LL1 Recognition Algorithm

1
LL(1) Recognition Algorithm

• place the start symbol and the EOL input symbol
  on the prediction stack with the start symbol on
  top
• read the leftmost symbol from input into current
  symbol
• Repeat until the EOL symbol is matched
• pop the top symbol off the prediction stack
• if a terminal, compare it to the current
  symbol
• if they match, read the next input symbol into
  the current symbol
• else an error has been discovered in the input
• else (a nonterminal) choose one of its RHS and
  push it on the prediction stack
• choose the RHS by looking at the current symbol

2
Parse i i EOL

• e t e_2_7.
• e_2_7 e_2_9 t e_2_7.
• e_2_7 .
• e_2_9 "".
• e_2_9 "-".
• f i.
• f "(" e ")".
• start e.
• t f t_3_8.
• t_3_10 "".
• t_3_10 "/".
• t_3_8 t_3_10 t.
• t_3_8 .

• i i EOL
• (current input)
• Stack
• start EOL
• Apply start e .
• e EOL
• Apply e t e_2_7 .
• e e_2_7 EOL

3
Parse i i EOL

• e t e_2_7.
• e_2_7 e_2_9 t e_2_7.
• e_2_7 .
• e_2_9 "".
• e_2_9 "-".
• f i.
• f "(" e ")".
• start e.
• t f t_3_8.
• t_3_10 "".
• t_3_10 "/".
• t_3_8 t_3_10 t.
• t_3_8 .

• Apply t f t_3_8.
• f t_3_8 e_2_7 EOL
• Apply f i.
• i t_3_8 e_2_7 EOL
• Have terminal i, get match, move current symbol
  pointer
• i i EOL
• t_3_8 e_2_7 EOL

4
Parse i i EOL

• e t e_2_7.
• e_2_7 e_2_9 t e_2_7.
• e_2_7 .
• e_2_9 "".
• e_2_9 "-".
• f i.
• f "(" e ")".
• start e.
• t f t_3_8.
• t_3_10 "".
• t_3_10 "/".
• t_3_8 t_3_10 t.
• t_3_8 .

• Apply t_3_8 .
• e_2_7 EOL
• Apply e_2_7 e_2_9 t e_2_7 .
• e_2_9 t e_2_7 EOL
• Apply e_2_9 "".
• t e_2_7 EOL

5
Parse i i EOL

• e t e_2_7.
• e_2_7 e_2_9 t e_2_7.
• e_2_7 .
• e_2_9 "".
• e_2_9 "-".
• f i.
• f "(" e ")".
• start e.
• t f t_3_8.
• t_3_10 "".
• t_3_10 "/".
• t_3_8 t_3_10 t.
• t_3_8 .

• Pop terminal , match, move current symbol
• i i EOL
• t e_2_7 EOL
• Apply t f t_3_8
• f t_3_8 e_2_7 EOL
• Apply f i.
• i t_3_8 e_2_7 EOL

6
Parse i i EOL

• e t e_2_7.
• e_2_7 e_2_9 t e_2_7.
• e_2_7 .
• e_2_9 "".
• e_2_9 "-".
• f i.
• f "(" e ")".
• start e.
• t f t_3_8.
• t_3_10 "".
• t_3_10 "/".
• t_3_8 t_3_10 t.
• t_3_8 .

• Pop terminal i, match, move current symbol
• i i EOL
• t_3_8 e_2_7 EOL
• Apply t_3_8 .
• e_2_7 EOL
• Apply e_2_7 .
• EOL
• Pop terminal EOL, match, move current symbol

7
Putting grammars into LL(1) form

• You may introduce new nonterminals.
• You may revise the definitions of existing
  nonterminals.
• You may delete nonterminals if they are no longer
  needed.
• Never change the meaning of a nonterminal.
• Never change the set of strings a nonterminal
  generates.

8
How a grammar fails to be LL(1)

• The parser picks one of the nonterminals right
  hand sides to replace it with
• To do this, it can look only at the next symbol
  in the input.
• Example
• t f "" t .
• t f "/" t .
• t f .
• f i .
• f "(" e ")" .

9
Concepts for making an LL(1) grammar

• The first set of a string of symbols, u, is the
  set of terminal symbols, First(u), that can occur
  leftmost in a string derived from u.
• First(u) a uÞ av , a is a terminal
  symbol, u and v are strings
• The follow set of a nonterminal, A, is the set of
  symbols, Follow(A), that can follow A in a
  sentential form.
• Follow(A) b sÞ v A b w , s is the start
  symbol, b is a terminal symbol, v and w are
  strings

10
Which RHS to choose?

• t f "" t .
• t f "/" t .
• t f .
• f i .
• f "(" e ")" .
• Given a production A u .
• What terminal symbol, t, would tell us to replace
  A with u?
• If symbol t is in First(u), choose u.
• If u is the empty string, or if it derives the
  empty string, and if t is in Follow(A) choose u.

11
LL(1) Grammar

• If any terminal symbol tells us to choose more
  than one right hand side for a nonterminal, the
  grammar is not LL(1).
• If no terminal symbol ever tells us to choose
  more than one right hand side for any
  nonterminal, the grammar is LL(1).