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Effective entropies and data compression. (English) Zbl 0715.68047
Summary: We introduce the formal definition of the concept of the (\({\mathcal C}_ 1,{\mathcal C}_ 2)\)-effective entropy of a language, where \({\mathcal C}_ 1\), \({\mathcal C}_ 2\) are complexity classes. The (\({\mathcal C}_ 1,{\mathcal C}_ 2)\)-effective entropy of a language L is used to measure how much a string x of length \(\leq n\) in L can be compressed to a string y by a \({\mathcal C}_ 1\) algorithm so that given y, x can be recovered by a \({\mathcal C}_ 2\) 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 absolute 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.

68Q45 Formal languages and automata
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
94A17 Measures of information, entropy
Full Text: DOI
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