Cocke–Younger–Kasami algorithm (CYK)

Class	Parsing withcontext-free grammars
Data structure	String
Worst-caseperformance	${\displaystyle \mathcal{O}\left(n^3\cdot \left|G\right|\right)}$, where: ${\displaystyle n}$ is length of the string ${\displaystyle |G|}$ is the size of the CNF grammar

Incomputer science, theCocke–Younger–Kasami algorithm (alternatively calledCYK, orCKY) is aparsingalgorithm forcontext-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, andJacob T. Schwartz. It employsbottom-up parsing anddynamic programming.

The standard version of CYK operates only on context-free grammars given inChomsky normal form (CNF). However any context-free grammar may be algorithmically transformed into a CNF grammar expressing the same language (Sipser 1997).

The importance of the CYK algorithm stems from its high efficiency in certain situations. UsingbigO notation, theworst case running time of CYK is${\displaystyle \mathcal{O}\left(n^3\cdot \left|G\right|\right)}$, where${\displaystyle n}$ is the length of the parsed string and${\displaystyle \left|G\right|}$ is the size of the CNF grammar${\displaystyle G}$ (Hopcroft & Ullman 1979, p. 140). This makes it one of the most efficient^{[citation needed]} parsing algorithms in terms of worst-caseasymptotic complexity, although other algorithms exist with better average running time in many practical scenarios.

Thedynamic programming algorithm requires the context-free grammar to be rendered intoChomsky 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 onlyproduction rules of the forms${\displaystyle A\to \alpha }$ and${\displaystyle A\to BC}$; to allow for the empty string, one can explicitly allow${\displaystyle S\to \epsilon }$, where${\displaystyle S}$ is the start symbol.^[3]

The algorithm inpseudocode is as follows:

let the input be a stringI consisting ofn characters:a1 ...an.let the grammar containr nonterminal symbolsR1 ...Rr, with start symbolR1.letP[n,n,r] be an array of booleans. Initialize all elements ofP to false.letback[n,n,r] be an array of lists of backpointing triples. Initialize all elements ofback to the empty list.for eachs = 1 tonfor each unit productionRv →assetP[1,s,v] = truefor eachl = 2 ton-- Length of spanfor eachs = 1 ton-l+1-- Start of spanfor eachp = 1 tol-1-- Partition of spanfor each productionRa    →RbRcifP[p,s,b] andP[l-p,s+p,c]thensetP[l,s,a] = true,                     append <p,b,c> toback[l,s,a]ifP[n,1,1] is truethenI is member of languagereturnback -- byretracing the steps through back, one can easily construct all possible parse trees of the string.elsereturn "not a member of language"

let the input be a stringI consisting ofn characters:a1 ...an.let the grammar containr nonterminal symbolsR1 ...Rr, with start symbolR1.letP[n,n,r] be an array of real numbers. Initialize all elements ofP to zero.letback[n,n,r] be an array of backpointing triples.for eachs = 1 tonfor each unit productionRv →assetP[1,s,v] = Pr(Rv →as)for eachl = 2 ton-- Length of spanfor eachs = 1 ton-l+1-- Start of spanfor eachp = 1 tol-1-- Partition of spanfor each productionRa →RbRc        prob_splitting = Pr(Ra →RbRc) *P[p,s,b] *P[l-p,s+p,c]if prob_splitting >P[l,s,a]thensetP[l,s,a] = prob_splittingsetback[l,s,a] = <p,b,c>ifP[n,1,1] > 0then    find the parse tree by retracing throughbackreturn the parse treeelsereturn "not a member of language"

In informal terms, this algorithm considers every possible substring of the input string and sets${\displaystyle P[l,s,v]}$ to be true if the substring of length${\displaystyle l}$ starting from${\displaystyle s}$ can be generated from the nonterminal${\displaystyle R_v}$. 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 parts, and checks to see if there is some production${\displaystyle A\to BC}$ such that${\displaystyle B}$ matches the first part and${\displaystyle C}$ matches the second part. If so, it records${\displaystyle A}$ as matching the whole substring. 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 sentenceshe eats a fish with a fork is analyzed using the CYK algorithm. In the following table, in${\displaystyle P[i,j,k]}$,i is the number of the row (starting at the bottom at 1), andj is the number of the column (starting at the left at 1).

For readability, the CYK table forP is represented here as a 2-dimensional matrixM containing a set of non-terminal symbols, such thatR_k is in⁠${\displaystyle M[i,j]}$⁠ if, and only if,⁠${\displaystyle P[i,j,k]}$⁠.In the above example, since a start symbolS is in⁠${\displaystyle M[7,1]}$⁠, the sentence can be generated by the grammar.

The above algorithm is arecognizer that will only determine if a sentence is in the language. It is simple to extend it into aparser that also constructs aparse 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 is sometimes done with a second table B[n,n,r] of so-calledbackpointers.The end result is then a shared-forest of possible parse trees, where common trees parts are factored between the various parses. This shared forest can conveniently be read as anambiguous 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 byLang (1994).

As pointed out byLange & Leiß (2009), the drawback of all known transformations into Chomsky normal form is that they can lead to an undesirable bloat in grammar size. The size of a grammar is the sum of the sizes of its production rules, where the size of a rule is one plus the length of its right-hand side. Using${\displaystyle g}$ to denote the size of the original grammar, the size blow-up in the worst case may range from${\displaystyle g^2}$ to${\displaystyle 2^{2g}}$, depending on the transformation algorithm used. 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 usingweighted andstochastic 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).

When the probabilistic CYK algorithm is applied to a long string, the splitting probability can become very small due to multiplying many probabilities together. This can be dealt with by summing log-probability instead of multiplying probabilities.

Theworst case running time of CYK is${\displaystyle \mathrm{\Theta }(n^3\cdot |G|)}$, wheren is the length of the parsed string and |G| is the size of the CNF grammarG. 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 tableas the CYK algorithm; yet he showed thatalgorithms for efficient multiplication ofmatrices with 0-1-entries can be utilized for performing this computation.

Using theCoppersmith–Winograd algorithm for multiplying these matrices, this gives an asymptotic worst-case running time of${\displaystyle O(n^{2.38}\cdot |G|)}$. However, the constant term hidden by theBig 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${\displaystyle O(n^{3-\epsilon }\cdot |G|)}$ can be effectively converted into an algorithm computing the product of${\displaystyle (n\times n)}$-matrices with 0-1-entries in time${\displaystyle O(n^{3-\epsilon /3})}$, and this was extended by Abboud et al.^[4] to apply to a constant-size grammar.

• GLR parser
• Earley parser
• Packrat parser
• Inside–outside algorithm

• Sakai, Itiroo (1962).Syntax in universal translation. 1961 International Conference on Machine Translation of Languages and Applied Language Analysis, Teddington, England. Vol. II. London: Her Majesty’s Stationery Office. pp. 593–608.
• Cocke, John; Schwartz, Jacob T. (April 1970).Programming languages and their compilers: Preliminary notes(PDF) (Technical report) (2nd revised ed.).CIMS,NYU.
• Hopcroft, John E.;Ullman, Jeffrey D. (1979).Introduction to Automata Theory, Languages, and Computation. Reading/MA: Addison-Wesley.ISBN 0-201-02988-X.
• Kasami, T. (1965).An efficient recognition and syntax-analysis algorithm for context-free languages (Technical report).AFCRL. 65-758.
• Knuth, Donald E. (November 14, 1997).The Art of Computer Programming Volume 2: Seminumerical Algorithms (3rd ed.). Addison-Wesley Professional. p. 501.ISBN 0-201-89684-2.
• Lang, Bernard (1994). "Recognition can be harder than parsing".Comput. Intell.10 (4):486–494.CiteSeerX 10.1.1.50.6982.doi:10.1111/j.1467-8640.1994.tb00011.x.S2CID 5873640.
• Lange, Martin; Leiß, Hans (2009)."To CNF or not to CNF? An Efficient Yet Presentable Version of the CYK Algorithm".Informatica Didactica.8.
• Lee, Lillian (2002). "Fast context-free grammar parsing requires fast Boolean matrix multiplication".J. ACM.49 (1):1–15.arXiv:cs/0112018.doi:10.1145/505241.505242.S2CID 1243491.
• Sipser, Michael (1997).Introduction to the Theory of Computation (1st ed.). IPS. p. 99.ISBN 0-534-94728-X.
• Valiant, Leslie G. (1975)."General context-free recognition in less than cubic time".J. Comput. Syst. Sci.10 (2):308–314.doi:10.1016/s0022-0000(75)80046-8.
• Younger, Daniel H. (February 1967)."Recognition and parsing of context-free languages in timen³".Inform. Control.10 (2):189–208.doi:10.1016/s0019-9958(67)80007-x.

• Interactive Visualization of the CYK algorithm
• CYK parsing demo in JavaScript
• Exorciser is a Java application to generate exercises in the CYK algorithm as well as Finite State Machines, Markov algorithms etc

• Parsing algorithms

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