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| In [[computer science]], a '''linear grammar''' is a [[context-free grammar]] that has at most one nonterminal in the [[Left-hand side and right-hand side of an equation|right hand side]] of its productions.
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| A '''linear language''' is a language generated by some linear grammar.
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| ==Example==
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| A simple linear grammar is ''G'' with ''N'' = {S}, Σ = {a, b}, ''P'' with start symbol ''S'' and rules
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| : S → aSb
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| : S → ε
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| It generates the language <math>\{ a^ib^i \; | \; i \geq 0\}</math>.
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| ==Relationship with regular grammars==
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| Two special types of linear grammars are the following:
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| * the '''left-linear''' or [[Regular grammar|left regular grammars]], in which '''all nonterminals''' in right hand sides are '''at the left ends''';
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| * the '''right-linear''' or right regular grammars, in which '''all nonterminals''' in right hand sides are '''at the right ends'''.
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| Collectively, these two special types of linear grammars are known as the [[regular grammar]]s; both can describe exactly the [[regular language]]s.
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| Another special type of linear grammar is the following:
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| * linear grammars in which all nonterminals in right hand sides are at the left or right ends, but not necessarily all at the same end.
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| By inserting new nonterminals, every linear grammar can be brought into this form without affecting the language generated.
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| For instance, the rules of ''G'' above can be replaced with
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| : S → aA
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| : A → Sb
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| : S → ε
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| Hence, linear grammars of this special form can generate all [[linear language]]s.
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| ==Expressive power==
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| We have seen that all regular languages are linear; the example gave a linear, non-regular language.
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| All linear languages are [[context-free language|context-free]] by definition; a simple example of a context-free, non-linear language is the [[Dyck language]] of well-balanced bracket pairs. Hence, the regular languages are a [[proper subset]] of the linear ones, which in turn are a proper subset of the context-free languages.
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| While the linear languages that are regular languages are deterministic, there exist linear languages that are nondeterministic. For example, the language of even-length [[palindrome]]s on the alphabet of 0 and 1 has the linear grammar S → 0S0 | 1S1 | ε. An arbitrary string of this language cannot be parsed without reading all its letters first which means that a pushdown automaton has to try alternative state transitions to accommodate for the different possible lengths of a semi-parsed string.<ref>{{cite book | last = [[John Hopcroft|Hopcroft]] | first = John | coauthors = [[Rajeev Motwani]] & [[Jeffrey Ullman]] | title = [[Introduction to automata theory, languages, and computation]] 2nd edition | year = 2001 | publisher = Addison-Wesley | pages = 249–253 }}</ref> This language is nondeterministic. Since nondeterministic context-free languages cannot be accepted in linear time, linear languages cannot be accepted in linear time in the general case. Furthermore, it is undecidable whether a given context-free language is a linear context-free language.<ref>{{cite journal | last=Greibach | first=Sheila | title=The Unsolvability of the Recognition of Linear Context-Free Languages |date=October 1966 | work=Journal of the ACM | volume=13 | issue=4}}</ref>
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| ==Closure properties== | |
| If ''L'' is a linear language and ''M'' is a regular language, then the intersection <math>L \cap M</math> is again a linear language; in other words, the linear languages are closed under intersection with regular sets. Furthermore, the linear languages are closed under [[String_operations#String_homomorphism|homomorphism]] and [[String_operations#String_homomorphism|inverse homomorphism]].<ref>John E. Hopcroft and Jeffrey D. Ullman, ''[[Introduction to Automata Theory, Languages, and Computation]]'', Addison-Wesley Publishing, Reading Massachusetts, 1979. ISBN 0-201-02988-X., Ex. 11.1, pp. 282f</ref> These three closure properties show that the linear languages form a [[cone (formal languages)|full trio]]. Full trios in general are language families that enjoy a couple of other desirable mathematical properties.
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| ==References==
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| <references/>
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| {{Formal languages and grammars}}
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| {{DEFAULTSORT:Linear Grammar}}
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| [[Category:Formal languages]]
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