Dimension, Entropy Rates, and Compression

This paper develops new 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 ∆E 3 , 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 ∆E 3 and of Lutz (2000) 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 chara...

• Type: Conference
• Year: 2004
• Where: COCO
• Authors: John M. Hitchcock, N. V. Vinodchandran
• Keywords: Circuit Complexity Classes, COCO 2004, Complexity Classes, Entropy Rates