Unrestricted grammar

In formal language theory, an unrestricted grammar is a formal grammar on which no restrictions are made on the left and right sides of the grammar's productions. This is the most general class of grammars in the Chomsky–Schützenberger hierarchy and can generate arbitrary recursively enumerable languages.

An unrestricted grammar is a formal grammar ${\displaystyle G=(N,\Sigma ,P,S)}$, where ${\displaystyle N}$ is a set of nonterminal symbols, ${\displaystyle \Sigma }$ is a set of terminal symbols, ${\displaystyle N}$ and ${\displaystyle \Sigma }$ are disjoint (actually, this is not strictly necessary since unrestricted grammars make no real distinction between the two; the designation exists purely so that one knows when to stop generating sentential forms of the grammar), ${\displaystyle P}$ is a set of production rules of the form ${\displaystyle \alpha \to \beta }$ where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are strings of symbols in ${\displaystyle N\cup \Sigma }$ and ${\displaystyle \alpha }$ is not the empty string, and ${\displaystyle S\in N}$ is a specially designated start symbol. As the name implies, there are no real restrictions on the types of production rules that unrestricted grammars can have.

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