1. LR(1) for Bill McKeeman Dartmouth April 18, 2008 E ← T ┤ T ← F
F ← i
F ← ( T )
Bill McKeeman
Dartmouth
April 18, 2008

2. a simple expression cfg
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
a simple expression cfg
end-of-file symbol

3. E ← ▫ T ┤
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
LR start state

4. marked rules closure for T lookahead kernel in red closure in green
E ← ▫ T ┤
T ← ▫ F
T ← ▫ T * F ┤
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
marked rules
closure for T
lookahead
kernel in red
closure in green

5. closure for F E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i F ← ▫ ( T ) ┤

6. another closure for T E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i

7. another closure for F E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i

8. merge lookaheads closure complete E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
merge lookaheads
closure complete

9. shift T new kernel E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i F ← ▫ ( T )┤*
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
E ← T ▫ ┤
T ← T ▫ * F ┤*
shift T
new kernel

10. shift F another new kernel E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i┤*
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
E ← T ▫ ┤
T ← T ▫ * F ┤*
T ← F ▫ ┤*
shift F
another new kernel

11. shift i another new kernel E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i┤*
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
E ← T ▫ ┤
T ← T ▫ * F ┤*
T ← F ▫ ┤*
shift i
F ← i ▫ ┤*
another new kernel

12. 1 finish state 1 shift ( another new kernel E ← ▫ T ┤ T ← ▫ F┤*
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T ) 1
E ← T ▫ ┤
T ← T ▫ * F ┤*
finish state 1
T ← F ▫ ┤*
shift (
F ← i ▫ ┤*
another new kernel
F ← ( ▫ T ) ┤*

13. 1 closure E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i F ← ▫ ( T ) ┤* E ← T ┤
) *

14. shift ┤ (final LR state)
E ← ▫ T ┤
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
E ← T ┤ ▫
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T ) 1
E ← T ▫ ┤
T ← T ▫ * F ┤*
shift ┤ (final LR state)
T ← F ▫ ┤*
F ← i ▫ ┤*
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) *

15. 1 2 shift * 3 finish states 2,3,4 4 E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F┤*
E ← T ┤ ▫
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
T ← T * ▫ F ┤* 1
E ← T ▫ ┤
T ← T ▫ * F ┤* 2
shift * 3
T ← F ▫ ┤*
finish states 2,3,4 4
F ← i ▫ ┤*
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) *

16. 1 2 closure 3 4 E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i F ← ▫ ( T ) ┤*
) *

17. 1 2 3 4 shift T E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i F ← ▫ ( T ) ┤*
) *
E ← T ▫ ┤
T ← T ▫ * F ┤* 2 3
T ← F ▫ ┤* 4
F ← i ▫ ┤*
shift T
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) *

18. 1 2 3 4 shift F E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i F ← ▫ ( T ) ┤*
) *
E ← T ▫ ┤
T ← T ▫ * F ┤* 2
T ← F ▫
) * 3
T ← F ▫ ┤* 4
F ← i ▫ ┤*
shift F
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) *

19. 1 2 3 4 shift i E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i F ← ▫ ( T ) ┤*
) *
E ← T ▫ ┤
T ← T ▫ * F ┤* 2
T ← F ▫
) * 3
T ← F ▫ ┤*
F ← i ▫
) * 4
F ← i ▫ ┤*
shift i
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) *

20. 1 2 3 4 shift ( 5 finish state 5 E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i┤*
E ← T ┤ ▫
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T ) ┤* 1
F ← ( T ▫ )
T ← T ▫ * F ┤*
) *
E ← T ▫ ┤
T ← T ▫ * F ┤* 2
T ← F ▫
) * 3
T ← F ▫ ┤*
F ← i ▫
) * 4
F ← i ▫ ┤*
shift (
F ← ( ▫ T )
) *
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) * 5
finish state 5

21. 1 2 3 4 closure 5 E ← ▫ T ┤ T ← ▫ F T ← ▫ T * F F ← ▫ i F ← ▫ ( T ) ┤*
) *
E ← T ▫ ┤
T ← T ▫ * F ┤* 2
T ← F ▫
) * 3
T ← F ▫ ┤*
F ← i ▫
) * 4
F ← i ▫ ┤*
closure
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T )
) *
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) * 5

22. 6 7 4 1 5 2 finish state 6 shift on F and I and ( finish state 7
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
E ← ▫ T ┤
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤* 6
E ← T ┤ ▫
T ← T * F ▫ ┤* 7
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T ) ┤* 4
F ← i ▫ ┤* 1 5
F ← ( ▫ T ) ┤*
F ← ( T ▫ )
T ← T ▫ * F ┤*
) *
E ← T ▫ ┤
T ← T ▫ * F ┤* 2
finish state 6
shift on F and I and (
finish state 7
repeat state 4 and 5
T ← F ▫
) * 3
T ← F ▫ ┤*
F ← i ▫
) * 4
F ← i ▫ ┤*
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T )
) *
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) * 5

23. 6 7 4 1 5 8 2 9 3 10 4 shift on ) and * finish states 8,9,10 5 E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
E ← ▫ T ┤
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤* 6
E ← T ┤ ▫
T ← T * F ▫ ┤* 7
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T ) ┤* 4
F ← i ▫ ┤* 1 5
F ← ( ▫ T ) ┤*
F ← ( T ▫ )
T ← T ▫ * F ┤*
) *
F ← ( T ) ▫ ┤*
E ← T ▫ ┤
T ← T ▫ * F ┤* 8 2
T ← T * ▫ F
) * 9
T ← F ▫
) * 3
T ← F ▫ ┤* 10
F ← i ▫
) * 4
F ← i ▫ ┤*
shift on ) and *
finish states 8,9,10
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T )
) *
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) * 5

24. 6 7 4 1 5 8 2 9 3 10 4 closure 5 E ← T ┤ T ← F T ← T * F F ← i┤* 6
E ← T ┤ ▫
T ← T * F ▫ ┤* 7
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T ) ┤* 4
F ← i ▫ ┤* 1 5
F ← ( ▫ T ) ┤*
F ← ( T ▫ )
T ← T ▫ * F ┤*
) *
F ← ( T ) ▫ ┤*
E ← T ▫ ┤
T ← T ▫ * F ┤* 8 2
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T )
) * 9
T ← F ▫
) * 3
T ← F ▫ ┤* 10
F ← i ▫
) * 4
F ← i ▫ ┤*
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T )
) *
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) *
closure 5

25. shift on T and F and i and (
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
E ← ▫ T ┤
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤* 6
E ← T ┤ ▫ 12
T ← T * F ▫ ┤* 7
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T ) ┤* 4
F ← i ▫ ┤* 1 5
F ← ( ▫ T ) ┤*
F ← ( T ▫ )
T ← T ▫ * F ┤*
) * 13
F ← ( T ) ▫ ┤*
E ← T ▫ ┤
T ← T ▫ * F ┤* 8 2
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T )
) * 9
T ← F ▫
) * 3
T ← F ▫ ┤* 10
F ← i ▫
) * 4
F ← i ▫ ┤* 11
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T )
) *
F ← ( T ▫ )
T ← T ▫ * F
) *
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) * 9
T ← F ▫
) * 5 10
F ← i ▫
) * 11
F ← ( ▫ T )
) *
shift on T and F and i and (
finish state 11, repeat states 9,10,11, finish states 12,13

26. finish state 14, repeat states 10,11
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
E ← ▫ T ┤
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤* 6
E ← T ┤ ▫ 12
T ← T * F ▫ ┤* 7
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T ) ┤* 4
F ← i ▫ ┤* 1 5
F ← ( ▫ T ) ┤*
F ← ( T ▫ )
T ← T ▫ * F ┤*
) * 13
F ← ( T ) ▫ ┤*
E ← T ▫ ┤
T ← T ▫ * F ┤* 8 2
T ← T * F ▫
) * 14
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T )
) * 9
T ← F ▫
) * 11
F ← i ▫
) * 3
T ← F ▫ ┤* 10
F ← i ▫
) * 10
F ← ( ▫ T )
) * 4
F ← i ▫ ┤* 11
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T )
) *
F ← ( T ▫ )
T ← T ▫ * F
) *
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) * 9
T ← F ▫
) * 5 10
F ← i ▫
) * 11
F ← ( ▫ T )
) *
shift on F and i and (
finish state 14, repeat states 10,11

27. shift ) and *, finish state 15, repeat state 14
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
E ← ▫ T ┤
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤* 6
E ← T ┤ ▫ 12
T ← T * F ▫ ┤* 7
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T ) ┤* 4
F ← i ▫ ┤* 1 5
F ← ( ▫ T ) ┤*
F ← ( T ▫ )
T ← T ▫ * F ┤*
) * 13
F ← ( T ) ▫ ┤*
E ← T ▫ ┤
T ← T ▫ * F ┤* 8 2
T ← T * F ▫
) * 14
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T )
) * 9
T ← F ▫
) * 11
F ← i ▫
) * 3
T ← F ▫ ┤* 10
F ← i ▫
) * 10
F ← ( ▫ T )
) * 4
F ← i ▫ ┤* 11
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T )
) *
F ← ( T ▫ )
T ← T ▫ * F
) *
F ← ( T ) ▫
) * 15
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) * 14
T ← T * ▫ F
) * 9
T ← F ▫
) * 5 10
F ← i ▫
) * 11
F ← ( ▫ T )
) *
shift ) and *, finish state 15, repeat state 14

28. E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
E ← ▫ T ┤
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤* 6
E ← T ┤ ▫ 12
T ← T * F ▫ ┤* 7
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T ) ┤* 4
F ← i ▫ ┤* 1 5
F ← ( ▫ T ) ┤*
F ← ( T ▫ )
T ← T ▫ * F ┤*
) * 13
F ← ( T ) ▫ ┤*
E ← T ▫ ┤
T ← T ▫ * F ┤* 8 2 16
T ← T * F ▫
) * 14
T ← T * ▫ F
F ← ▫ i
F ← ▫ ( T )
) * 9
T ← F ▫
) * 11
F ← i ▫
) * 3
T ← F ▫ ┤* 10
F ← i ▫
) * 10
F ← ( ▫ T )
) * 4
F ← i ▫ ┤* 11
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T )
) *
F ← ( T ▫ )
T ← T ▫ * F
) * 17
F ← ( T ) ▫
) * 15
F ← ( ▫ T )
T ← ▫ F
T ← ▫ T * F
F ← ▫ i
F ← ▫ ( T ) ┤*
) * 14
T ← T * ▫ F
) * 9
T ← F ▫
) * 5 10
F ← i ▫
) * 11
F ← ( ▫ T )
) *
finish states 16,17

29. The LR Matrix symbols stack input states i * i ┤ 1 1i4 * i ┤ 1F * i ┤( ) 1 2 3 4 5 6 7 -2 -4 8 9 10 11 -1 12 14 13 15 -3 -5 16 17
-1:
-2:
-3:
-4:
-5:
E ← T ┤
T ← F
T ← T * F
F ← i
F ← ( T )
i * i ┤ 1
1i4
* i ┤ 1F
* i ┤
1F3
* i ┤ 1T
* i ┤
1T2
* i ┤
1T2*7
i ┤
states
1T2*7i4 ┤
1T2*7F ┤
1T2*7F12 ┤ 1T ┤
1T2 ┤
1T2 ┤6 E