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 has a grammar of length , 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 where is the length of the given string and 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 would also improve certain algorithms for approximate addition chains.[5]
^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
^ abLohrey, Markus (2012). "Algorithmics on SLP-compressed strings: A survey" (PDF). Groups Complexity Cryptology. 4 (2): 241–299. doi:10.1515/GCC-2012-0016.
^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.
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