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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 ^[1] and SEQUITUR, 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 $G$ .

Examples and Characteristics

The class of grammar-based codes is very broad. It includes block codes, variations of the incremental parsing Lempel-Ziv code ^[2], the multilevel pattern matching (MPM) algorithm^[3], and many other new universal lossless compression algorithms.

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, J. C.; Yang, E.-H. (2000), "Grammar-based codes: A new class of universal lossless source codes", IEEE Trans. Inform. Theory, 46: 737–754, doi:10.1109/18.841160
^ Kieffer, J. C.; Yang, E.-H.; Nelson, G.; Cosman, P. (2000), "Universal lossless compression via multilevel pattern matching", IEEE Trans. Inform. Theory, 46: 1227–1245, doi:10.1109/18.850665
^ Ziv, J.; Lempel, A. (1978), "Compression of individual sequences via variable rate coding", IEEE Trans. Inform. Theory, 24: 530–536, doi:10.1109/TIT.1978.1055934

External links

Description of grammar-based codes with example

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[1] Kieffer, J. C.; Yang, E.-H. (2000), "Grammar-based codes: A new class of universal lossless source codes", IEEE Trans. Inform. Theory, 46: 737–754, doi:10.1109/18.841160

[2] Kieffer, J. C.; Yang, E.-H.; Nelson, G.; Cosman, P. (2000), "Universal lossless compression via multilevel pattern matching", IEEE Trans. Inform. Theory, 46: 1227–1245, doi:10.1109/18.850665

[3] Ziv, J.; Lempel, A. (1978), "Compression of individual sequences via variable rate coding", IEEE Trans. Inform. Theory, 24: 530–536, doi:10.1109/TIT.1978.1055934

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 '''Grammar-based codes''' are [[Data compression|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
-<ref>{{Citation | last = Kieffer  | first =  J. C.  | last2 = Yang | first2 = E.-H. | title = Grammar-based codes: A new class of universal lossless source codes | journal = IEEE Trans. Inform. Theory | volume = 46 | pages = 737 – 754 | year = 2000 }}</ref> and [[SEQUITUR algorithm|SEQUITUR]], among others. 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>.
+<ref>{{Citation | last = Kieffer  | first =  J. C.  | last2 = Yang | first2 = E.-H. | title = Grammar-based codes: A new class of universal lossless source codes | journal = IEEE Trans. Inform. Theory | volume = 46 | pages = 737 – 754 | year = 2000 | doi = 10.1109/18.841160 }}</ref> and [[SEQUITUR algorithm|SEQUITUR]], among others. 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>.
 == Examples and Characteristics ==
-The class of grammar-based codes is very broad. It includes block codes, variations of the incremental parsing [[LZ77 and LZ78|Lempel-Ziv code]] <ref> {{Citation | last = Kieffer | first = J. C. | last2 = Yang | first2 = E.-H.  | last3 = Nelson | first3 = G. | last4 = Cosman | first4 = P. | title = Universal lossless compression via multilevel pattern matching | journal = IEEE Trans. Inform. Theory | volume = 46 | pages = 1227 –1245 | year = 2000 }}</ref>, the multilevel pattern matching (MPM) algorithm<ref>{{Citation | last = Ziv | first = J. | last2 = Lempel | first2 = A. | title = Compression of individual sequences via variable rate coding | journal = IEEE Trans. Inform. Theory | volume = 24 | pages = 530 – 536 | year = 1978 }}</ref>, and many other new universal lossless compression algorithms.
+The class of grammar-based codes is very broad. It includes block codes, variations of the incremental parsing [[LZ77 and LZ78|Lempel-Ziv code]] <ref> {{Citation | last = Kieffer | first = J. C. | last2 = Yang | first2 = E.-H.  | last3 = Nelson | first3 = G. | last4 = Cosman | first4 = P. | title = Universal lossless compression via multilevel pattern matching | journal = IEEE Trans. Inform. Theory | volume = 46 | pages = 1227 –1245 | year = 2000 | doi = 10.1109/18.850665 }}</ref>, the multilevel pattern matching (MPM) algorithm<ref>{{Citation | last = Ziv | first = J. | last2 = Lempel | first2 = A. | title = Compression of individual sequences via variable rate coding | journal = IEEE Trans. Inform. Theory | volume = 24 | pages = 530 – 536 | year = 1978 | doi = 10.1109/TIT.1978.1055934 }}</ref>, and many other new universal lossless compression algorithms.
 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.