Base invariance of feasible dimension

Abstract

Effective fractal dimensions were introduced by in order to study the dimensions of individual sequences and quantitatively analyze the structure of complexity classes. Interesting connections of effective dimensions with information theory were also found, implying that constructive dimension as well as polynomial-space dimension are invariant under base-change while finite-state dimension is not.We consider the intermediate case, polynomial-time dimension, and prove that it is indeed invariant under base-change by a nontrivial argument which is quite different from the Kolmogorov complexity ones used in the other cases.Polynomial-time dimension can be characterized in terms of prediction-loss-rate, entropy, and compression algorithms. Our result implies that in an asymptotic way each of these concepts is invariant under base-change.A corollary of the main theorem is any polynomial-time dimension 1 number (which may be established in any base) is an absolutely normal number, providing an interesting source of absolute normality.

Algorithmic Fractal Dimensions in Geometric Measure Theory

“…Resourcebounded dimension has a natural extension Σ ∞ for other finite alphabets Σ and the first question is therefore whether the choice of alphabet is relevant for the study of Euclidean space. A satisfactory answer is given in [36] where it is proven that polynomial-time dimension is invariant under base change, that is, for every base b and set X ⊆ R the set of base-b-representations of all elements in X has a polynomial-time dimension independent of b.…”

Effective Hausdorff Dimension in General Metric Spaces

“…For instance in the case of Finite-State dimension in R n , that is, restriction to gales that can be computed by Finite State Automata (done for Cantor space in [2]), it matters whether we use dyadic intervals, triadic intervals, etc and this is related to the existence of normal sequences that are not absolutely normal [2]. On the other hand certain invariance properties are known for constructive and polynomial-time dimension [8].…”

Feasible Analysis, Randomness, and Base Invariance

“…Let E, G and M be as in the proof of Lemma 6. Using L we can rewrite its definitions: It follows from Hitchcock and Mayordomo [14,Corollary 3.3] that any polynomial time random is absolutely normal. We show that n 4 -randomness suffices:…”