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In fact, the language defined by a grammar is precisely the set of terminal strings that can be so derived. Context-free grammars are those grammars in which the left-hand side of each production rule consists of only a single nonterminal symbol. This restriction is non-trivial; not all languages can be generated by context-free grammars.
In particular, an alternative containing a useless nonterminal symbol can be deleted from the right-hand side of a rule. Such rules and alternatives are called useless. [24] In the depicted example grammar, the nonterminal D is unreachable, and E is unproductive, while C → C causes a cycle.
Some do not permit the second form of rule and cannot transform context-free grammars that can generate the empty word. For one such construction the size of the constructed grammar is O(n 4) in the general case and O(n 3) if no derivation of the original grammar consists of a single nonterminal symbol, where n is the size of the original ...
By deleting in this grammar each ε-rule, unless its left-hand side is the start symbol, the transformed grammar is obtained. [4]: 90 For example, in the following grammar, with start symbol S 0, S 0 → AbB | C B → AA | AC C → b | c A → a | ε. the nonterminal A, and hence also B, is nullable, while neither C nor S 0 is.
The language { www : w ∈ {a,b} *} is generable by an indexed grammar, but not by a linear indexed grammar, while both { ww : w ∈ {a,b} *} and { a n b n c n : n ≥ 1 } are generable by a linear indexed grammar. If both the original and the modified production rules are admitted, the language class remains the indexed languages. [16]
The standard version of CYK operates only on context-free grammars given in Chomsky normal form (CNF). However any context-free grammar may be algorithmically transformed into a CNF grammar expressing the same language (Sipser 1997). The importance of the CYK algorithm stems from its high efficiency in certain situations.
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It is named after Sige-Yuki Kuroda, who originally called it a linear bounded grammar, a terminology that was also used by a few other authors thereafter. [ 3 ] Every grammar in Kuroda normal form is noncontracting , and therefore, generates a context-sensitive language .
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