Membership problem CYK Algorithm

Membership problem CYK Algorithm Project presentation CS 5800 Spring 2013 Professor : Dr. Elise de Doncker Presented by : Savithaparurvenkitachalam

Membership problem • To determine if the given string is a member of the language defined by a context free grammar. • Given a context-free grammar G and a string w • G = (V, ∑ ,P , S) where • V finite set of variables • ∑ (the alphabet) finite set of terminal symbols • P finite set of rules • S start symbol (distinguished element of V) • V and ∑ are assumed to be disjoint Is W in the language of G?

CYK Algorithm • Developed by J. Cocke D. Younger, T. Kasamito answer the membership problem • Input should be in Chomsky Normal form • A  BC • A  a • S  λ where B, C Є V – {S} • Uses bottom up parsing • Uses dynamic programming or table filling algorithm • Complexity - O(n3)

CYK basic Ideas • CYK works on two basic ideas 1 . Consider rules satisfying substrings of length from 1 to N Let the string to search be abca First consider substring of length 1 – a , b , c, a Next step length 2– ab , bc , ca Next step length 3– abc , bca Final length 4 – abca

CYK basic ideas 2. longer substrings can be parsed from parsing shorter ones Eg: abc can be split as a . bc or ab . c if we know rules to form a and bc (or ab and c) then we know the rules to form abc A substring can be given as Si,j = (Si, i , Si+1, j), (Si, i+1 , Si+2, j) … (Si, j-1 , Sj, j) i – start index and j- end index bcd can be formed from abcd as S2,4 = (S2,2 , S3,4) , (S2,3 , S4,4) = (b . cd) , (bc . d)

CYK table filling • Wi,j= (Wi, i , Wi+1, j), (Wi, i+1 , Wi+2, j) …… (Wi, j-1 , Wj, j) • Fill the table with the rules satisfying the substrings • If the final box contains the start symbol then the string is a member of the language W1 W2 W3 W4

Table filling example Search string ‘cbba’ c b b a

To fill the next row of the table consider Wi,j= (Wi, i , Wi+1, j), (Wi, i+1 , Wi+2, j) …… (Wi, j-1 , Wj, j) W 1,2 = (W1,1 , W2,2) = {A,C} {B} = {AB , CB} Rules to form AB or CB = {S, C} W 2,3= (W2,2 , W3,3) = {B} {B} = {B B} Rules to form BB =∅ W 3,4= (W3,3 , W4,4) = {B} {A} = {B A} Rules to form BA = {C }

Table :

Wi,j= (Wi, i , Wi+1, j), (Wi, i+1 , Wi+2, j) …… (Wi, j-1 , Wj, j) W1,3 = (W1,1, W2, 3), (W1, 2, W3, 3) = {A,C}U{S,C} {B} = { A , C , SB , CB} Rules to form A or C or SB or CB = {C} W2,4 = (W2,2, W3, 4), (W2, 3, W4, 4) = {B} {C}U{A} = { BC, A} Rules to form BC or A = {B}

Table :

Wi,j = (Wi, i , Wi+1, j), (Wi, i+1 , Wi+2, j) …… (Wi, j-1 , Wj, j) W1,4 = (W1, 1, W2, 4), (W1, 2, W3, 4) ,(W1, 3, W4, 4) = {A,C} {B} U {S,C} {C} U {C} {A} = { AB, CB , SC , CC , CA} Rules to form AB or CB or SC or CC or CA = {S,C,A}

Final Table : The first cell represents the original string and contains the start symbol ‘S’ . Result : ‘cbba’ is a member of the language.

Design • Read the input grammar from a file or prompt user to input the rules • Check if the grammar is in CNF • If grammar is in CNF, start filling the table • Output : ‘String is a member of the input grammar’ Or ‘String is not a member of the input grammar’

References • References • http://en.wikipedia.org/wiki/CYK_algorithm#Algorithm • http://www.cs.ucdavis.edu/~rogaway/classes/120/winter12/CYK.pdf • Languages and Machines, An Introduction to the Theory of Computer Science - Thomas A. Sudkamp • “Parsing” Internet: http://qntm.org/top • http://en.wikipedia.org/wiki/Parsing • http://en.wikipedia.org/wiki/Dynamic_programming • http://en.wikipedia.org/wiki/Bottom-up_parsing

Questions

Membership problem CYK Algorithm