Straight-line grammar

A straight-line grammar (sometimes abbreviated as SLG) is a formal grammar that generates exactly one string. Consequently, it does not branch (every non-terminal has only one associated production rule) nor loop (if non-terminal A appears in a derivation of B, then B does not appear in a derivation of A).

Areas of usefulness

Straight-line grammars are widely used in the development of algorithms that execute directly on compressed structures (without prior decompression).^: 212

SLGs are of interest in fields like Kolmogorov complexity, Lossless data compression, Structure discovery and Compressed data structures.

The problem of finding a context-free grammar (equivalently: an SLG) of minimal size that generates a given string is called the smallest grammar problem.

Straight-line grammars (more precisely: straight-line context-free string grammars) can be generalized to Straight-line context-free tree grammars. The latter can be used conveniently to compress trees.^: 212

Formal Definition

A context-free grammar G is an SLG if:

1. for every non-terminal N, there is at most one production rule that has N as its left-hand side, and

2. the directed graph G=<V,E>, defined by V being the set of non-terminals and (A,B) ∈ E whenever B appears at the right-hand side of a production rule for A, is acyclic.

A mathematical definition of the more general formalism of straight-line context-free tree grammars can be found in Lohrey et al.^: 215

An SLG in Chomsky normal form is equivalent to a straight-line program.

A list of algorithms using SLGs

• The Sequitur algorithm constructs a straight-line grammar for a given string.
• The Lempel-Ziv-Welch algorithm creates a context-free grammar in such a deterministic way that it is necessary to store only the start rule of the generated grammar.
• Byte pair encoding

See also

• Grammar-based code – Lossless data compression algorithm
• Non-recursive grammar - a grammar that does not loop, but may branch; generating a finite rather than a singleton language

References