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**CYK )Cocke-Younger-Kasami) Parsing Algorithm**

دانشگاه صنعتی امیر کبیر دانشکده مهندسی کامپیوتر CYK )Cocke-Younger-Kasami) Parsing Algorithm سید محمد حسین معطر پردازش زبان طبیعی

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Parsing Algorithms CFGs are basis for describing (syntactic) structure of NL sentences Thus - Parsing Algorithms are core of NL analysis systems Recognition vs. Parsing: Recognition - deciding the membership in the language: Parsing – Recognition+ producing a parse tree for it Parsing is more “difficult” than recognition? (time complexity) Ambiguity - an input may have exponentially many parses

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**Parsing Algorithms Parsing General CFLs vs. Limited Forms Efficiency:**

Deterministic (LR) languages can be parsed in linear time A number of parsing algorithms for general CFLs require O(n3) time Asymptotically best parsing algorithm for general CFLs requires O(n2.37), but is not practical Utility - why parse general grammars and not just CNF? Grammar intended to reflect actual structure of language Conversion to CNF completely destroys the parse structure

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**CYK )Cocke-Younger-Kasami)**

One of the earliest recognition and parsing algorithms The standard version of CYK can only recognize languages defined by context-free grammars in Chomsky Normal Form (CNF). It is also possible to extend the CYK algorithm to handle some grammars which are not in CNF Harder to understand Based on a “dynamic programming” approach: Build solutions compositionally from sub-solutions Store sub-solutions and re-use them whenever necessary Uses the grammar directly (no PDA is used) Recognition version: decide whether S == > w ?

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CYK Algorithm The CYK algorithm for the membership problem is as follows: Let the input string be a sequence of n letters a1 ... an. Let the grammar contain r terminal and nonterminal symbols R1 ... Rr, and let R1 be the start symbol. 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 P[1,n,1] is true Then string is member of language Else string is not member of language

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**CYK Pseudocode On input x = x1x2 … xn :**

for (i = 1 to n) //create middle diagonal for (each var. A) if(Axi) add A to table[i-1][i] for (d = 2 to n) // d’th diagonal for (i = 0 to n-d) for (k = i+1 to i+d-1) for(each var. B in table[i][k]) for(each var. C in table[k][k+d]) if(ABC) add A to table[i][k+d] return Stable[0][n] ? ACCEPT : REJECT

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CYK Algorithm this algorithm considers every possible consecutive subsequence of the sequence of letters and sets P[i,j,k] to be true if the sequence of letters starting from i of length j can be generated from Rk. Once it has considered sequences of length 1, it goes on to sequences of length 2, and so on. For subsequences of length 2 and greater, it considers every possible partition of the subsequence into two halves, and checks to see if there is some production P -> Q R such that Q matches the first half and R matches the second half. 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 string is matched by the start symbol

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**CYK Algorithm for Deciding Context Free Languages**

Q: Consider the grammar G given by S e | AB | XB T AB | XB X AT A a B b Is x = aaabb in L(G ) Is x = aaabbb in L(G )

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**CYK Algorithm for Deciding Context Free Languages**

The algorithm is “bottom-up” in that we start with bottom of derivation tree. S e | AB | XB T AB | XB X AT A a B b a a a b b

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**CYK Algorithm for Deciding Context Free Languages**

1) Write variables for all length 1 substrings S e | AB | XB T AB | XB X AT A a B b a a a b b A A A B B

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**CYK Algorithm for Deciding Context Free Languages**

2) Write variables for all length 2 substrings S e | AB | XB T AB | XB X AT A a B b a a a b b A A A B B S,T T

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**CYK Algorithm for Deciding Context Free Languages**

3) Write variables for all length 3 substrings S e | AB | XB T AB | XB X AT A a B b a a a b b A A A B B S,T T X

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**CYK Algorithm for Deciding Context Free Languages**

4) Write variables for all length 4 substrings S e | AB | XB T AB | XB X AT A a B b a a a b b A A A B B S,T T X S,T

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**CYK Algorithm for Deciding Context Free Languages**

Write variables for all length 5 substrings. S e | AB | XB T AB | XB X AT A a B b REJECT! a a a b b A A A B B S,T T X S,T X

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**CYK Algorithm for Deciding Context Free Languages**

Now look at aaabbb : S e | AB | XB T AB | XB X AT A a B b a a a b b b

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**CYK Algorithm for Deciding Context Free Languages**

1) Write variables for all length 1 substrings. S e | AB | XB T AB | XB X AT A a B b a a a b b b A A A B B B

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**CYK Algorithm for Deciding Context Free Languages**

2) Write variables for all length 2 substrings. S e | AB | XB T AB | XB X AT A a B b a a a b b b A A A B B B S,T

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**CYK Algorithm for Deciding Context Free Languages**

3) Write variables for all length 3 substrings. S e | AB | XB T AB | XB X AT A a B b a a a b b b A A A B B B S,T T X

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**CYK Algorithm for Deciding Context Free Languages**

4) Write variables for all length 4 substrings. S e | AB | XB T AB | XB X AT A a B b a a a b b b A A A B B B S,T T X S,T

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**CYK Algorithm for Deciding Context Free Languages**

5) Write variables for all length 5 substrings. S e | AB | XB T AB | XB X AT A a B b a a a b b b A A A B B B S,T T X S,T X

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**CYK Algorithm for Deciding Context Free Languages**

6) Write variables for all length 6 substrings. S e | AB | XB T AB | XB X AT A a B b S is included so aaabbb accepted! a a a b b b A A A B B B S,T T X S,T X S,T

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**CYK Algorithm for Deciding Context Free Languages**

Can also use a table for same purpose. end at start at 1: aaabbb 2: aaabbb 3: aaabbb 4: aaabbb 5: aaabbb 6: aaabbb 0:aaabbb 1:aaabbb 2:aaabbb 3:aaabbb 4:aaabbb 5:aaabbb

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**CYK Algorithm for Deciding Context Free Languages**

1. Variables for length 1 substrings. end at start at 1: aaabbb 2: aaabbb 3: aaabbb 4: aaabbb 5: aaabbb 6: aaabbb 0:aaabbb A 1:aaabbb 2:aaabbb 3:aaabbb B 4:aaabbb 5:aaabbb

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**CYK Algorithm for Deciding Context Free Languages**

2. Variables for length 2 substrings. end at start at 1: aaabbb 2: aaabbb 3: aaabbb 4: aaabbb 5: aaabbb 6: aaabbb 0:aaabbb A - 1:aaabbb 2:aaabbb S,T 3:aaabbb B 4:aaabbb 5:aaabbb

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**CYK Algorithm for Deciding Context Free Languages**

3. Variables for length 3 substrings. end at start at 1: aaabbb 2: aaabbb 3: aaabbb 4: aaabbb 5: aaabbb 6: aaabbb 0:aaabbb A - 1:aaabbb X 2:aaabbb S,T 3:aaabbb B 4:aaabbb 5:aaabbb

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**CYK Algorithm for Deciding Context Free Languages**

4. Variables for length 4 substrings. end at start at 1: aaabbb 2: aaabbb 3: aaabbb 4: aaabbb 5: aaabbb 6: aaabbb 0:aaabbb A - 1:aaabbb X S,T 2:aaabbb 3:aaabbb B 4:aaabbb 5:aaabbb

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**CYK Algorithm for Deciding Context Free Languages**

5. Variables for length 5 substrings. end at start at 1: aaabbb 2: aaabbb 3: aaabbb 4: aaabbb 5: aaabbb 6: aaabbb 0:aaabbb A - X 1:aaabbb S,T 2:aaabbb 3:aaabbb B 4:aaabbb 5:aaabbb

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**CYK Algorithm for Deciding Context Free Languages**

6. Variables for aaabbb. ACCEPTED! end at start at 1: aaabbb 2: aaabbb 3: aaabbb 4: aaabbb 5: aaabbb 6: aaabbb 0:aaabbb A - X S,T 1:aaabbb 2:aaabbb 3:aaabbb B 4:aaabbb 5:aaabbb

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**Parsing results We keep the results for every wij in a table.**

Note that we only need to fill in entries up to the diagonal – the longest substring starting at i is of length n-i+1

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**Constructing parse tree**

we need to construct parse trees for string w: Idea: Keep back-pointers to the table entries that we combine At the end - reconstruct a parse from the back-pointers This allows us to find all parse trees

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**Ambiguity Efficient Representation of Ambiguities**

Local Ambiguity Packing : a Local Ambiguity - multiple ways to derive the same substring from a non-terminal All possible ways to derive each non-terminal are stored together When creating back-pointers, create a single back-pointer to the “packed” representation Allows to efficiently represent a very large number of ambiguities (even exponentially many) Unpacking - producing one or more of the packed parse trees by following the back-pointers.

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References Hopcroft and Ullman,“Intro. to Automata Theory, Lang. and Comp.”Section 6.3, pp “CYK algorithm ” , Wikipedia, the free encyclopedia A representation by Zeph Grunschlag

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