Dimension, entropy rates, and compression

This paper develops relationships between resource-bounded dimension, entropy rates, and compression. New tools for calculating dimensions are given and used to improve previous results about circuit-size complexity classes. Approximate counting of SpanP functions is used to prove that the NP-entropy rate is an upper bound for dimension in Δ₃^E, the third level of the exponential-time hierarchy. This general result is applied to simultaneously improve the results of Mayordomo (1994) on the measure on P/poly in Δ₃^E and of Lutz (2003) on the dimension of exponential-size circuit complexity classes in ESPACE. Entropy rates of efficiently rankable sets, sets that are optimally compressible, are studied in conjunction with time-bounded dimension. It is shown that rankable entropy rates give upper bounds for time-bounded dimensions. We use this to improve results of Lutz (1992) about polynomial-size circuit complexity classes from resource-bounded measure to dimension. Exact characterizations of the effective dimensions in terms of Kolmogorov complexity rates at the polynomial-space and higher levels have been established, but in the time-bounded setting no such equivalence is known. We introduce the concept of polynomial-time superranking as an extension of ranking. We show that superranking provides an equivalent definition of polynomial-time dimension. From this superranking characterization we show that polynomial-time Kolmogorov complexity rates give a lower bound on polynomial-time dimension.