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Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one [note 1] which is in Chomsky normal form and has a size no larger than the square of the original grammar's size.
Note that the set of grammars corresponding to recursive languages is not a member of this hierarchy; these would be properly between Type-0 and Type-1. Every regular language is context-free, every context-free language is context-sensitive, every context-sensitive language is recursive and every recursive language is recursively enumerable.
The complement of a context-sensitive language is itself context-sensitive [10] a result known as the Immerman–Szelepcsényi theorem. Membership of a string in a language defined by an arbitrary context-sensitive grammar, or by an arbitrary deterministic context-sensitive grammar, is a PSPACE-complete problem.
A context-sensitive grammar (CSG) is a formal grammar in which the left-hand sides and right-hand sides of any production rules may be surrounded by a context of terminal and nonterminal symbols. Context-sensitive grammars are more general than context-free grammars , in the sense that there are languages that can be described by a CSG but not ...
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The pumping lemma for context-free languages (called just "the pumping lemma" for the rest of this article) describes a property that all context-free languages are guaranteed to have. The property is a property of all strings in the language that are of length at least p {\displaystyle p} , where p {\displaystyle p} is a constant—called the ...
Their book Generalized Phrase Structure Grammar, published in 1985, is the main monograph on GPSG, especially as it applies to English syntax. GPSG was in part a reaction against transformational theories of syntax. In fact, the notational extensions to context-free grammars (CFGs) developed in GPSG are claimed to make transformations redundant ...
Generalized context-free grammar (GCFG) is a grammar formalism that expands on context-free grammars by adding potentially non-context-free composition functions to rewrite rules. [1] Head grammar (and its weak equivalents) is an instance of such a GCFG which is known to be especially adept at handling a wide variety of non-CF properties of ...