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. The size of a grammar is defined by some authors as the number of symbols on the right side of the production rules.
Others also add the number of rules to that. A grammar that generates only a single string, as required for the solution to this problem, is called a straight-line grammar.
Every binary string of length has a grammar of length, as expressed using big O notation. For binary de Bruijn sequences, no better length is possible.
The smallest grammar problem is NP-complete.
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.