1. COP4620 – Programming Language Translators Dr. Manuel E. Bermudez
LALR(1) parsing
COP4620 – Programming Language Translators Dr. Manuel E. Bermudez

2. topics Non SLR(1) Grammars Calculating LALR(1) lookahead sets
LALR(1) examples.
Summary of Parsing
(top-down vs. bottom-up)

3. LALR(1) parsing LR(0) Automaton: SLR(1) Analysis: (State 5)2 S 6 3 7 A 10 b a
A → a 4 8 B
A → AbAa 9 11 5
S → Ba
B → a
Grammar is not LR(0): reduce-reduce conflict. 
Grammar:
S’ → S 
S → AbAa
→ Ba
A → a
B → a
LR(0)
Automaton:
SLR(1) Analysis: (State 5)
Follow(A) = {a, b}
Follow(B) = {a}
Conflict not
resolved.
Grammar not SLR(1).
Need Follow(A)
’in the context of’
State 1, not state 7.
State a b  5
R/A→a
R/B→a
State a b  5
R/A→a
R/B→a
LR(0) table.
SLR(1) table.

4. LR Parsing history LR parsing: D. Knuth seminal paper, 1965.
SLR(1): F. DeRemer, Ph.D. thesis, MIT, 1969.
LALR(1): various algorithms (from LR(1) DFA), 1970’s.
LALR(1): Efficient Computation of LALR(1) Lookahead sets,
(from LR(0) DFA) F. DeRemer, T. Pennello, UCSC, 1981.
5. LALR(k): M. Bermudez, Ph.D. thesis, UCSC, 1984.
6. LALR(1): Simple Computation of LALR(1) Lookahead sets,
M. Bermudez and G. Logothetis, UF, 1989.

5. Simple LALR(1) parsing
I. For each conflicting reduction A → ω at each conflicting state q, find all nonterminal transitions (pi, A) such that
II. Need the union of Follow(pi, A) for all i. p1 A q pn
A → ω ω .

6. Simple LALR(1) parsing Build G’, an ’expanded’ version of G:
For each (p, A) and for each A → w1w2…wn, we have
For each such situation, G’ has a production of the form:
(p, A) → (p, w1)(p2, w2)…(pn, wn)
G’ structurally similar to G.
Uses vocabulary of LR(0) transitions, rather than symbols in G.
Follow sets in G’ are the key ! p A p2
A → w1…wn … w2 wn w1

7. Simple LALR(1) parsing Grammar: Not LR(0). Not SLR(1). S’ → S 
S → AbAa
→ Ba
A → a
B → a
Not LR(0). Not SLR(1). 1 2 S 6 3 7 A 10 b a
A → a 4 8 B
A → AbAa 9 11 5
S → Ba
B → a 
For the conflict in state 5, we need
Follow(1, A) = {(3, b)}
Follow(1, B) = {(4, a)}.
Extract the symbols.
{b}
{a}
Disjoint. Grammar is LALR(1)!
G’: (1, S) → (1, A)(3, b)(7, A)(9, a) → (1, B)(4, a) (1, A) → (1, a) (7, A) → (7, a) (1, B) → (1, a)
These have
split !
State a b  5
R/B→a
R/A→a
LALR(1) table.

8. Simple LALR(1) parsing 1 2 5 8 11 7 4 6 3 12 10 9 13 Grammar:
S’ → S  B → A
S → bBb A → c
→ aBa
→ acb
Simple LALR(1) parsing 1 2 S 5 8 11 b
A → c 7 4 6 3 a
S → bBb
S → aBa 12 10 9 A c B
B → A 13
S → acb 
LALR(1) Analysis:
Need Follow(4, A).
G’: (1,S) → (1, b)(3, B)(6, b)
→ (1, a)(4, B)(9, a)
→ (1, a)(4, c)(10, b)
(3, B) → (3, A)
(3, A) → (3, c)
(4, B) → (4, A)
(4, A) → (4, c)
Follow(4, A)⊇Follow(4, B) = {(9, a)}.
The lookahead set is {a}.
{a}
State 10: shift-reduce conflict.
Grammar is not LR(0).
SLR(1) Analysis, state 10:
Follow(A) ⊇ Follow(B) ={a,b}.
Grammar is not SLR(1).
Since b ∉ {a}, the grammar is LALR(1).

9. Summary of parsing S part of tree known stack wβ
part of tree known
stack
part of tree left to predict
remaining input
input already parsed α
Top-Down Parsing
Hand-written or
Table Driven:LL(1) S w β
unknown
(left to predict)
stack
known
remaining input
input already parsed α
Bottom-up Parsing
Usually Table-Driven: LR(0), SLR(1), LALR(1).

10. summary Non SLR(1) Grammars Calculating LALR(1) lookahead sets
LALR(1) examples.
Summary of Parsing
(top-down vs. bottom-up)