Definition: Type3 (Linear) Grammars and Regular Languages
A type3 (or linear) grammar is a grammar $G=(V,T,R,S)$ in which every rule $P\to Q\in R$ allows the replacement of only one variable and the variable is at one end (leftside or rightside) of a production rule. There are two types of linear grammars:
 leftlinear if and only if all grammar rules $P\to Q\in R$ consist of $P\in V$ (the premise $P$ is a variable) and $Q\in V T^+\cup \{\epsilon\}$ (the conclusion $Q$ is either the empty string $\epsilon$ or it is a concatenation starting with a variable, followed by some nonempty string of terminals).
 rightlinear if and only if all grammar rules $P\to Q\in R$ consist of $P\in V$ (the premise $P$ is a variable) and $Q\in T^+V\cup \{\epsilon\}$ (the conclusion $Q$ is either the empty string or it is a concatenation of some nonempty string of terminals ending with a variable).
Formal languages generated by type3 grammars are called regular.
Notes
 The characteristics of type3 grammars is that they always replace a single variable by the empty word or a concatenation of terminals ending or starting with a single variable.
 The notation $T^+$ denotes the Kleene plus of all terminal symbols.
Mentioned in:
Explanations: 1
Proofs: 2
Theorems: 3
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References
Bibliography
 Erk, Katrin; Priese, Lutz: "Theoretische Informatik", Springer Verlag, 2000, 2nd Edition
 Hoffmann, Dirk: "Theoretische Informatik, 3. Auflage", Hanser, 2015