Finite-State Dimension and Real Arithmetic

We use entropy rates and Schur concavity to prove that, for every integer k >= 2, every nonzero rational number q, and every real number alpha, the base-k expansions of alpha, q+alpha, and q*alpha all have the same finite-state dimension and the same finite-state strong dimension. This extends, and gives a new proof of, Wall's 1949 theorem stating that the sum or product of a nonzero rational number and a Borel normal number is always Borel normal.Comment: 15 page

Finite-State Dimension and Lossy Decompressors

This paper examines information-theoretic questions regarding the difficulty of compressing data versus the difficulty of decompressing data and the role that information loss plays in this interaction. Finite-state compression and decompression are shown to be of equivalent difficulty, even when the decompressors are allowed to be lossy. Inspired by Kolmogorov complexity, this paper defines the optimal *decompression *ratio achievable on an infinite sequence by finite-state decompressors (that is, finite-state transducers outputting the sequence in question). It is shown that the optimal compression ratio achievable on a sequence S by any *information lossless* finite state compressor, known as the finite-state dimension of S, is equal to the optimal decompression ratio achievable on S by any finite-state decompressor. This result implies a new decompression characterization of finite-state dimension in terms of lossy finite-state transducers.Comment: We found that Theorem 3.11, which was basically the motive for this paper, was already proven by Sheinwald, Ziv, and Lempel in 1991 and 1995 paper

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

Dimensions of Copeland-Erdos Sequences

The base-$k$ {\em Copeland-Erd\"os sequence} given by an infinite set $A$ of positive integers is the infinite sequence \CE_k(A) formed by concatenating the base-$k$ representations of the elements of $A$ in numerical order. This paper concerns the following four quantities. The {\em finite-state dimension} \dimfs (\CE_k(A)), a finite-state version of classical Hausdorff dimension introduced in 2001. The {\em finite-state strong dimension} \Dimfs(\CE_k(A)), a finite-state version of classical packing dimension introduced in 2004. This is a dual of \dimfs(\CE_k(A)) satisfying \Dimfs(\CE_k(A)) \geq \dimfs(\CE_k(A)). The {\em zeta-dimension} \Dimzeta(A), a kind of discrete fractal dimension discovered many times over the past few decades. The {\em lower zeta-dimension} \dimzeta(A), a dual of \Dimzeta(A) satisfying \dimzeta(A)\leq \Dimzeta(A). We prove the following. \dimfs(\CE_k(A))\geq \dimzeta(A). This extends the 1946 proof by Copeland and Erd\"os that the sequence \CE_k(\mathrm{PRIMES}) is Borel normal. \Dimfs(\CE_k(A))\geq \Dimzeta(A). These bounds are tight in the strong sense that these four quantities can have (simultaneously) any four values in $[0,1]$ satisfying the four above-mentioned inequalities.Comment: 19 page

Dimension Extractors and Optimal Decompression

A *dimension extractor* is an algorithm designed to increase the effective dimension -- i.e., the amount of computational randomness -- of an infinite binary sequence, in order to turn a "partially random" sequence into a "more random" sequence. Extractors are exhibited for various effective dimensions, including constructive, computable, space-bounded, time-bounded, and finite-state dimension. Using similar techniques, the Kucera-Gacs theorem is examined from the perspective of decompression, by showing that every infinite sequence S is Turing reducible to a Martin-Loef random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S, which is shown to be the optimal ratio of query bits to computed bits achievable with Turing reductions. The extractors and decompressors that are developed lead directly to new characterizations of some effective dimensions in terms of optimal decompression by Turing reductions.Comment: This report was combined with a different conference paper "Every Sequence is Decompressible from a Random One" (cs.IT/0511074, at http://dx.doi.org/10.1007/11780342_17), and both titles were changed, with the conference paper incorporated as section 5 of this new combined paper. The combined paper was accepted to the journal Theory of Computing Systems, as part of a special issue of invited papers from the second conference on Computability in Europe, 200