Grammar-based codes
Grammar-based codes are universal lossless data compression algorithms proposed in Kieffer and Yang 2000. To compress a data sequence<math>x = x_1 \cdots x_n</math>, a grammar-based code first transforms <math>x</math> into a context-free grammar <math>G</math>, and then uses an arithmetic coding algorithm to compress the grammar <math>G</math>. For a detailed description of grammar-based codes and context-free grammars, the read is referred to Kieffer and Yang 2000.
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Examples
The class of grammar-based codes is very broad. It includes block codes, variations of the incremental parsing Lempel-Ziv code Ziv and Lempel 1978 (LZ78), the multilevel pattern matching (MPM) algorithm Kieffer et. al. 2000, and many other new universal lossless compression algorithms.
Coding Theorems
Grammar-based codes are universal in the sense that they can achieve asymptotically the entropy rate of any stationary, ergodic source with a finite alphabet.
References
Kieffer-Yang 2000. J. C. Kieffer and E.-H. Yang, "Grammar-based codes: A new class of universal lossless source codes," IEEE Trans. Inform. Theory, vol. 46, pp. 737--754, 2000.
Ziv and Lempel 1978. J. Ziv and A. Lempel, "Compression of individual sequences via variable rate coding," IEEE Trans. Inform. Theory, vol. 24, pp. 530--536, 2000.
Kieffer et. al. 2000. J. C. Kieffer, E.-H. Yang, G. Nelson, and P. Cosman, "Universal lossless compression via multilevel pattern matching," IEEE Trans. Inform. Theory, vol. 46, pp. 1227--1245, 2000.