View source for Grothendieck–Katz p-curvature conjecture

'''Grammar-based codes''' or '''Grammar-based compression''' are [[Data compression|compression]] algorithms based on the idea of constructing a [[context-free grammar]] (CFG) 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 | doi = 10.1109/18.841160 | issue = 3 }}</ref> and [[SEQUITUR algorithm|SEQUITUR]], among others. To compress a data sequence <math>x = x_1 \cdots x_n</math>, a grammar-based code transforms <math>x</math> into a context-free CFG <math>G</math>.
The problem to find a smallest grammar for an input sequence is known to be NP-hard,<ref>{{Citation | last = Charikar  | first =  M.  | last2 = Lehman | first2 = E. | last3 = Liu | first3 = D. | last4 = Panigrapy | first4 = R. | last5 = Prabharakan | first5 = M. | last6 = Sahai | first6 = A. | last7 = Shelat | first7 = A. |  title = The Smallest Grammar Problem | journal = IEEE Trans. Inform. Theory | volume = 51 | pages = 2554–2576 | year = 2005 | issue = 7 }}</ref> so many grammar-transform algorithms are proposed from theoretical and practical viewpoints.
Generally, the produced grammar <math>G</math> is further compressed by statistical encoders like [[arithmetic coding]].

== 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 | doi = 10.1109/18.850665 | issue = 4 }}</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 | issue = 5 }}</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.

== Practical algorithms ==
The compression programs of following are available from external links.
* [[SEQUITUR algorithm|Sequitur]]<ref>{{Citation | last = Nevill-Manning  | first =  C. G.  | last2 = Witten | first2 = I. H. | title = Identifying Hierarchical Structure in Sequences: A linear-time algorithm | journal =  Journal of Artificial Intelligence Research | volume = 7 | pages = 67–82 | year = 1997 | doi = | issue = 4 | id={{hdl|10289/1186}}}}</ref> is a classical grammar compression algorithm that sequentially translates an input text into a CFG, and then the produced CFG is encoded by an arithmetic coder.
* [[Re-Pair]]<ref>{{Citation | last = Larsson  | first =  N. J.  | last2 = Moffat | first2 = A. | title = Offline Dictionary-Based Compression | journal = IEEE | volume = 88 | pages = 1722–1732| year = 2000 | issue = 11 }}</ref> is a greedy algorithm by the strategy of most-frequent-first substitution. The compressive performance is powerful, although the main memory space is very large.

==See also==
* [[Smallest grammar problem]]
* [[Straight-line grammar]]
== References ==
{{reflist|30em}}

==External links==
*[http://www.mit.edu/~6.454/www_fall_2002/emin/summary.pdf Description of grammar-based codes with example]
*[http://sequitur.info Sequitur codes] 
*[http://www.cbrc.jp/~rwan/en/restore.html Re-Pair codes]
*[http://www.dcc.uchile.cl/~gnavarro/software/index.html Re-Pair codes] a version of Gonzalo Navarro.

[[Category:Data compression]]
[[Category:Coding theory]]
[[Category:Information theory]]