# Linear grammar

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

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 $\{a^{i}b^{i}\mid i\geq 0\}$.

## Relationship with regular grammars

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

All regular languages are linear; conversely, an example of a linear, non-regular language is { a$^n$b$^n$ }. 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. 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.
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
### Positive cases

Linear languages are closed under union. Construction is the same as the construction for the union of context-free languages. Let $L_{1},L_{2}$ be two linear languages, then $L_{1}\cup L_{2}$ is constructed by a linear grammar with $S\to S_{1}|S_{2}$, and $S_{1},S_{2}$ playing the role of the linear grammars for $L_{1},L_{2}$.
If L is a linear language and M is a regular language, then the intersection $L\cap M$ 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.
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

Linear languages are not closed under intersection. For example, let $L_{1}=\{a^{n}b^{n}c^{m}\mid n,m\geq 0\},L_{2}=\{a^{n}b^{m}c^{m}\mid n,m\geq 0\}$, 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).