## 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.

### MSC:

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

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