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Linear grammar

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In computer science, a linear grammar is a context-free grammar that has at most one nonterminal in the right-hand side of each of its productions.

A linear language is a language generated by some linear grammar.

Example

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An example of a linear grammar is G with N = {S}, Σ = {a, b}, P with start symbol S and rules

S → aSb
S → ε

It generates the language .

Relationship with regular grammars

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Two special types of linear grammars are the following:

  • the left-linear or left-regular grammars, in which all rules are of the form A → αw where α is either empty or a single nonterminal and w is a string of terminals;
  • the right-linear or right-regular grammars, in which all rules are of the form A → wα where w is a string of terminals and α is either empty or a single nonterminal.

Each of these can describe exactly the regular languages. A regular grammar is a grammar that is left-linear or right-linear.

Observe that by inserting new nonterminals, any linear grammar can be replaced by an equivalent one where some of the rules are left-linear and some are right-linear. For instance, the rules of G above can be replaced with

S → aA
A → Sb
S → ε

However, the requirement that all rules be left-linear (or all rules be right-linear) leads to a strict decrease in the expressive power of linear grammars.

Expressive power

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All regular languages are linear; conversely, an example of a linear, non-regular language is { anbn }. as explained above. All linear languages are context-free; conversely, an 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 languages, which in turn are a proper subset of the context-free languages.

While regular languages are deterministic, there exist linear languages that are nondeterministic. For example, the language of even-length palindromes 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.[1] This language is nondeterministic. Since nondeterministic context-free languages cannot be accepted in linear time [clarification needed], 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.[2]

A language is linear iff it can be generated by a one-turn pushdown automaton – a pushdown automaton that, once it starts popping, never pushes again.

Closure properties

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Positive cases

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Linear languages are closed under union. Construction is the same as the construction for the union of context-free languages. Let be two linear languages, then is constructed by a linear grammar with , and playing the role of the linear grammars for .

If L is a linear language and M is a regular language, then the intersection is again a linear language; in other words, the linear languages are closed under intersection with regular sets.

Linear languages are closed under homomorphism and inverse homomorphism.[3]

As a corollary, linear languages form a full trio. Full trios in general are language families that enjoy a couple of other desirable mathematical properties.

Negative cases

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Linear languages are not closed under intersection. For example, let , then their intersection is not only not linear, but also not context-free. See pumping lemma for context-free languages.

As a corollary, linear languages are not closed under complement (as intersection can be constructed by de Morgan's laws out of union and complement).

References

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  1. ^ Hopcroft, John; Rajeev Motwani; Jeffrey Ullman (2001). Introduction to automata theory, languages, and computation 2nd edition. Addison-Wesley. pp. 249–253.
  2. ^ Greibach, Sheila (October 1966). "The Unsolvability of the Recognition of Linear Context-Free Languages". Journal of the ACM. 13 (4): 582–587. doi:10.1145/321356.321365. S2CID 37003419.
  3. ^ 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