Constructing canonical LR(1) Parsing Tables (Aho, Sethi, & Ullman p230)

Constructing canonical LR(1) Parsing Tables(Aho, Sethi, & Ullman p230) Example 4.42. S' S, S  CC, C  cC, C  d State 0. Closure({[S' S , $]}) = { S' S , $ S  CC, FIRST($)  S  CC, $ C  cC, FIRST(C$) C  cC, c / d C  d, FIRST(C$)  C  d, c / d } State 1. Closure({[S' S , $]}) = {S'  S, $ }

Constructing canonical LR Parsing Tables(Aho, Sethi, & Ullman p230) State 2. Closure({[S CC, $]}) = { S CC, $ C  cC, FIRST($)  C  cC, $ C  d, FIRST($)  C  d, $ } State 3. Closure({[C cC , c/d]}) = {C cC , c/d C  cC , FIRST(c/d)  C  cC, c/d C  d , FIRST(c/d)  C  d, c/d }

Constructing canonical LR Parsing Tables(Aho, Sethi, & Ullman p230) State 4. Closure({[C d, c/d]}) = {C d, c/d} State 5. Closure({[S CC, $]}) = {S CC, $} State 6. Closure({[C cC, $]}) = { C cC, $ C  cC, FIRST($)  C  cC, $ C  d, FIRST($)  C  d, $ }

Constructing canonical LR Parsing Tables(Aho, Sethi, & Ullman p230) State 7. Closure({[C d, $]}) = {C d, $} State 8. Closure({[C cC, c/d]}) = {C cC, c/d } State 9. Closure({[C cC, $]}) = {C cC, $ }

Constructing canonical LR Parsing Tables(Aho, Sethi, & Ullman p230) State 0. {S'S, $; SCC, $; C cC, c/d; Cd, c/d} State 1. {S'  S, $ } State 2. { S CC, $; C  cC, $; C  d, $ } State 3. {C cC, c/d; C  cC, c/d; C  d, c/d} State 4. {C d, c/d} State 5. {S CC, $} State 6. { C cC, $; C  cC, $; C  d, $ } State 7. {C d, $} State 8. {C cC, c/d } State 9. {C cC, $ }

Efficient Construction of LALR Parsing Tables(Aho, Sethi, & Ullman p240) 1. Kernel set: Represent a set of items I by its kernel, i.e. by those items that are either the initial item [S' S, $], or that have the dot somewhere other than at the beginning of the right side. 2. Parsig actions: Any item calling for a reduction by A will be in the kernel unless  = . Reduction by A  is called for on input a iff there is a kernel item [B C, b] such that C ⇒*A for some , and a is in FIRST(b).

Efficient Construction of LALR Parsing Tables(Aho, Sethi, & Ullman p240) 3. Shift actions: The shift actions generated by I can be determined from the kernel of I as follows: We shift on input a if there is a kernel item [B C, b] where C ⇒*ax in derivation in which the last step does not use an -production. 4. Goto transition: If [B X, b] is in the kernel of I, then [B X, b] is in the kernel of goto(I, X). Item [A X, a] is also in the kernel of goto(I, X) if there is an item [B C, b] in the kernel of I, and C ⇒*A for some .

Efficient Construction of LALR Parsing Tables(Aho, Sethi, & Ullman p240) 5. Expand the proper lookahead symbols to kernels Spontaneously: Consider item [B C, b] in the kernel of I. Suppose C ⇒*A for some , and A X is a production. Then A X is in goto(I, X). The lookahead symbols for A X are the set of FIRST(). By definition, $ is generated spontaneously as a lookahead for the item S' S in the initial set of items.

Efficient Construction of LALR Parsing Tables(Aho, Sethi, & Ullman p240) 5. Expand the proper lookahead symbols to kernels Propagate: Another source of lookaheads for item A X, if  ⇒*, then the set b is also the lookaheaks of A X. (B C, C A, FOLLOW(A)  FOLLOW(C)  FOLLOW(B))

Efficient Construction of LALR Parsing Tables(Aho, Sethi, & Ullman p240) Algorithm 4.12 Determining lookaheads Input. The kernel K of a set of LR(0) items I and a grammar symbol X Output. The lookaheads: propagated and spontaneously generated Method. for each item B  in K { J' = closure({[B  , #]}); if [A X, a] is in J' where a is not # then lookahead a is generated spontaneously for item A X in goto(I, X) if [A X, #] is in J' then lookaheads propagate from B  in I to A X in goto(I, X) }

Efficient Construction of LALR Parsing Tables(Aho, Sethi, & Ullman p240) • Example. S' S, S L = R | R,L  *R | id, R  L The Kernels of the sets of LR(0) items are: 0. S'  S 6. S  L= R 1. S'  S 7. L  *R 2. S  L=R 8. R  L R  L 9. S  L=R 3. S  R 4. L  *R 5. L  id

Efficient Construction of LALR Parsing Tables(Aho, Sethi, & Ullman p240) • Example. S' S, S L = R | R, L  *R | id, R  L 0. S'  S, S  L=R, S R, L *R, Lid, RL 1. S'  S 2. S  L=R, RL 3. S R 4. L *R 5. Lid 6. S  L=R 7. L *R 8. S  L=R

Efficient Construction of LALR Parsing Tables(Aho, Sethi, & Ullman p240) Propagation of lookaheads (by algorithm 4.12) 0. S'  S on S 1. S'  S on L 2. S  L=R, R  L on R 3. S  R on * 4. L  *R on id 5. L  id 2. S  L=R on = 6. S  L= R 6. S  L= R on R 9. S  L=R on L 8. R  L on * 4. L  *R on id 5. L  id 4. L  *R on R 7. L  *R on L 8. R  L on id 5. L  id on * 4. L  *R

Efficient Construction of LALR Parsing Tables(Aho, Sethi, & Ullman p240)

Constructing canonical LR Parsing Tables(Aho, Sethi, & Ullman p230) Example 4.42. S' S, S  CC, C  cC, C  d Kernels: 0. S'S 1. S'  S 2. S CC 3. C cC 4. C d 5. S CC 6. C cC

Constructing canonical LR Parsing Tables(Aho, Sethi, & Ullman p230) Propagation of lookaheads (by algorithm 4.12) 0. S'S on S 1. S'  S on C 2. S CC on c 3. C cC on d 4. C d 2. S CC on C 5. S CC 3. C cC on C 6. C cC

Efficient Construction of LALR Parsing Tables(Aho, Sethi, & Ullman p240)

Constructing canonical LR(1) Parsing Tables (Aho, Sethi, & Ullman p230)