In computer science, the Cocke–Younger–Kasami algorithm (alternatively called CYK, or CKY) is a parsing algorithm for context-free grammars published by Itiroo Sakai in 1961.

[1][2] The algorithm is named after some of its rediscoverers: John Cocke, Daniel Younger, Tadao Kasami, and Jacob T. Schwartz.

It employs bottom-up parsing and dynamic programming.

The standard version of CYK operates only on context-free grammars given in Chomsky normal form (CNF).

The importance of the CYK algorithm stems from its high efficiency in certain situations.

Using big O notation, the worst case running time of CYK is

This makes it one of the most efficient [citation needed] parsing algorithms in terms of worst-case asymptotic complexity, although other algorithms exist with better average running time in many practical scenarios.

The dynamic programming algorithm requires the context-free grammar to be rendered into Chomsky normal form (CNF), because it tests for possibilities to split the current sequence into two smaller sequences.

Any context-free grammar that does not generate the empty string can be represented in CNF using only production rules of the forms

In informal terms, this algorithm considers every possible substring of the input string and sets

Once this process is completed, the input string is generated by the grammar if the substring containing the entire input string is matched by the start symbol.

This is an example grammar: Now the sentence she eats a fish with a fork is analyzed using the CYK algorithm.

For readability, the CYK table for P is represented here as a 2-dimensional matrix M containing a set of non-terminal symbols, such that Rk is in ⁠

The above algorithm is a recognizer that will only determine if a sentence is in the language.

It is simple to extend it into a parser that also constructs a parse tree, by storing parse tree nodes as elements of the array, instead of the boolean 1.

The node is linked to the array elements that were used to produce it, so as to build the tree structure.

Only one such node in each array element is needed if only one parse tree is to be produced.

However, if all parse trees of an ambiguous sentence are to be kept, it is necessary to store in the array element a list of all the ways the corresponding node can be obtained in the parsing process.

This shared forest can conveniently be read as an ambiguous grammar generating only the sentence parsed, but with the same ambiguity as the original grammar, and the same parse trees up to a very simple renaming of non-terminals, as shown by Lang (1994).

As pointed out by Lange & Leiß (2009), the drawback of all known transformations into Chomsky normal form is that they can lead to an undesirable bloat in grammar size.

For the use in teaching, Lange and Leiß propose a slight generalization of the CYK algorithm, "without compromising efficiency of the algorithm, clarity of its presentation, or simplicity of proofs" (Lange & Leiß 2009).

It is also possible to extend the CYK algorithm to parse strings using weighted and stochastic context-free grammars.

The worst case running time of CYK is

, 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 algorithms for recognizing general context-free languages in practice.

Valiant (1975) gave an extension of the CYK algorithm.

Using the Coppersmith–Winograd algorithm for multiplying these matrices, this gives an asymptotic worst-case running time of

However, the constant term hidden by the Big O Notation is so large that the Coppersmith–Winograd algorithm is only worthwhile for matrices that are too large to handle on present-day computers (Knuth 1997), and this approach requires subtraction and so is only suitable for recognition.

The dependence on efficient matrix multiplication cannot be avoided altogether: Lee (2002) has proved that any parser for context-free grammars working in time

can be effectively converted into an algorithm computing the product of

, and this was extended by Abboud et al.[4] to apply to a constant-size grammar.