Grammar-based code

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 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.^[1] To compress a data sequence $x=x_{1}\cdots x_{n}$ , a grammar-based code transforms $x$ into a context-free grammar $G$ . The problem of finding a smallest grammar for an input sequence (smallest grammar problem) is known to be NP-hard,^[2] so many grammar-transform algorithms are proposed from theoretical and practical viewpoints. Generally, the produced grammar $G$ 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, the multilevel pattern matching (MPM) algorithm,^[3] variations of the incremental parsing Lempel-Ziv code,^[4] 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^[5] 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^[6] 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,^[7] 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.)

References

^ Kieffer, J. C.; Yang, E.-H. (2000), "Grammar-based codes: A new class of universal lossless source codes", IEEE Trans. Inf. Theory, 46 (3): 737–754, doi:10.1109/18.841160
^ Charikar, M.; Lehman, E.; Liu, D.; Panigrahy, R.; Prabharakan, M.; Sahai, A.; Shelat, A. (2005), "The Smallest Grammar Problem", IEEE Trans. Inf. Theory, 51 (7): 2554–2576, doi:10.1109/tit.2005.850116, S2CID 6900082
^ Kieffer, J. C.; Yang, E.-H.; Nelson, G.; Cosman, P. (2000), "Universal lossless compression via multilevel pattern matching", IEEE Trans. Inf. Theory, 46 (4): 1227–1245, doi:10.1109/18.850665, S2CID 8191526
^ Ziv, J.; Lempel, A. (1978), "Compression of individual sequences via variable rate coding", IEEE Trans. Inf. Theory, 24 (5): 530–536, doi:10.1109/TIT.1978.1055934, hdl:10338.dmlcz/142945
^ Nevill-Manning, C. G.; Witten, I. H. (1997), "Identifying Hierarchical Structure in Sequences: A linear-time algorithm", Journal of Artificial Intelligence Research, 7 (4): 67–82, arXiv:cs/9709102, doi:10.1613/jair.374, hdl:10289/1186, S2CID 2957960
^ Larsson, N. J.; Moffat, A. (2000), "Offline Dictionary-Based Compression" (PDF), Proceedings of the IEEE, 88 (11): 1722–1732, doi:10.1109/5.892708
^ Conrad, Kennon J.; Wilson, Paul R. (2016). "Grammatical Ziv-Lempel Compression: Achieving PPM-Class Text Compression Ratios with LZ-Class Decompression Speed". 2016 Data Compression Conference (DCC). p. 586. doi:10.1109/DCC.2016.119. ISBN 978-1-5090-1853-6. S2CID 3116024.