# Natural Language Processing

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1 Natural Language Processing Anoop Sarkar 9/18/18 1

2 Ambiguity An input is ambiguous with respect to a CFG if it can be derived with two different parse trees A parser needs a mechanical definition of ambiguity as it parses the input string Is a parser choice really ambiguous, i.e. does it lead to ambiguous parse trees? or not? We can formally define ambiguity in terms of the derivations possible in a CFG 9/18/18 2

3 Arithmetic xpressions + * ( ) - id 9/18/18 3

4 Leftmost derivations for id + id * id + * Þ + Þ id + ( ) Þ id + * + - id Þ id + id * Þ id + id * id id * id id 9/18/18 4

5 Leftmost derivations for id + id * id + Þ * * Þ + * ( ) Þ id + * * - id Þ id + id * Þ id + id * id + id id id 9/18/18 5

6 Rightmost derivation for id + id * id + Þ + * ( ) Þ + * Þ + * id + - id Þ + id * id Þ id + id * id id * id id 9/18/18 6

7 Rightmost derivation for id + id * id + Þ * * Þ * id ( ) Þ + * id * - id Þ + id * id Þ id + id * id + id id id 9/18/18 7

8 Ambiguity We can now define ambiguity for a contextfree parser If a parser has a choice of two different leftmost derivations, or if a parser has a choice of two different rightmost derivations, for a particular input then that input is ambiguous 9/18/18 8

9 CKY Recognition Algorithm The Cocke-Kasami-Younger algorithm As we shall see it runs in time that is polynomial in the size of the input It takes space polynomial in the size of the input Remarkable fact: it can find all possible parse trees (exponentially many) in polynomial time 9/18/18 9

10 Chomsky Normal Form Before we can see how CKY works, we need to convert the input CFG into Chomsky Normal Form CNF is one of many grammar transformations that preserve the language CNF means that the input CFG G is converted to a new CFG G in which all rules are of the form: A B C A a 9/18/18 10

11 psilon Removal First step, remove epsilon rules A B C C e C D a D b B b After e-removal: A B B C D B a BC C D C D D a D C D a D b B b 9/18/18 11

12 Removal of Chain Rules Second step, remove chain rules A B C C D C C D a D d B b After removal of chain rules: A B a B D a D a a D D D D a D D D D d B b 9/18/18 12

13 liminate terminals from RHS Third step, remove terminals from the rhs of rules A B a C d After removal of terminals from the rhs: A B N 1 C N 2 N 1 a N 2 d 9/18/18 13

14 Binarize RHS with Nonterminals Fourth step, convert the rhs of each rule to have two non-terminals A B N 1 C N 2 N 1 a N 2 d After converting to binary form: A B N 3 N 3 N 1 N 4 N 4 C N 2 N 1 a N 2 d 9/18/18 14

15 CKY algorithm We will consider the working of the algorithm on an example CFG and input string xample CFG: S A X Y B X A B B A A a B a Y B A xample input string: aaa 9/18/18 15

16 CKY Algorithm A, B X, Y A a X A B B A B a Y B A A, B A a B a S A (0,1) X (1,3) S Y (0,2) B (2,3) X, Y X A B B A Y B A A, B A a B a a a a 9/18/18 16 S

17 Parse trees S S S Y B A X A X B A A B B A a a a a a a a a a 9/18/18 17

18 CKY Algorithm Input string input of size n Create a 2D table chart of size n 2 for i=0 to n-1 chart[i][i+1] = A if there is a rule A a and input[i]=a for j=2 to N for i=j-2 downto 0 for k=i+1 to j-1 chart[i][j] = A if there is a rule A B C and chart[i][k] = B and chart[k][j] = C return yes if chart[0][n] has the start symbol else return no 9/18/18 18

19 CKY algorithm summary Parsing arbitrary CFGs For the CKY algorithm, the time complexity is O( G 2 n 3 ) The space requirement is O(n 2 ) The CKY algorithm handles arbitrary ambiguous CFGs All ambiguous choices are stored in the chart For compilers we consider parsing algorithms for CFGs that do not handle ambiguous grammars 9/18/18 19

20 Parsing - Summary Parsing arbitrary CFGs: O(n 3 ) time complexity Top-down vs. bottom-up Recursive-descent parsing Shift-reduce parsing arley parsing Ambiguous grammars result in parser output with multiple parse trees for a single input string 9/18/18 20

21 Parsing - Additional Results O(n 2 ) time complexity for linear grammars All rules are of the form S asb or S a Reason for O(n 2 ) bound is the linear grammar normal form: A ab, A Ba, A B, A a Left corner parsers extension of top-down parsing to arbitrary CFGs arley s parsing algorithm O(n 3 ) worst case time for arbitrary CFGs just like CKY O(n 2 ) worst case time for unambiguous CFGs O(n) for specific unambiguous grammars (e.g. S asa bsb e) 9/18/18 21

22 CKY algorithm for PCFGs We will consider the working of the algorithm on an example PCFG and input string xample PCFG: S A X (0.3) Y B (0.7) X A B (0.1) B A (0.9) Y B A (1.0) A a (1.0) B a (1.0) xample input string: aaa 9/18/18 22

23 Max(0.1, 0.9) CKY Algorithm A 1.0, B 1.0 A a 1.0 B a 1.0 X 0.9, Y 1.0 X A B 0.1 B A 0.9 Y B A 1.0 A 1.0, B 1.0 A a 1.0 B a 1.0 S 0.7 S A (0,1) X (1,3) 0.3 S Y (0,2) B (2,3) 0.7 X 0.9, Y 1.0 X A B 0.1 B A 0.9 Y B A 1.0 A 1.0, B 1.0 A a * 0.9 = 0.27 Max(0.27, 0.7) B a 1.0 a a a 9/18/18 23

24 Parse trees PCFG is consistent: = 1.0 S S S Y B A X A X B A A B B A a a a a a a a a a /18/18 24

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