1. LR Parsing – The Items
Lecture 10
Fri, Feb 13, 2004

2. LR Parsers
A bottom-up parser follows a rightmost derivation from the bottom up.
Such parsers typically use the LR algorithm and are called LR parsers.
L means process tokens from Left to right.
R means follow a Rightmost derivation.

3. LR Parsers
Furthermore, in LR parsing, the production is applied only after the pattern has been matched.
In LL (predictive) parsing, the production was selected, and then the tokens were matched to it.

4. Rightmost Derivations
Let the grammar be
E  E + T | T
T  T * F | F
F  (E) | id | num

5. Rightmost Derivations
A rightmost derivation of (id + num)*id is
E  T
 T*F
 T*id
 F*id
 (E)*id
 (E + T)*id
 (E + F)*id
 (E + num)*id
 (T + num)*id
 (F + num)*id
 (id + num)*id.

6. LR Parsers
An LR parser uses a parsing table, an input buffer, and a stack of “states.”
If performs three operations.
Shift a token from the input buffer to the stack.
Reduce the content of the stack by applying a production.
Go to a new state.

7. LR(0) Items
To build an LR parsing table, we must first find the LR(0) items.
An LR(0) item is a production with a special marker () marking a position within the string on the right side of the production.
LR(0) parsing is also called SLR parsing (“simple” LR).

8. Example: LR(0) Items If the production is E  E + T,
then the possible LR(0) items are
[E   E + T]
[E  E  + T]
[E  E +  T]
[E  E + T ]

9. LR(0) Items The interpretation of [A    ] is
“We have processed  and
we might process  next.”
Whether we do actually process  will be borne out by the subsequent sequence of tokens.

10. LR Parsing We will build a PDA whose states are sets of LR(0) items.
First we augment the grammar with a new start symbol S'.
S'  S.
This guarantees that the start symbol will not recurse.

11. States of the PDA The initial state is called I0.
State I0 is the closure of the set
{[S'   S]}.
To form the closure of a set of items
For each item [A    B] in the set and for each production B  , add the item [B   ] to the set.
Let us call [B   ] an initial B-item.
Continue in this manner until there is no further change.

12. Example: LR Parsing
Continuing with the example, the augmented grammar is
E'  E
E  E + T | T
T  T * F | F
F  (E) | id | num

13. Example: LR Parsing [E'   E]
The state I0 consists of the items the closure of item [E'   E].
[E'   E]
[E   E + T]
[E   T]
[T   T * F]
[T   F]
[F   (E)]
[F   id]
[F   num]

14. Transitions
There will be a transition from one state to another state for each grammar symbol in an item that immediately follows the marker  in an item in that state.
If an item in the state is [A    X], then
The transition from that state occurs when the symbol X is processed.
The transition is to the state that is the closure of the item [A  X  ].

15. Example: LR Parsing
Thus, from the state I0, there will be transitions for the symbols E, T, F, (, id, and num.
On processing E, the items
[E'   E] and [E   E + T]
become
[E'  E ] and [E  E  + T].

16. Example: LR Parsing Let state I1 be the closure of these items.
I1: [E'  E ]
[E  E  + T]
Thus the PDA has the transition I0 I1 E

17. Example: LR Parsing
Similarly we determine the other transitions from I0.
Process T:
I2: [E  T ]
[T  T  * F]
Process F:
I3: [T  F ]

18. Example: LR Parsing I4: [F  (  E)] Process (: [E   E + T]
[T   T + F]
[T   F]
[F   (E)]
[F   id]
[F   num]

19. Example: LR Parsing Process id: I5: [F  id ] Process num:
I6: [F  num ]

20. Example: LR Parsing
Now find the transitions from states I1 through I6 to other states, and so on, until no new states appear.

21. Example: LR Parsing I7: [E  E +  T] [T   T * F] [T   F]
[F   (E)]
[F   id]
[F   num]

22. Example: LR Parsing I8: [T  T *  F] [F   (E)] [F   id]
[F   num]
I9: [F  (E  )]
[E  E  + T]

23. Example: LR Parsing I10: [E  E + T ] [T  T  * F]
I11: [T  T * F ]
I12: [F  (E) ]