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Possibly negative range concatenation languages are also closed under set complement. A consequence of the above is that it is undecidable whether a (positive) range concatenation language is nonempty, because it is undecidable whether the intersection of two context-free languages is nonempty. Hence range concatenation grammars are not generative.
In formal language theory and pattern matching (including regular expressions), the concatenation operation on strings is generalised to an operation on sets of strings as follows: For two sets of strings S 1 and S 2 , the concatenation S 1 S 2 consists of all strings of the form vw where v is a string from S 1 and w is a string from S 2 , or ...
The collection of regular languages over an alphabet Σ is defined recursively as follows: The empty language Ø is a regular language. For each a ∈ Σ (a belongs to Σ), the singleton language {a } is a regular language. If A is a regular language, A* (Kleene star) is a regular language. Due to this, the empty string language {ε} is also ...
Let A be the set of all regular languages over Σ (or the set of all context-free languages over Σ; or the set of all recursive languages over Σ; or the set of all languages over Σ). Then the union (written as +) and the concatenation (written as ·) of two elements of A again belong to A, and so does the Kleene star operation applied to any ...
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]
Later, Jeż [12] showed that non-regular unary languages can be defined by language equations with union, intersection and concatenation, equivalent to conjunctive grammars. By this method Jeż and Okhotin [13] proved that every recursive unary language is a unique solution of some equation.
Regular languages are closed under string substitution. That is, if each character in the alphabet of a regular language is substituted by another regular language, the result is still a regular language. [2] Similarly, context-free languages are closed under string substitution. [3] [note 1]
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 ...