Introduction to LR Parsing

1. Introduction to LR Parsing Aggelos Kiayias Computer Science & Engineering Department The University of Connecticut 371 Fairfield Road, Box U-1155 Storrs, CT 06269 aggelos@cse.uconn.edu http://www.cse.uconn.edu/~akiayias

2. Example Consider: S  aABe A  Abc | b B  d Rightmost Derivation of the string abbcde: S  aABe  aAde  aAbcde  abbcde The (unique) handle is underlined for each step. A viable prefix is (1) a string that equals a prefix of a right-sentential form up to (and including) its unique handle. (2) any prefix of a string that satisfies (1) Examples: a, aA, aAd, aAbc, ab, aAb,… Not viable prefixes: aAde, Abc, aAA,…

3. Shift/Reduce Parser STACK INPUT Remark $ abbcde$ SHIFT $a bbcde$ SHIFT $ab bcde$ REDUCE $aA bcde$ SHIFT $aAb cde$ SHIFT (?) $aAbc de$ REDUCE $aA de$ SHIFT $aAd e$ REDUCE $aAB e$ SHIFT $aABe$ REDUCE $S$ ACCEPT Observe: all Strings in the stack are viableprefixes

4. When to shift? When to Reduce? • Sometimes on top of the stack something appears to be a handle (i.e., matches the RHS of a production). • But: maybe we have not shifted enough elements to identify the handle. • Observe the correct sequence of Shift and Reducesteps preserves the property that the stack IS a viable prefix. Example $aAb cde$ Shift or Reduce? • If we shift we obtain aAbc in the stack. • Recall that Abc is a handle. • Instead if we reduce we obtain aAA in the stack. (this is NOT a viable prefix!!!)

5. When to Shift? When to Reduce? II • In order to make shift/reduce decisions: • We need to look to perhaps a few elements inside the stack. • We need to make sure that the way we modify the stack preserves the “viable prefix condition.” • For our previous example: • Any b appearing to the right of “A” should not be reduced. • In fact we can come up with heuristic decisions based on the grammar structure: • A “b” is reduced only if it is to the right of “a” • PROBLEM: what kind of information do we need to store inside the stack so that we can make decisions as above just by looking at the top element?

6. LR Parsing • LR (left-to-right, rightmost derivation). • LR(1) = 1 lookahead symbol. • Use stack • Stack contains “more information” (in a compressed form) compared to a Top-Down Table-driven parser. • LR(1): • Decisions are taken looking at the top of the stack + 1 input element.

7. Anatomy of an LR parser a + b $ s X s X Input (String + terminator) “States” Stack NT + T symbols of CFG LR Parsing Program Output What actions parser should take based on stack / input Parsing Table action[.,.] goto[.,.] • General parser behavior: s : top of stack a : current input • 1. If action[s,a]=“accept” halt, accept, success • If action[s,a]=“reduce by production A ” do the following:2a. Pop 2*|| elements from the stack.2b. Push A2c. Push goto[s*,A] • If action[s,a]=“shift and goto state s*”Shift; push s*

8. Example 1. S  aABe 2. A  Abc 3. A  b 4. B  d action goto

9. Example, II STACK INPUT Remark $0 abbcde$

10. Interesting Fact + LR Parsing Table Construction Methods HOW TO CONSTRUCT SUCH TABLES? • The set of all viable prefixes is Regular. • It is possible to write a DFA that recognizes it! • Use the DFA as an aid to construction of the table. Design Methodologies: • SLR (simple LR) “short table but limited methodology.” • Canonical LR“general methodology but big table.” • LALR (lookahead LR)“in between”