LL(1) Grammar Parsing Methods and Techniques

1. Chapter 3 Chang Chi-Chung 2015.6.8

2. LL(1) Grammar • Predictive parsers, that is, recursive-descent parsers needing no backtracking, can be constructed for a class of grammars called LL(1) • First “L” means the input from left to right. • Second “L” means leftmost derivation. • “1” for using one input symbol of lookahead at each step tp make parsing action decisions. • No left-recursive. • No ambiguous.

3. LL(1) 文法 • 明確性文法 (No Ambiguity) • 不可以有左遞迴 (No Left Recursion) • 不可以有左因子 (No Left Factor)

4. E E E T T T T T T + id + id + id Top-Down Parsing • LL methods and recursive-descent parsing • Left-to-right, Leftmost derivation • Creating the nodes of the parse tree in preorder ( depth-first ) GrammarE  T+TT  (E)T  -ET  id Leftmost derivationE lmT + T lmid+T lm id + id E

5. Top-down Parsing • Give a Grammar G E → TE’ E’ → + TE’ | ε T → FT’ T’ → * F T’ | ε F → ( E ) | id E  E + T | TT  T * F | F F  ( E ) | id

6. Elimination of Left Recursion • Productions of the formA A | are left recursive • Non-left-recursions A  A’ A’   A’ | ε • When one of the productions in a grammar is left recursive then a predictive parser loops forever on certain inputs

7. Immediate Left-Recursion Elimination • Group the Productions as AA1 | A2 | … | Am| 1 | 2 | … | n Whereno i begins with an A • Replace the A-Productions by A 1A’ | 2 A’| … | n A’ A’  1A’ | 2 A’ | … | mA’ | ε

8. Example • Left-recursive grammar A A  |  |  | A  • Into a right-recursive production A  AR|  AR AR AR|  AR| 

9. Non-Immediate Left-Recursion • The Grammar S A a | b A  A c |S d | ε • The nonterminal S is left recursive, because S A a  Sda But S is not immediately left recursive.

10. Elimination of Left Recursion • Eliminating left recursion algorithm Arrange the nonterminals in some order A1, A2, …, Anfor (each i from 1 to n) {for (each j from 1 to i-1){ replace each productionAi Aj withAi 1 | 2 | … | k whereAj 1 | 2 | … | k } eliminate the immediate left recursion in Ai}

11. Example A BC | aB C A | AbC A B | C C | a

12. Exercise • The grammar S A a | b A  A c |S d | ε • Answer • A A c | A a d | b d | 

13. Left Factoring • Left Factoring is a grammar transformation. • Predictive Parsing • Top-down Parsing • Replace productionsA   1|  2| … |  n| withA  AR| AR 1| 2| … | n

14. Example • The Grammar stmt  ifexpr then stmt | if expr then stmtelse stmt • Replace with stmt ifexpr then stmt stmts stmts  else stmt | ε

15. Exercise • The following grammar S iEtS | iEtSeS | a E  b • Answer S iEtSS’ | a S’ e S | ε E  b

16. LL(1)

17. LL(1) • A grammar G is LL(1) if it is not left recursive and for each collection of productionsA  1 |2 |… | nfor nonterminal A the following holds: • FIRST(i)  FIRST(j) =  for all i  j如果交集不是空集合，會如何？ • if i *  then • j *  for all i  j • FIRST(j)  FOLLOW(A) =  for all i  j

18. Example

19. Top-down Parsing • Give a Grammar G E → TE’ E’ → + TE’ | ε T → FT’ T’ → * F T’ | ε F → ( E ) | id E  E + T | TT  T * F | F F  ( E ) | id

20. Example • Give a Grammar G E → TE’ E’ → + TE’ | ε T → FT’ T’ → * F T’ | ε F → ( E ) | id

21. Non-Recursive Predictive Parsing • Table-Driven Parsing • Given an LL(1) grammar G = <N, T, P, S> construct a table M[A,a] for A N, a  Tand use a driver program with a stack input stack Predictive parsingprogram (driver) output Parsing tableM

22. Predictive Parsing Table Algorithm

23. Example E T E’E’ +TE’ |  T F T ’T’*FT’ |  F ( E ) | id

24. Example Table-Driven Parsing E T E’E’ +TE’ |  T F T ’T’*FT’ |  F ( E ) | id

25. Exercise • Give a Grammar G as below S iEtSS’ S’  e S | ε E  b • Calculate the FIRST and FOLLOW • Create a predictive parsing table

26. Answer Ambiguous grammar S i E t S S’ | aS’eS |  E  b Error: duplicate table entry