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Unformatted text preview: Chapter 4 Normal Forms for CFGs 2 4.5 Chomsky Normal Form Defn 4.4.1 A CFG G = ( V , , P , S ) is in chomsky normal form if each rule in G has one of the following forms: i) A BC ii) A a iii) S where A , B , C V and B , C V  { S } and a A simplified normal form which restricts the length & composition of the R.H.S. of a rule in CFG The derivation tree for a string generated by a CFG in chomsky normal form is a binary tree 3 4.5 Chomsky Normal Form Theorem 4.4.2 Let G = ( V , , P , S ) be a CFG. There is an algorithm to construct a grammar G = ( V , , P , S ) in chomsky normal form that is equivalent to G Proof (sketch) : (i) For each rule A w , where  w  > 1, replace each terminal a in w by a distinct variable Y & create new rule Y a (ii) For each modified rule X w , w is either a terminal or a string in V + . Rules in the latter form must be broken into a sequence of rules, each of whose R.H.S. consists of two variables. Example 4.4.1 One of the applications of using CFGs that are in Chomsky Normal Form Constructing binary search trees to accomplish optimal time & space search complexity for parsing an input string 4 4.1 Grammar Transformations Lemma 4.1.1 Let G = ( V , , P , S ) be a CFG. There is a CFG G = ( V , , P , S ) that satisfies i) L( G ) = L( G ) ii) Rules in P are of the form A w where A V and w (( V { S }) )*. Proof . If S is a recursive variable , then construct G by creating a new start symbol S & adding S S to P , i.e., G = ( V { S } , , P { S S } , S ). If S u , then S S u , where u *. Example. 4.1.1 Assume that P in G includes S a S  AB  AC , then P in G should include S S, S a S  AB  AC G * G G * 5 4.2 Elimination of rules Nullable variables are variables that can derive ....
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 Winter '12
 DennisNg

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