environment. We determine, for every "size" of game, the minimum number of periods of play that stochastic (randomized) players must recall in order for uncoupled dynamics to be self-stabilizing. We also do this for the special case when the game is guaranteed to have unique best replies. For deterministic players, we demonstrate two self-stabilizing uncoupled protocols. One applies to all games and uses three steps of recall. The other uses two steps of recall and applies to games where each player has at least four available actions. For uncoupled deterministic players, we prove that a single step of recall is insufficient to achieve self-stabilization, regardless of the number of available actions.

cs.GT↗