Smallest grammar problem

In data compression and the theory of formal languages, the smallest grammar problem is the problem of finding the smallest context-free grammar that generates a given string of characters (but no other string). The size of a grammar is defined by some authors as the number of symbols on the right side of the production rules.^[1] Others also add the number of rules to that.^[2] A grammar that generates only a single string, as required for the solution to this problem, is called a straight-line grammar.^[3]

Every binary string of length $n$ has a grammar of length $O(n/\log n)$ , as expressed using big O notation.^[3] For binary de Bruijn sequences, no better length is possible.^[4]

The (decision version of the) smallest grammar problem is NP-complete.^[1] It can be approximated in polynomial time to within a logarithmic approximation ratio; more precisely, the ratio is $O(\log {\tfrac {n}{g}})$ where $n$ is the length of the given string and $g$ is the size of its smallest grammar. It is hard to approximate to within a constant approximation ratio. An improvement of the approximation ratio to $o(\log n/\log \log n)$ would also improve certain algorithms for approximate addition chains.^[5]

References

^ ^a ^b Charikar, Moses; Lehman, Eric; Liu, Ding; Panigrahy, Rina; Prabhakaran, Manoj; Sahai, Amit; Shelat, Abhi (2005). "The Smallest Grammar Problem". IEEE Transactions on Information Theory. 51 (7): 2554–2576. CiteSeerX 10.1.1.185.2130. doi:10.1109/TIT.2005.850116. S2CID 6900082. Zbl 1296.68086.
^ Florian Benz and Timo Kötzing, “An effective heuristic for the smallest grammar problem,” Proceedings of the fifteenth annual conference on Genetic and evolutionary computation conference - GECCO ’13, 2013. ISBN 978-1-4503-1963-8 doi:10.1145/2463372.2463441
^ ^a ^b Lohrey, Markus (2012). "Algorithmics on SLP-compressed strings: A survey" (PDF). Groups Complexity Cryptology. 4 (2): 241–299. doi:10.1515/GCC-2012-0016.
^ Domaratzki, Michael; Pighizzini, Giovanni; Shallit, Jeffrey (2002). "Simulating finite automata with context-free grammars". Information Processing Letters. 84 (6): 339–344. doi:10.1016/S0020-0190(02)00316-2. MR 1937222.
^ Charikar, Moses; Lehman, Eric; Liu, Ding; Panigrahy, Rina; Prabhakaran, Manoj; Rasala, April; Sahai, Amit; Shelat, Abhi (2002). "Approximating the Smallest Grammar: Kolmogorov Complexity in Natural Models" (PDF). Proceedings of the thirty-fourth annual ACM symposium on theory of computing (STOC 2002), Montreal, Quebec, Canada, May 19–21, 2002. New York, NY: ACM Press. pp. 792–801. doi:10.1145/509907.510021. ISBN 978-1-581-13495-7. S2CID 282489. Zbl 1192.68397.