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 -> aAb.BcC suggests that we’ve “seen” input that could replace aAb. If, by following the rules we get A -> aAbBcC. we can reduce by A -> aAbBcC

2. Building LR(0) Itemsets
Start with an augmented grammar; if S is the grammar start symbol add S’ -> S
The first set of items includes the closure of S’ -> 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  α.Bβ is in closure(J) and
B  φ is a production, add
B  .φ to closure(J)

4. Closure Example Grammar: A  aBC A  aA B  bB B bC C cC C  λ
Closure(J)
A  a.BC
A-> 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 αX.β for A  α.Xβ 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  λ}
{S  (S), S  SS, S  λ}

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 [J][X] 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  λ}
{S  (S), S  SS, S  λ}

10. Building SLR Table from Itemsets
One row for each Itemset
One column for each terminal or non-terminal symbol, and one for $
Table [J][X] 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  α.Xβ, {terminals}
A  α.Xβ, a/b/c ≠ A  α.Xβ, 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  α.Bβ, a in closure(J) and
B  φ is a production, add
B  .φ, First(β,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  λ}

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 [J][X] 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