# Prefix grammar

In theoretical computer science and formal language theory, a prefix grammar is a type of string rewriting system, consisting of a set of string rewriting rules, and similar to a formal grammar or a semi-Thue system. What is specific about prefix grammars is not the shape of their rules, but the way in which they are applied: only prefixes are rewritten. The prefix grammars describe exactly all regular languages.[1]

## Formal definition

A prefix grammar G is a 3-tuple, (Σ, S, P), where

• Σ is a finite alphabet
• S is a finite set of base strings over Σ
• P is a set of production rules of the form uv where u and v are strings over Σ

For strings x, y, we write x →G y (and say: G can derive y from x in one step) if there are strings u, v, w such that x = vu, y = wu, and v → w is in P. Note that G is a binary relation on the strings of Σ.

The language of G, denoted L(G), is the set of strings derivable from S in zero or more steps: formally, the set of strings w such that for some s in S, s R w, where R is the transitive closure of G.

## Example

The prefix grammar

• Σ = {0, 1}
• S = {01, 10}
• P = {0 → 010, 10 → 100}

describes the language defined by the regular expression

$01(01)^* \cup 100^*$