Grammar-based code

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 ${\displaystyle x=x_{1}\cdots x_{n}}$, a grammar-based code transforms ${\displaystyle x}$ into a context-free grammar ${\displaystyle G}$. 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 ${\displaystyle G}$ is further compressed by statistical encoders like arithmetic coding.

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.

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.)

See also

• Dictionary coder
• Grammar induction
• Straight-line grammar