Effective entropies and data compression

• Published in: Information and computation Vol. 90; no. 1; pp. 67 - 85
• Main Author: Huyn, Dung T.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 1991
• ISSN: 0890-5401; 1090-2651
• DOI: 10.1016/0890-5401(91)90060-F

In this paper, we introduce the formal definition of the concept of the ( C 1, C 2)-effective entropy of a language, where C 1, C 2 are complexity classes. The ( C 1, C 2)-effective entropy of a language L is used to measure how much a string x of length ≤ n in L can be compressed to a string y by a C 1 algorithm so that given y, x can be recovered by a C 2 algorithm. We also relate this concept to issues in data compression. The main results are: (1) The SC j -effective entropy of a 2 O(log n) -sparse regular language is equal its abosolute entropy, i.e., it is optimal; (2) the ( DET, SC ( j) )-effective entropy of a 2 O(log n) -sparse, 2 O(log n) -ambiguous linear context-free language is, up to a constant factor, equal its absolute entropy ( DET denotes the class of problems which are NC (1)-reducible to computing the determinants of integer matrices); and (3) the ( NC (2), P)-effective entropy of a 2 O(log n) -sparse, 2 O(log n) -ambiguous context-free language is, up to a constant factor, equal its absolute entropy.