1. Lecture # 9 Chap 4: Ambiguous Grammar

2. 2 Chomsky Hierarchy: Language Classification A grammar G is said to be – Regular if it is right linear where each production is of the form A  w Bor A  w or left linear where each production is of the form A  B wor A  w – Context free if each production is of the form A   where A  N and   (N  T)* – Context sensitive if each production is of the form  A      where A  N, , ,   (N  T)*, |  | > 0 – Unrestricted

3. 3 Chomsky Hierarchy L (regular)  L (context free)  L (context sensitive)  L (unrestricted) Where L (T) = { L(G) | G is of type T } That is: the set of all languages generated by grammars G of type T L 1 = { a n b n | n  1 } is context free L 2 = { a n b n c n | n  1 } is context sensitive Every finite language is regular! (construct a FSA for strings in L(G)) Examples:

4. 4 Parse Trees The root of the tree is labeled by the start symbol Each leaf of the tree is labeled by a terminal (=token) or  Each interior node is labeled by a nonterminal If A  X 1 X 2 … X n is a production, then node A has immediate children X 1, X 2, …, X n where X i is a (non)terminal or  (  denotes the empty string)

5. 5 Parse Tree for the Example Grammar Parse tree of the string 9-5+2 using grammar G list digit 9-5+2 list digit The sequence of leafs is called the yield of the parse tree

6. Example of Parse Tree Suppose we have the following grammar E → E + E E → E * E E → ( E ) E → - E E → id Perform Left most derivation, right most derivation and construct a parse tree for the string id+id*id

7. Two possible Parse Trees using Leftmost derivation

8. Parse Tree via Right most derivation

9. Ambiguity Grammar is ambiguous if more than one parse tree is possible for some string as shown in the previous example. If there are more than one left most derivations or more than one right most derivations. Ambiguity is not acceptable – Unfortunately, it’s undecidable to check whether a given CFG is ambiguous – Some CFLs are inherently ambiguous (do not have an unambiguous CFG)

10. 10 Ambiguity (cont’d) string  string + string | string - string | 0 | 1 | … | 9 G = with production P = Consider the following context-free grammar: This grammar is ambiguous, because more than one parse tree represents the string 9-5+2

11. 11 Two Parse Trees for the same string string 9-5+2 9-5+2

12. Practice Show that the following grammar is ambiguous: (Find out strings and two parse trees) 1) S  AB | aaB 2) S  a | abSb |aAb A  a | Aa A  bS | aAAb B  b 3) S  aSb | SS | ε

13. How to eliminate Ambiguity? If ambiguity is of the form: S  α S β S  | α1 |……| αn Rewrite: S  α S β S’  | S’ S’  α1 |……| αn Make the following grammar unambiguous: E  E+E | E*E | (E) | id

14. Eliminate Left Recursion If S  S α for some α then there exists left recursion If there are productions: S  S α | β Rewrite: S  β S’ S’  α S’ | 

15. Example Remove Left Recursion: S  A A  S d | b