Cocke-Younger-Kasami algorithm (CYK) is an example of dynamic programming. Its purpose is to efficiently parse a string belonging to a context-free language.
The standard version of CYK can only recognize languages defined by context-free grammars in Chomsky Normal Form (CNF). However, 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 grammars which are not in CNF, although at the cost of making the algorithm significantly harder to understand.
The worst case asympotic time complexity of CYK is O(n3), which makes it the most efficient (in those terms) algorithm for recognizing a context-free language. However, in many cases, other algorithms will give better performance.
The CYK algorithm is important theoretically, since it can be used to constructively prove that the membership problem for context-free languages is decidable.
The CYK algorithm is a type of active chart parsing (is this correct?).
The CYK algorithm for the membership problem is as follows:
- Let the input be a sequence of n words w1 ... wn.
- Let the grammar contain r terminal and nonterminal symbols R1 ... Rr, and let R1 be the start symbol.
- Let P[n,n,r] be an array of booleans. Initialize all elements of P to false.
- For each i = 1 to n
- For each unit production Rj -> wi, set P[i,1,j] = 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 j-1 -- Partition of span
- For each production RA -> RB RC
- If P[j,k,B] and P[j+k,i-k,C] then set P[j,i,A] = true
- For each production RA -> RB RC
- For each k = 1 to j-1 -- Partition of span
- For each j = 1 to n-i+1 -- Start of span
- If R[1,n,1] is true
- Then sentence is member of language
- Else sentence is not member of language
In informal terms, this algorithm considers every possible consecutive subsequence of the sequence of words, starting with those of length 1. It sets P[i,j,k] to be true if the sequence of words starting from i of length j matches a production whose left hand side is symbol number k. Once it has considered sequences of length 1, it goes on to sequences of length 2, and so on. For subsequences of length 2 and greater, it considers every possible partition of the subsequence 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 subsequence. Once this process is completed, the sentence is recognized by the grammar if the subsequence containing the entire sentence is matched by the start symbol.
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 neccessary to store a list of nodes (unless one wishes to only pick one possible parse); the end result is a forest of possible parse trees.
It is also possible to extend the CYK algorithm to parse Probabilised Context Free Grammars (PCFG).