CYK algorithm

related topics
{math, number, function}
{language, word, form}
{rate, high, increase}
{group, member, jewish}
{company, market, business}
{island, water, area}

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 (Sipser 1997).

In the theory of computation, the importance of the CYK algorithm stems from the fact that it constructively proves that it is decidable whether a given string belongs to the formal language described by a given context-free grammar, and the fact that it does so quite efficiently. Using Landau symbols, the worst case running time of CYK is \Theta(n^3 \cdot |G|), where n is the length of the parsed string and |G| is the size of the CNF grammar G.


Standard form

The algorithm requires the context-free grammar to be rendered into Chomsky normal form (CNF), because it tests for possibilities to split the current sequence in half. CNF only allows rules of the forms A\rightarrow \alpha and A\rightarrow B C, and so such splitting is always possible for CNF.

As pseudocode

The algorithm in pseudocode is as follows:

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

[edit] 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 Rk. 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
NP V, VP Det. N P Det N
she eats a fish with a fork

[edit] Extensions

[edit] 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.

[edit] Parsing non-CNF context-free grammars

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. Using g to denote the size of the original grammar, the size blow-up in the worst case may range from g2 to 22g, 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 & Leiß 2009).

[edit] 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).

[edit] Valiant's algorithm

Using Landau symbols, the worst case running time of CYK is \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 algorithms for recognizing general context-free languages in practice. Valiant (1975) gave an extension of the CYK algorithm. His algorithm computes the same parsing table as the CYK algorithm; yet he showed that algorithms for efficient multiplication of matrices with 0-1-entries can be utilized for performing this computation.

Using the Coppersmith–Winograd algorithm for multiplying these matrices, this gives an asymptotic worst-case running time of O(n^{2.38} \cdot |G|). 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). The dependence on efficient matrix multiplication cannot be avoided altogether: Lee (2002) has proved that any parser for context-free grammars working in time O(n^{3-\varepsilon} \cdot |G|) can be effectively converted into an algorithm computing the product of (n \times n)-matrices with 0-1-entries in time O(n^{3 - \varepsilon/3}).

Full article ▸

related documents
Toeplitz matrix
Convolution theorem
Normal subgroup
Congruence relation
Hidden Markov model
Identity element
Box-Muller transform
Quaternion group
De Moivre's formula
Parity (mathematics)
Nash embedding theorem
List of logarithmic identities
Interior (topology)
Euphoria (programming language)
Uncountable set
Event (probability theory)
Bézout's theorem
Chomsky normal form
Inverse element
Linear congruential generator
Simple LR parser
Lagrange's theorem (group theory)