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The set of all context-free languages is identical to the set of languages accepted by pushdown automata, which makes these languages amenable to parsing.Further, for a given CFG, there is a direct way to produce a pushdown automaton for the grammar (and thereby the corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.
An extended context-free grammar (or regular right part grammar) is one in which the right-hand side of the production rules is allowed to be a regular expression over the grammar's terminals and nonterminals. Extended context-free grammars describe exactly the context-free languages. [36]
The general idea of a hierarchy of grammars was first described by Noam Chomsky in "Three models for the description of language". [1] Marcel-Paul Schützenberger also played a role in the development of the theory of formal languages; the paper "The algebraic theory of context free languages" [2] describes the modern hierarchy, including context-free grammars.
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 ...
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.
A language L over the alphabet is context-free if and only if there exists . a matched alphabet ¯; a regular language over ¯,; and a homomorphism : (¯); such that = ().. We can interpret this as saying that any CFG language can be generated by first generating a typed Dyck language, filtering it by a regular grammar, and finally converting each bracket into a word in the CFG language.
Deterministic context-free grammars were particularly useful because they could be parsed sequentially by a deterministic pushdown automaton, which was a requirement due to computer memory constraints. [4] In 1965, Donald Knuth invented the LR(k) parser and proved that there exists an LR(k) grammar for every deterministic context-free language. [5]
Despite Ogden's lemma being a strengthening of the pumping lemma, it is insufficient to fully characterize the class of context-free languages. [2] This is in contrast to the Myhill-Nerode theorem, which unlike the pumping lemma for regular languages is a necessary and sufficient condition for regularity.