Items and Itemsets

1
Items and Itemsets

• An itemset is merely a set of items
• In LR parsing terminology an item
• Looks like a production with a . in it
• The . indicates how far the parse has gone in
  recognizing a string that matches this production
• e.g. A -gt aAb.BcC suggests that weve seen
  input that could replace aAb. If, by following
  the rules we get A -gt aAbBcC. we can reduce by A
  -gt aAbBcC

2
Building LR(0) Itemsets

• Start with an augmented grammar if S is the
  grammar start symbol add S -gt S
• The first set of items includes the closure of S
  -gt S
• Itemset construction requires two functions
• Closure
• Goto

3
Closure of LR(0) Itemset

• If J is a set of items for Grammar G, then
  closure(J) is the set of items constructed from G
  by two rules
• 1) Each item in J is added to closure(J)
• 2) If A ? a.Bß is in closure(J) and
• B ? f is a production, add
• B ? .f to closure(J)

4
Closure Example
Grammar A ? aBC A ? aA B ? bB B? bC C? cC C ? ?
Closure(J) A ? a.BC A-gt
a.A A ? .aBC A ? .aA B ? .bB B ? .bC
J A ? a.BC A? a.A
5
GoTo

• Goto(J,X) where J is a set of items and X is a
  grammar symbol either terminal or non-terminal
  is defined to be closure of
• A? aX.ß for A ? a.Xß in J
• So, in English, Goto(J,X) is the closure of all
  items in J which have a . immediately preceding
  X

6
Set of Items Construction

• Procedure items(G)
• Begin
• C closure(S ? .S)
• repeat
• for each set of items J in C and each
• grammar symbol X such that GoTo(J,X)
• is not empty and not in C do
• add GoTo(J,X) to C
• until no more sets of items can be added to C

7
Build LR(0) Itemsets for

• S ? (S), S ? ?
• S ? (S), S ? SS, S ? ?

8
Building LR(0) Table from Itemsets

• One row for each Itemset
• One column for each terminal or non-terminal
  symbol, and one for
• Table JX is
• Rn if J includes A ? rhs., A ? rhs is rule
  number n, and X is a terminal
• Sn if Goto(J,X) is itemset n

9
LR(0) Parse Table for

• S ? (S), S ? ?
• S ? (S), S ? SS, S ? ?

10
Building SLR Table from Itemsets

• One row for each Itemset
• One column for each terminal or non-terminal
  symbol, and one for
• Table JX is
• Rn if J includes A ? rhs., A ? rhs is rule
  number n, X is a terminal, AND
• X is in Follow(A)
• Sn if Goto(J,X) is itemset n

11
LR(0) and LR(1) Items

• LR(0) item is a production with a . in it.
• LR(1) item has a kernel that looks like LR(0),
  but also has a lookahead e.g.
• A ? a.Xß, terminals
• A ? a.Xß, a/b/c ? A ? a.Xß, a/b/d

12
Closure of LR(1) Itemset

• If J is a set of LR(1) items for Grammar G, then
  closure(J) includes
• 1) Each LR(1) item in J
• 2) If A ? a.Bß, a in closure(J) and
• B ? f is a production, add
• B ? .f, First(ß,a) to closure(J)

13
LR(1) Itemset Construction

• Procedure items(G)
• Begin
• C closure(S ? .S, )
• repeat
• for each set of items J in C and each
• grammar symbol X such that GoTo(J,X)
• is not empty and not in C do
• add GoTo(J,X) to C
• until no more sets of items can be added to C

14
Build LR(1) Itemsets for

• S ? (S), S ? SS, S ? ?

15
S ? CC, S ? cC, C ?d

• Is this grammar
• LR(0)?
• SLR?
• LR(1)?
• How can we tell?

16
LR(1) Table from LR(1) Itemsets

• One row for each Itemset
• One column for each terminal or non-terminal
  symbol, and one for
• Table JX is
• Rn if J includes A ? rhs., a A ? rhs is rule
  number n X a
• Sn if Goto(J,X) in LR(1) itemset n