Definition: two derivations are similar if one of them precedes the other.

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1 Parse Trees and Ambiguity (Chapter 3, ection 3.2) Cmc 365 Theory of Computation 1. Derivations and similarity Let G be a CFG. A string w L(G) may have many derivations, corresponding to how we choose the rules to apply and how we choose which symbol to expand. At a given step we may be able to expand two or more non-terminal symbols. The order in which we expand the non-terminal symbols will determine the derivation, i.e. different orders will result in different derivations, even if we choose same rules. At a given step for a given non-terminal symbol there may be several rules to choose. Choosing different rules will result in different derivations. ome derivations are inherently different, others are not. To distinguish between these two cases, we will define a precedence relation on the set of the derivations of a given string, and similarity between derivations. Definition: Let D and D be two derivations of some string in L(G) D = x 1 x 2 x n D = x 1 x 2 x n where x i, x i V* for i = 1, 2,, n x 1, x 1 V -, x n, x n * We say that D precedes D, written as D < D, if N > 2 and there is an integer 1 < k < n, uch that: a) x i = x i for i k b) x k-1 = x k-1 = uavbw, where u, v, w V*, A, B V - c) x k = uyvbw, where A y R d) x k = uavzw, where B z R e) x k+1 = x k+1 = uyvzw The difference is in the order of expanding non-terminal symbols. Definition: two derivations are similar if one of them precedes the other. The similarity relation is reflexive, symmetric and transitive, therefore it defines classes of equivalence. Each derivation can be depicted using a derivation tree, also called a parse tree. Example: Consider the string (()()) and the grammar with rules given below: 1

2 Rule1: Rule2: () Rule3: ( ) One possible derivation is: () () (( )) by Rule1 by Rule2 by Rule3 (( )( )) by Rule3 The process of derivation is depicted in the following parse tree Rule 1 ( ) Rule 2 Rule 3 ( ) ( ) Each non-terminal symbol is expanded by applying a grammar rule that contains the symbol in its left- hand side. Its children are the symbols in the right-hand side of the rule. Note: The order of applying the rules depends on the symbols to be expanded. At each tree level we may apply several different rules corresponding to the nodes to be expanded. Derivations within the same class of similarity yield the same parse tree. 2

3 Within a given class, we have a derivation that is not preceded by any other derivation. This is the left-most derivation. There is a derivation is preceded by other derivations in a given class. This is the right-most derivation. Thus we can use either left-most or right-most derivation when building the parse tree. 2. Parse Trees A parse tree for a string in L(G) is a tree where the root is the start symbol for G the interior nodes are the nonterminals of G the leaf nodes are the terminal symbols of G. the children of a node T (from left to right) correspond to the symbols on the right hand side of some rule for T in G. Every terminal string generated by a grammar has a corresponding parse tree; every valid parse tree represents a string generated by the grammar (called the yield of the parse tree). Example: Given the grammar G = (V,, R, E), Where V = { E, D, 1,2,3,4,5,6,7,8,9,0,+,-,*,/,(,)}, = { 1,2,3,4,5,6,7,8,9,0,+,-,*,/,(,)}, and R contains the following rules: 1. E D 2. E ( E ) 3. E E + E 4. E E - E 5. E E * E 6. E E / E 7. D find a parse tree for the string * 3: The parse tree is: E E \ / \ \ \ E E E D D D * 3 (from 3

4 3. Formal definition of a parse tree 1. The parse tree for a terminal symbol a is just one node - it is both a root and a leaf. 2. If there is a rule A e. its parse tree has a root A and a leaf e. 3. If T1, Tn are parse trees with roots A1,, An, and there is a rule: A A1,..An, then the tree with a root A, and subtrees T1, T2, Tn rooted at the children of A: A1, A2,.,An, is a parse tree. 4. Nothing else is a parse tree. 4. Ambiguity Definition: A grammar G is ambiguous if there is a string L(G) with two different left-most (right-most) derivations. Derivations, where we apply different rules at same point, have different parse trees. If the grammar yields more than one parse tree for a given string, the grammar is called ambiguous, and the corresponding language - ambiguous language. Example: Let denote a statement, Exp - an expression. Consider the following rules for if statements (the words if, then, else are terminal symbols):: Rule1: IF_statement if Exp then else Rule2: IF_statement if Exp then Consider now the statement if a < b then if c < y then write(yes) else write(no); Following the grammar rules above, there are two possible interpretations: If a < b then if c < y then write(yes) else write(no); 4

5 5

6 econd interpretation: If a < b then if c < y then write(yes) else write(no); 6

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