Straight-line grammar

A straight-line grammar (sometimes with "straight-line" in scare quotes, also 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).

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

The problem of finding an SLG of minimal size that generates a given string is called the Smallest grammar problem.

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 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.

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 a such deterministic way that it is necessary to store only the start rule of the generated grammar.
• Byte pair encoding

See also

• Grammar-based code
• Non-recursive grammar - a grammar that doesn't loop, but may branch; generating a finite rather than a singleton language