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{{Short description|Lossless data compression algorithm}}
[[File:IndpGrm.gif|thumb|600px|[[Straight-line grammar]] (with start symbol ß) for the second sentence of the [[United States Declaration of Independence]]. Each blue character denotes a [[nonterminal symbol]]; they were obtained from a [[gzip]]-compression of the sentence.]]
'''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 | last1 = Kieffer | first1 = J. C. | last2 = Yang | first2 = E.-H. | title = Grammar-based codes: A new class of universal lossless source codes | journal = IEEE Trans. Inf. Theory | volume = 46 | pages = 737–754 | year = 2000 | doi = 10.1109/18.841160 | issue = 3 }}</ref> 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 grammar <math>G</math>.
The problem of finding a smallest grammar for an input sequence ([[smallest grammar problem]]) is known to be NP-hard,<ref>{{Citation | last1 = Charikar | first1 = M. | last2 = Lehman | first2 = E. | last3 = Liu | first3 = D. | last4 = Panigrahy | first4 = R. | last5 = Prabharakan | first5 = M. | last6 = Sahai | first6 = A. | last7 = Shelat | first7 = A. | title = The Smallest Grammar Problem | journal = IEEE Trans. Inf. Theory | volume = 51 | pages = 2554–2576 | year = 2005 | issue = 7 | doi=10.1109/tit.2005.850116| s2cid = 6900082 }}</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 code]]s, the multilevel pattern matching (MPM) algorithm,<ref>{{Citation | last1 = Kieffer | first1 = 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. Inf. Theory | volume = 46 | pages = 1227–1245 | year = 2000 | doi = 10.1109/18.850665 | issue = 4 | s2cid = 8191526 | url = https://escholarship.org/uc/item/39k54514 }}</ref> variations of the incremental parsing [[LZ77 and LZ78|Lempel-Ziv code]],<ref>{{Citation | last1 = Ziv | first1 = J. | last2 = Lempel | first2 = A. | title = Compression of individual sequences via variable rate coding | journal = IEEE Trans. Inf. Theory | volume = 24 | pages = 530–536 | year = 1978 | doi = 10.1109/TIT.1978.1055934 | issue = 5 | hdl = 10338.dmlcz/142945 | hdl-access = free }}</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 the following are available from external links.

* [[SEQUITUR algorithm|Sequitur]]<ref>{{Citation | last1 = Nevill-Manning | first1 = 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 = 10.1613/jair.374| issue = 4 | hdl=10289/1186| arxiv = cs/9709102 | s2cid = 2957960 }}</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 | last1 = Larsson | first1 = N. J. | last2 = Moffat | first2 = A. | title = Offline Dictionary-Based Compression | journal = Proceedings of the IEEE| volume = 88 | pages = 1722–1732| year = 2000 | issue = 11 | doi=10.1109/5.892708| url = http://www.davidsalomon.name/DC4advertis/BPE.pdf }}</ref> is a greedy algorithm using the strategy of most-frequent-first substitution. The compressive performance is powerful, although the main memory space requirement is very large.
* [[GLZA]],<ref>{{Cite book| last1 = Conrad | first1 = Kennon J. | last2 = Wilson | first2 = Paul R. | title = 2016 Data Compression Conference (DCC) | chapter = Grammatical Ziv-Lempel Compression: Achieving PPM-Class Text Compression Ratios with LZ-Class Decompression Speed | pages = 586 | year = 2016 | doi=10.1109/DCC.2016.119| isbn = 978-1-5090-1853-6 | s2cid = 3116024 }}</ref> which constructs a grammar that may be reducible, i.e., contain repeats, where the entropy-coding cost of "spelling out" the repeats is less than the cost creating and entropy-coding a rule to capture them. (In general, the compression-optimal SLG is not irreducible, and the Smallest Grammar Problem is different from the actual SLG compression problem.)

==See also==
* [[Dictionary coder]]
*[[Grammar induction]]
* [[Straight-line grammar]]

== References ==
{{reflist|30em}}

==External links==
*[https://encode.su/threads/2427-GLZA GLZA discussion and paper]
*[https://www.mit.edu/~6.454/www_fall_2002/emin/summary.pdf Description of grammar-based codes with example]
*[http://sequitur.info Sequitur codes] {{Webarchive|url=https://web.archive.org/web/20081013024323/http://www.sequitur.info/ |date=2008-10-13 }}
*[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.
*[https://github.com/GrammarViz2/grammarviz2_src GrammarViz 2.0] - implementation of Sequitur, Re-Pair, and parallel Re-Pair in Java.

{{Compression methods}}

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

Grammar-based code - Revision history

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