Cyk Algorithmus Beispiel Essay

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カテゴリー: 子育て, 引越 | 投稿日: | 投稿者: bactereal

今の家に引越して次の冬で一年になる 引越し当時の記事 さすがにもう落ち着いたから、引越した理由を書く。 前の家は 木造3階建てで1階部分は車庫の札幌ではよく見かけるタイプ。 2階部分にトラブった老夫婦が住み、自分らはその ・・・続きを読む

カテゴリー: 金, 家, 引越 | 投稿日: | 投稿者: bactereal

少し前にも書いた が、家賃が2万あがったんだが、 ちょうど同じ時期に給料も2万あがったから良い。 とも言えず、前までがマイナスだったのでこの給料アップでやっと0、少し余裕出るかな、と思ってた位なので家賃2万アップはマイナ ・・・続きを読む

カテゴリー: 車, 引越 | 投稿日: | 投稿者: bactereal

引っ越して2ヶ月程 少し前にやっと全部の手続きが終わった。 長かった。 学んだのは、 早く手続しなきゃ、とか思って頑張ると疲れて生活が厳しくなるという事。 日々の生活が1番大時なので引越に伴う手続はマイペースに仕事や家事 ・・・続きを読む

カテゴリー: 金, 家, 引越 | 投稿日: | 投稿者: bactereal

旧居の敷金が戻ってきた事 を書いたけど、新居のその辺の事が少しモヤモヤ。 更新料というものを初めて知った 本州では割と一般的らしい。 しかも家賃分とられたりして負担になってるっぽい。 ウチは幸い?更新料は1万。 それ込み ・・・続きを読む

カテゴリー: 金, 引越 | 投稿日: | 投稿者: bactereal

敷金は戻ってくるもんだと思ってる 以前は違ったけど。 前のそのまた前の家の退去時は 色々あって、 少し焦り、でも相殺だけで済んだからまだよかったかな、と。 調べるきっかけにもなったし。 前の家ではその経験を生かし布団を上 ・・・続きを読む

カテゴリー: 車, 引越 | 投稿日: | 投稿者: bactereal

今回の引越の住所変更で NHK 車 はすんなりいかないと思ってたがやはり。 でも意外にNHKはすんなり ただの住所変更だけじゃなく衛生契約→地上契約に変更もするからてこずると思ってたが、 あとは口座に多く払った金が無事に ・・・続きを読む

カテゴリー: 金, 雑記, 引越 | 投稿日: | 投稿者: bactereal

引越たのでその事で少し。 家賃が2万上がる。 でも給料も2万上がってる。 給料上がる前はマイナスで、上がってやっと少し余裕できるかなと思ってたから残念、もっと頑張らねば。 家賃以外にも保育園が遠くなったのでガソリン代、部 ・・・続きを読む

カテゴリー: PC・ネット・WEB関係, 金, 引越 | 投稿日: | 投稿者: bactereal

前の家ではjcomは入ってたんだが、 速度が不安でやめた。 WiMAX少し使ってみたかったが不安定との事を聞きやめた。 携帯がauなのでauひかりにしてみたかったが対応エリアじゃない。 結局やっぱNTT有線に。マンション ・・・続きを読む

カテゴリー: 引越 | 投稿日: | 投稿者: bactereal

ウチの洗濯機が少しやっかいでして、 ドラム式洗濯機が出始めた頃のビッグドラム。 結構、大きい。 引越が困る。 まず部屋探し。 スペースに入らない。 入っても、通路が狭い。 今回の引越、 一番の目的は音だったんですが、一応 ・・・続きを読む

カテゴリー: 家, 引越 | 投稿日: | 投稿者: bactereal

入居時に入居チェック表なるものを管理会社からもらう。 退去時にトラブルにならない為のもの。 少し細かい所まで書いた。 早めに直してもらいたい部分も知らせた。 最低限。 ・玄関のストッパー破損 ・ドアがちゃんと閉まらない/ ・・・続きを読む

In computer science, the Cocke–Younger–Kasami algorithm (alternatively called CYK, or CKY) is a parsingalgorithm for context-free grammars, named after its inventors, John Cocke, Daniel Younger and Tadao Kasami. 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). However any context-free grammar may be transformed to a CNF grammar expressing the same language (Sipser 1997).

The importance of the CYK algorithm stems from its high efficiency in certain situations. Using Landau symbols, the worst case running time of CYK is Ο, where n is the length of the parsed string and |G| is the size of the CNF grammar G (Hopcroft & Ullman 1979, p. 140). This makes it one of the most efficient parsing algorithms in terms of worst-case asymptotic complexity, although other algorithms exist with better average running time in many practical scenarios.

Standard form[edit]

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

Algorithm[edit]

As pseudocode[edit]

The algorithm in pseudocode is as follows:

let the input be a string I consisting of n characters: a1 ... an. let the grammar contain r nonterminal symbols R1 ... Rr, with start symbol R1. letP[n,n,r] be an array of booleans. Initialize all elements of P to false. for eachs = 1 to nfor each unit production Rv -> as set P[1,s,v] = true for eachl = 2 to n-- Length of spanfor eachs = 1 to n-l+1 -- Start of spanfor eachp = 1 to l-1 -- Partition of spanfor each production Ra -> RbRcifP[p,s,b] and P[l-p,s+p,c] then set P[l,s,a] = true ifP[n,1,1] is true thenI is member of language elseI is not member of language
Probabilistic CYK (for finding the most probable parse) Allows to recover the most probable parse given the probabilities of all productions.
let the input be a string I consisting of n characters: a1 ... an. let the grammar contain r nonterminal symbols R1 ... Rr, with start symbol R1. letP[n,n,r] be an array of real numbers. Initialize all elements of P to zero. letback[n,n,r] be an array of backpointing triples. for eachs = 1 to nfor each unit production Rv -> as set P[1,s,v] = Pr(Rv -> as) for eachl = 2 to n-- Length of spanfor eachs = 1 to n-l+1 -- Start of spanfor eachp = 1 to l-1 -- Partition of spanfor each production Ra -> RbRc prob_splitting = Pr(Ra -> RbRc) * P[p,s,b] * P[l-p,s+p,c] ifP[p,s,b] > 0 and P[l-p,s+p,c] > 0 and P[l,s,a] < prob_splitting then set P[l,s,a] = prob_splitting set back[l,s,a] = <p,b,c>

As prose[edit]

In informal terms, this algorithm considers every possible substring of the input string and sets to be true if the substring of length starting from can be generated from nonterminal variable . 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 such that matches the first part and matches the second part. If so, it records as matching the whole substring. Once this process is completed, the sentence is recognized by the grammar if the substring containing the entire input string is matched by the start symbol.

Example[edit]

This is an example grammar:

Now the sentence she eats a fish with a fork is analyzed using the CYK algorithm. In the following table, in , i is the number of the row (starting at the bottom at 1), and j is the number of the column (starting at the left at 1).

S
VP
 
S
VPPP
SNPNP
NPV, VPDet.NPDetN
sheeatsafishwithafork

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 if, and only if, . In the above example, since a start symbol S is in , the sentence can be generated by the grammar.

Extensions[edit]

Generating a parse tree[edit]

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 is sometimes done with a second table B[n,n,r] of so-called backpointers. 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 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).

Parsing non-CNF context-free grammars[edit]

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. 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 to denote the size of the original grammar, the size blow-up in the worst case may range from to , 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).

Parsing weighted context-free grammars[edit]

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).

Valiant's algorithm[edit]

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. 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 . 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 -matrices with 0-1-entries in time .

See also[edit]

References[edit]

  • 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. 
  • 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. doi:10.1145/505241.505242. 
  • 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 time n3". Inform. Control. 10 (2): 189–208. doi:10.1016/s0019-9958(67)80007-x. 

External links[edit]

Sentence parsing using the CYK algorithm
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