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In [[theoretical computer science]] and [[formal language theory]], a '''prefix grammar''' is a type of [[string rewriting system]], consisting of a set of [[string (computer science)|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 [[prefix (computer science)|prefixes]] are rewritten. The prefix grammars describe exactly all [[regular language]]s.<ref>[http://portal.acm.org/citation.cfm?id=185820 M. Frazier and C. D. Page. Prefix grammars: An alternative characterization of the regular languages. Information Processing Letters, 51(2):67–71, 1994.]</ref> | |||
==Formal definition== | |||
A prefix grammar ''G'' is a [[n-tuple|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 ''u'' → ''v'' where ''u'' and ''v'' are strings over Σ | |||
For strings ''x'', ''y'', we write ''x →<sub>G</sub> 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 ''→<sub>G</sub>'' 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 ''→<sub>G</sub>''. | |||
==Example== | |||
The prefix grammar | |||
*Σ = {0, 1} | |||
*''S'' = {01, 10} | |||
*''P'' = {0 → 010, 10 → 100} | |||
describes the language defined by the [[regular expression]] | |||
:<math> 01(01)^* \cup 100^* </math> | |||
==See also == | |||
* [[Regular grammar]] | |||
==References== | |||
{{reflist}} | |||
[[Category:Formal languages]] | |||
Revision as of 19:34, 11 March 2013
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 u → v 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
See also
References
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