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A weighted context-free grammar (WCFG) is a more general category of context-free grammar, where each production has a numeric weight associated with it. The weight of a specific parse tree in a WCFG is the product [7] (or sum [8]) of all rule weights in the tree. Each rule weight is included as often as the rule is used in the tree.
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.
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.
Sequitur (or Nevill-Manning–Witten algorithm) is a recursive algorithm developed by Craig Nevill-Manning and Ian H. Witten in 1997 [1] that infers a hierarchical structure (context-free grammar) from a sequence of discrete symbols. The algorithm operates in linear space and time. It can be used in data compression software applications. [2]
To compress a data sequence =, a grammar-based code transforms into a context-free grammar . The problem of finding a smallest grammar for an input sequence ( smallest grammar problem ) is known to be NP-hard, [ 2 ] so many grammar-transform algorithms are proposed from theoretical and practical viewpoints.
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 ...
To do so technically would require a more sophisticated grammar, like a Chomsky Type 1 grammar, also termed a context-sensitive grammar. However, parser generators for context-free grammars often support the ability for user-written code to introduce limited amounts of context-sensitivity.
The theorem can be used in analytic combinatorics to estimate the number of words of length n generated by a given unambiguous context-free grammar, as n grows large. The following example is given by Gruber, Lee & Shallit (2012): the unambiguous context-free grammar G over the alphabet {0,1} has start symbol S and the following rules