1. LR Parsing – The Items Lecture 10 Mon, Feb 14, 2005

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 parse table, an input buffer, and a stack of “states.” It 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 parse 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 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 I 0 (item 0). State I 0 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   in the grammar, 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 our standard example, the augmented grammar is E'  E E  E + T | T T  T * F | F F  (E) | id | num

13. Example: LR Parsing The state I 0 consists of the items in 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 I 0, there will be transitions for the symbols E, T, F, (, id, and num. For example, on processing E, the items [E'   E] and [E   E + T] become [E'  E  ] and [E  E  + T].

16. Example: LR Parsing Let state I 1 be the closure of these items. I 1 :[E'  E  ] [E  E  + T] Thus the PDA has the transition I0I0 I1I1 E

17. Example: LR Parsing Similarly we determine the other transitions from I 0. Process T: I 2 : [E  T  ] [T  T  * F] Process F: I 3 :[T  F  ]

18. Example: LR Parsing Process (: I 4 : [F  (  E)] [E   E + T] [E   T] [T   T + F] [T   F] [F   (E)] [F   id] [F   num]

19. Example: LR Parsing Process id: I 5 :[F  id  ] Process num: I 6 :[F  num  ]

20. Example: LR Parsing Now find the transitions from states I 1 through I 6 to other states, and so on, until no new states appear.

21. Example: LR Parsing I 7 :[E  E +  T] [T   T * F] [T   F] [F   (E)] [F   id] [F   num]

22. Example: LR Parsing I 8 :[T  T *  F] [F   (E)] [F   id] [F   num] I 9 :[F  (E  )] [E  E  + T]

23. Example: LR Parsing I 10 :[E  E + T  ] [T  T  * F] I 11 :[T  T * F  ] I 12 :[F  (E)  ]