Information AboutCyk Algorithm |
| CATEGORIES ABOUT CYK ALGORITHM | |
| parsing algorithms | |
| SHOPPER'S DELIGHT | |
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String can be generated by a given Context-free Grammar and, if so, how it can be generated. This is known as Parsing the string. The algorithm is an example of Dynamic Programming . The standard version of CYK recognizes languages defined by context-free grammars written in Chomsky Normal Form (CNF). Since any context-free grammar can be converted to CNF without too much difficulty, CYK can be used to recognize any context-free language. It is also possible to extend the CYK algorithm to handle some context-free grammars which are not written in CNF; this may be done to improve performance, although at the cost of making the algorithm harder to understand. The worst case Asymptotic time complexity of CYK is Θ (n3), where ''n'' is the length of the parsed string. This makes it one of the most efficient (in those terms) algorithms for recognizing any context-free language. However, there are other algorithms that will perform better for certain subsets of the context-free languages. THE ALGORITHM The CYK algorithm is a bottom up algorithm and is important theoretically, since it can be used to constructively prove that the membership problem for context-free languages is decidable. The CYK algorithm for the membership problem is as follows: Let the input string consist of ''n'' letters, ''a''1 ... ''a''''n''. Let the grammar contain ''r'' terminal and nonterminal symbols ''R''1 ... ''R''''r''. This grammar contains the subset Rs which is the set of start symbols. Let P {Link without Title} be an array of booleans. Initialize all elements of P to false. For each i = 1 to n For each unit production Rj → ai, '''set''' P {Link without Title} = true. For each i = 2 to n -- Length of span For each j = 1 to n-i+1 -- Start of span For each k = 1 to i-1 -- Partition of span For each production RA -> RB RC If P and P[j+k,i-k,C '''then''' set P[j,i,A] = true If any of P {Link without Title} is true (x is iterated over the set s, where s are all the indices for Rs) Then string is member of language Else string is not member of language Informally In informal terms, this algorithm considers every possible substring of the string of letters and sets P {Link without Title} to be true if the substring of letters starting from i of length j can be generated from Rk. Once it has considered substrings of length 1, it goes on to substrings of length 2, and so on. For substrings of length 2 and greater, it considers every possible partition of the substring into two halves, and checks to see if there is some production P → Q R such that Q matches the first half and R matches the second half. If so, it records P as matching the whole substring. Once this process is completed, the sentence is recognized by the grammar if the substring containing the entire string is matched by the start symbol. Algorithm extension It is simple to extend the above algorithm to not only determine if a sentence is in a language, but to also construct a parse tree, by storing parse tree nodes as elements of the array, instead of booleans. Since the grammars being recognized can be ambiguous, it is necessary to store a list of nodes (unless one wishes to only pick one possible parse tree); the end result is then a forest of possible parse trees. An alternative formulation employs a second table B {Link without Title} of so-called ''backpointers''. It is also possible to extend the CYK algorithm to parse strings using Weighted and Stochastic Context-free Grammar s. Weights (probabilities) are then stored in the table P instead of booleans, so P {Link without Title} will contain the minimum weight (maximum probability) that the substring from i to j can be derived from A. Further extensions of the algorithm allow all parses of a string to be enumerated from lowest to highest weight (highest to lowest probability). REFERENCES
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