CYK algorithm

The Cocke-Younger-Kasami (CYK) algorithm (alternatively called CKY) determines whether a 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 employs bottom-up parsing and dynamic programming.

The standard version of CYK operates on context-free grammars given in Chomsky normal form (CNF). Any context-free grammar may be transformed to a CNF grammar expressing the same language.^[1]

The importance of the CYK algorithm stems from the fact that it constructively proves that the membership problem for context-free languages is decidable (see also: theory of computation) and the fact that it does so quite efficiently. Using Landau symbols, the worst case asymptotic time complexity of CYK is ${\displaystyle \Theta (n^{3}\cdot |G|)}$, where n is the length of the parsed string and |G| is the size of the CNF grammar G. This makes it one of the most efficient (in those terms) algorithms for recognizing general context-free languages. Specialized and faster algorithms exist for certain subsets of the context-free languages.

Standard form

The algorithm requires the context-free grammar to be rendered into Chomsky normal form.

The algorithm as given in pseudocode is as follows:

As pseudocode

Let the input be a string S consisting of n characters: a1 ... an.
Let the grammar contain r nonterminal symbols R1 ... Rr.
This grammar contains the subset Rs which is the set of start symbols.
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 -> ai, 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 i-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
If any of P[1,n,x] is true (x is iterated over the set s, where s are all the indices for Rs)
  Then S is member of language
Else S is not member of language

As prose

In informal terms, this algorithm considers every possible subsequence of the sequence of words and sets P[i,j,k] to be true if the subsequence of words starting from i of length j can be generated from R_k. Once it has considered subsequences of length 1, it goes on to subsequences of length 2, and so on. For subsequences of length 2 and greater, it considers every possible partition of the subsequence into two parts, and checks to see if there is some production P → Q R such that Q matches the first part and R matches the second part. 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.

CYK table

S
	VP
S
	VP			PP
S		NP			NP
NP	V, VP	Det.	N	P	Det	N
she	eats	a	fish	with	a	fork

Extensions

Generating a parse tree

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[n,n,r] of so-called backpointers.

Parsing non-CNF context-free grammars

As pointed out by Lange and Leiß (2009), the drawback of these transformations is that they can lead to an undesirable bloat in grammar size. Using ${\displaystyle |G|}$  to denote the size of the original grammar ${\displaystyle G}$ , the size blow-up in the worst case may range from ${\displaystyle |G|^{2}}$  to ${\displaystyle 2^{2|G|}}$ , depending on the used transformation algorithm. For the use in teaching, they propose a slight generalization of the CYK algorithm, "without compromising efficiency of the algorithm, clarity of its presentation, or simplicity of proofs" (Lange and Leiß 2009).

Parsing weighted context-free grammars

It is also possible to extend the CYK algorithm to parse strings using weighted and stochastic context-free grammars. Weights (probabilities) are then stored in the table P instead of booleans, so P[i,j,A] 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).

See also

• Earley parser
• Packrat parser

References

1. Sipser, Michael (1997), Introduction to the Theory of Computation (1st ed.), IPS, p. 99, ISBN 0-534-94728-X

• John Cocke and Jacob T. Schwartz (1970). Programming languages and their compilers: Preliminary notes. Technical report, Courant Institute of Mathematical Sciences, New York University.
• T. Kasami (1965). An efficient recognition and syntax-analysis algorithm for context-free languages. Scientific report AFCRL-65-758, Air Force Cambridge Research Lab, Bedford, MA.
• Daniel H. Younger (1967). Recognition and parsing of context-free languages in time n³. Information and Control 10(2): 189–208.

External links

• CYK parsing demo in JavaScript
  
• Interactive Applet from the University of Leipzig to demonstrate the CYK-Algorithm (Site is in german)
  
• Exorciser is a Java application to generate exercises in the CYK algorithm as well as Finite State Machines, Markov algorithms etc (this page is in English)