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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.
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.
Context-free languages—or rather its subset of deterministic context-free languages—are the theoretical basis for the phrase structure of most programming languages, though their syntax also includes context-sensitive name resolution due to declarations and scope.
To convert a grammar to Chomsky normal form, a sequence of simple transformations is applied in a certain order; this is described in most textbooks on automata theory. [4]: 87–94 [5] [6] [7] The presentation here follows Hopcroft, Ullman (1979), but is adapted to use the transformation names from Lange, Leiß (2009).
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.
One way to do this is by using a probabilistic context-free grammar (PCFG) which has a probability of each constituency rule, and modifying CKY to maximise probabilities when parsing bottom-up. [6] [7] [8] A further modification is the lexicalized PCFG, which assigns a head to each constituent and encodes rule for each lexeme in that head slot.
The Earley parser executes in cubic time in the general case (), where n is the length of the parsed string, quadratic time for unambiguous grammars (), [4] and linear time for all deterministic context-free grammars.
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.
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