Definitions

# Grammar-based code

Grammar-based codes are compression algorithms based on the idea of constructing a context-free grammar for the string to be compressed. Examples include universal lossless data compression algorithms proposed in Kieffer and Yang 2000, and SEQUITUR(http://sequitur.info/), among others. To compress a data sequence$x = x_1 cdots x_n$, a grammar-based code first transforms $x$ into a context-free grammar $G$, and then uses an arithmetic coding algorithm to compress the grammar $G$. For a detailed description of grammar-based codes and context-free grammars, the reader is referred to Kieffer and Yang 2000.

## 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.
• Great description of grammar-based codes with easy to follow example

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