# What is the importance of linear languages?

What is the point of linear languages? They appear to be an intermediate set of languages in between regular and context-free languages, but do they have any useful or nice properties that either have been studied, or make them worthwhile to study?

• What is a linear language? – Tyson Williams Sep 23 '14 at 11:58
• @Tyson Williams: Maybe this one, I guess? – Tsuyoshi Ito Sep 23 '14 at 14:42
• An obvious thing to try is to look for papers showing some results about linear languages and check their introductions. Maybe you already did this and found nothing useful, but if so, I cannot see it from the question. – Tsuyoshi Ito Sep 23 '14 at 14:46
• In general, when you receive comments, you should consider to edit the question. – Tsuyoshi Ito Sep 23 '14 at 22:29
• (1) Have you checked the paper cited in the Wikipedia page? (Disclaimer: I have not.) (2) Have you tried searching "linear language" "context-free language"? – Tsuyoshi Ito Sep 23 '14 at 22:34

A language $L$ of $A^*$ is linear if and only if there exists a rational subset $R$ of $A^* \times A^*$ such that $$L = \{ u\tilde v \mid (u, v) \in R \}$$ where $\tilde v$ denotes the reversal of $v$. This result shows that, in a very loose sense, linear languages are a generalization of the language of palindromes.