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Other typical examples include the language consisting of all strings over the alphabet {a, b} which contain an even number of a's, or the language consisting of all strings of the form: several a's followed by several b's. A simple example of a language that is not regular is the set of strings {a n b n | n ≥ 0}. [4]
The left-regular grammars describe the reverses of all such languages, that is, exactly the regular languages as well. Every strict right-regular grammar is extended right-regular, while every extended right-regular grammar can be made strict by inserting new non-terminals, such that the result generates the same language; hence, extended right ...
In the theory of formal languages, the pumping lemma for regular languages is a lemma that describes an essential property of all regular languages. Informally, it says that all sufficiently long strings in a regular language may be pumped —that is, have a middle section of the string repeated an arbitrary number of times—to produce a new ...
Regular languages are commonly used to define search patterns and the lexical structure of programming languages. For example, the regular language = {| >} is generated by the Type-3 grammar = ({}, {,},,) with the productions being the following. S → aS S → a
Given a set of strings (also called "positive examples"), the task of regular language induction is to come up with a regular expression that denotes a set containing all of them. As an example, given {1, 10, 100}, a "natural" description could be the regular expression 1⋅0 * , corresponding to the informal characterization " a 1 followed by ...
Some classes of regular languages can only be described by deterministic finite automata whose size grows exponentially in the size of the shortest equivalent regular expressions. The standard example here is the languages L k consisting of all strings over the alphabet {a,b} whose kth-from-last letter equals a.
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.
Context-free languages are closed under the various operations, that is, if the languages K and L are context-free, so is the result of the following operations: union K ∪ L; concatenation K ∘ L; Kleene star L * [11] substitution (in particular homomorphism) [12] inverse homomorphism [13] intersection with a regular language [14]