Predictive Parsing

1. Predictive Parsing Lecture 9 Wed, Feb 9, 2005

2. Predictive Parsing Table • The parse table has • one row for each nonterminal, • one column for each terminal and $. • $ is the end-of-file marker. • Each cell in the table represents a combination (A, a) of a nonterminal A and a terminal a.

3. Nullability and FIRST for Strings • We need to extend the definitions of nullable and FIRST to strings of grammar symbols. •  is nullable. • Let  = X1X2…Xn be a string of grammar symbols. •  is nullable if each Xi is nullable.

4. Nullability and FIRST for Strings • By definition, FIRST() is the empty set. • For the string  = X1X2…Xn , let k be the largest integer for which X1, X2, …, and Xk are nullable. • Then FIRST() = FIRST(X1)  …  FIRST(Xk + 1). • We will not need FOLLOW for strings.

5. Parse Table Entries • Each table entry is a production A . • Rules for entering A  in the table are • Write A  in the (A, a) cell, for every a  FIRST(). • If  is nullable, then write A  in the (A, a) cell for every symbol a in FOLLOW(A). • Write “error” in all cells that do not contain a production.

6. Parse Table Entries • The interpretation of A  in cell (A, a) is that if A is on top of the stack and we encounter a in the input, then we should replace A by  on the stack. • Push the symbols of  onto the stack in reverse order, from right to left, so that the first (leftmost) symbol is on the top of the stack.

7. Example: Parse Table • Let the grammar be ET E' E'+T E' | . TF T' T'*F T' | . F(E) | id | num

8. Recall

9. Example: Parse Table • Consider the production ET E'. • FIRST(TE') = FIRST(T) = {(, num, id}. • Therefore, enter ET E' in cells (E, ( ), (E, num), and (E, id). • T E' is not nullable.

10. Example: Parse Table • Consider the production E'. • FIRST() = { }. •  is nullable and FOLLOW(E') = {$, )}. • Therefore, enter E' in cells (E', $) and (E', ) ).

11. Example: Parse Table • Handle the other productions similarly. • In which cells do we enter E'+T E' ? • In which cells do we enter TF T' ? • In which cells do we enter T'*F T' ? • In which cells do we enter T'? • In which cells do we enter F(E) ? • In which cells do we enter Fid ? • In which cells do we enter Fnum ?

12. Example: Parse Table • The parse table:

13. Predictive Parsing • A grammar is called LL(1) if its predictive parse table does not contain any multiple entries. • A multiple entry would indicate that we couldn’t decide which production to apply.

14. Predictive Parsing Algorithm • The predictive parsing algorithm uses • The parse table, • An input buffer containing a sequence of tokens, • A stack of grammar symbols. • Initially • The input buffer contains the input followed by $. • The stack contains $ and S, with S on the top.

15. Predictive Parsing Algorithm • Consider the top stack symbol X. • There are three possibilities. • X is a terminal. • X is a nonterminal. • X is $.

16. Predictive Parsing Algorithm • If X is a terminal, then • If X matches the current token, • Pop X from the stack. • Advance to the next token. • If X does not match the current token, then that is an error.

17. Predictive Parsing Algorithm • If X is a nonterminal, then • Use X together with the current token to get the entry from the parse table. • If the entry is a production, • Pop X from the stack. • Push the symbols on the right-hand side of the production, from right to left, onto the stack. • If the entry is not a production, then that is an error.

18. Predictive Parsing Algorithm • If X is $, then • If the current token is also $, • Accept the input. • If not, then that is an error.

19. Example: Predictive Parsing • Parse the string (id + num)*id.

20. Example: Predictive Parsing

21. Example: Predictive Parsing

22. Exercise • The grammar RRR | RR | R* | (R) | a | b generates all regular expressions on the alphabet {a, b}. • Using FIRST and FOLLOW from the exercise of the previous lecture, construct a parse table for this grammar. • Parse the expression ab*(a | ).