1. Fast SAT through Incremental Unit Propagation
Truth Maintenance and the ITMS
Brian C. Williams Oct. 30th, J/6.834J
9/20/2000
copyright Brian Williams

2. copyright Brian Williams
Outline
Motivation: Fast Mode Estimation and Reconfiguration
Review of unit propagation
Incremental unit propagation as truth maintenance
9/20/2000
copyright Brian Williams

3. Reconfiguring for a Failed Engine
Oxidizer tank
Fuel tank
Open four
valves
Valve fails
stuck closed
Fire backup
engine
9/20/2000
copyright Brian Williams

4. Diagnosis and Reconfiguration Performance on Cassini
Number of components: 80
Number of clauses: 11,101
9/20/2000
copyright Brian Williams

5. copyright Brian Williams
Outline
Motivation: Fast Mode Estimation and Reconfiguration
Review of unit propagation
Incremental unit propagation as truth maintenance
9/20/2000
copyright Brian Williams

6. Unit Propagation Examples
C1: Not A or B
C2: Not C or A
C3: Not B or C
C4: A A
True B
True C
True C4 C1 C3
9/20/2000
copyright Brian Williams

7. Unit Propagation Examples
C1: Not A or B
C2: Not C or A
C3: Not B or C
C4: A
C4’: Not B A
True B
True C
True C4 C1 C3
C4’ A
False B
False C
False C2 C1
9/20/2000
copyright Brian Williams

8. copyright Brian Williams
Unit Propagation
true
false t r q
C2: ¬ p ¬ t
C1 : ¬r  q p p
procedure propagate(C) // C is a clause
if all literals in C are false except l, and l is unassigned
then assign true to l and
record C as a support for l and
for each clause C’ mentioning “not l”,
propagate(C’)
end propagate
9/20/2000
copyright Brian Williams

9. copyright Brian Williams
Unit Propagation
true
false t r q
C2: ¬ p ¬ t
true
C1 : ¬r  q p p
procedure propagate(C) // C is a clause
if all literals in C are false except l, and l is unassigned
then assign true to l and
record C as a support for l and
for each clause C’ mentioning “not l”,
propagate(C’)
end propagate
9/20/2000
copyright Brian Williams

10. copyright Brian Williams
Unit Propagation
true
false t r q
C2: ¬ p ¬ t
true
C1 : ¬r  q p p
procedure propagate(C) // C is a clause
if all literals in C are false except l, and l is unassigned
then assign true to l and
record C as a support for l and
for each clause C’ mentioning “not l”,
propagate(C’)
end propagate
9/20/2000
copyright Brian Williams

11. copyright Brian Williams
Unit Propagation
true
false t r q
C2: ¬ p ¬ t
true
C1 : ¬r  q p p
procedure propagate(C) // C is a clause
if all literals in C are false except l, and l is unassigned
then assign true to l and
record C as a support for l and
for each clause C’ mentioning “not l”,
propagate(C’)
end propagate
9/20/2000
copyright Brian Williams

12. copyright Brian Williams
Unit Propagation
true
false
false t r q
C2: ¬ p ¬ t
true
C1 : ¬r  q p p
procedure propagate(C) // C is a clause
if all literals in C are false except l, and l is unassigned
then assign true to l and
record C as a support for l and
for each clause C’ mentioning “not l”,
propagate(C’)
end propagate
9/20/2000
copyright Brian Williams

13. copyright Brian Williams
DPLL Example
Example:
C1: Not A or B
C2: Not C or A
C3: Not B or C A
Propagate:
C = F
B = F
A = F
Propagate:
B = T
C = T
A = T F T
9/20/2000
copyright Brian Williams

14. DPLL Procedure [Davis, Putnam Logmann, Loveland, 1962]
DPLL(phi,A)
Input: A cnf theory phi, an assignment A to propositions in phi
Output: A decision of whether phi is satisfiable.
1. propagate(phi);
2. If a clause is violated return(false);
3. Else if all propositions are assigned return(true);
4. Else Q = some unassigned proposition in phi;
6. Return (DPLL(phi, A[Q = True]) or
DPLL(phi, A[Q = False])
9/20/2000
copyright Brian Williams

15. copyright Brian Williams
Outline
Motivation: Fast Mode Estimation and Reconfiguration
Review of unit propagation
Incremental unit propagation as truth maintenance
LTMS
ITMS
9/20/2000
copyright Brian Williams

16. copyright Brian Williams
Satisfiability with Incremental Unit Propagation (aka Truth Maintenance)
When adding an assignment incrementally introduce additional unit propagations (already done for DPLL)
When removing an assignment (backtracking), only remove propagations of that assignment.
9/20/2000
copyright Brian Williams

17. Propagating Incrementally: Adding Clause C1
true
false
false t r q
C2: ¬ p ¬ t
true
C1 : ¬r  q p p
procedure propagate(C) // C is a clause
if all literals in C are false except l, and l is unassigned
then assign true to l and
record C as a support for l and
for each clause C’ mentioning “not l”,
propagate(C’)
end propagate
9/20/2000
copyright Brian Williams

18. Undoing Propagation Incrementally: Deleting Clause Dq
true s
C: ¬ p ¬ t
D: ¬r q p p
A: ¬ s ¬ p
procedure unsupport(D) // D is a clause
if D supports some proposition p
then delete p’s support and truth assignment;
for each support C containing p // Delete consequences
unsupport(C);
for each clause A containing p // Resupport p
then propagate(A);
end unsupport
9/20/2000
copyright Brian Williams

19. Undoing Propagation Incrementally: Deleting Clause Dq
true s
C: ¬ p ¬ t
D: ¬r q p p
A: ¬ s ¬ p
procedure unsupport(D) // D is a clause
if D supports some proposition p
then delete p’s support and truth assignment;
for each support C containing p // Delete consequences
unsupport(C);
for each clause A containing p // Resupport p
then propagate(A);
end unsupport
9/20/2000
copyright Brian Williams

20. Undoing Propagation Incrementally: Deleting Clause Dq s
C: ¬ p ¬ t
D: ¬r q p p
A: ¬ s ¬ p
procedure unsupport(D) // D is a clause
if D supports some proposition p
then delete p’s support and truth assignment;
for each support C containing p // Delete consequences
unsupport(C);
for each clause A containing p // Resupport p
then propagate(A);
end unsupport
9/20/2000
copyright Brian Williams

21. Undoing Propagation Incrementally: Deleting Clause Dq s
C: ¬ p ¬ t
D: ¬r q p p
A: ¬ s ¬ p
procedure unsupport(D) // D is a clause
if D supports some proposition p
then delete p’s support and truth assignment;
for each support C containing p // Delete consequences
unsupport(C);
for each clause A containing p // Resupport p
then propagate(A);
end unsupport
9/20/2000
copyright Brian Williams

22. Undoing Propagation Incrementally: Deleting Clause Dq s
C: ¬ p ¬ t
D: ¬r q p p
A: ¬ s ¬ p
procedure unsupport(D) // D is a clause
if D supports some proposition p
then delete p’s support and truth assignment;
for each support C containing p // Delete consequences
unsupport(C);
for each clause A containing p // Resupport p
then propagate(A);
end unsupport
9/20/2000
copyright Brian Williams

23. Undoing Propagation Incrementally: Deleting Clause Dq s
C: ¬ p ¬ t
D: ¬r q p p
A: ¬ s ¬ p
procedure unsupport(D) // D is a clause
if D supports some proposition p
then delete p’s support and truth assignment;
for each support C containing p // Delete consequences
unsupport(C);
for each clause A containing p // Resupport p
then propagate(A);
end unsupport
9/20/2000
copyright Brian Williams

24. Undoing Propagation Incrementally: Deleting Clause Dq
true s
C: ¬ p ¬ t
D: ¬r q p p
A: ¬ s ¬ p
procedure unsupport(D) // D is a clause
if D supports some proposition p
then delete p’s support and truth assignment;
for each support C containing p // Delete consequences
unsupport(C);
for each clause A containing p // Resupport p
then propagate(A);
end unsupport
9/20/2000
copyright Brian Williams

25. copyright Brian Williams
Cassini Performance
Number of components: 80
Number of clauses: 11101
no TMS
TMS
9/20/2000
copyright Brian Williams

26. Unit Propagation in Real-Time: LTMS
Significant, unnecessary re-labeling during each switch
Outliers dominate ability to react.
9/20/2000
copyright Brian Williams

27. Unit Propagation in Real-Time: LTMS
Significant, unnecessary re-labeling during each switch
Outliers dominate ability to react.
9/20/2000
copyright Brian Williams

28. Example: DS-1 Bus Communication
PDE
SRU
PDU
GDE
PASM
DSEU
PEPE
Commands BC
1553 bus
Flight
Computer
Data
Clauses that describe the bus controller (bc):
C1: ¬ nci ¬ a  nco
C2: ¬ ia  nco
C3: ¬ ok  a
C4: ¬ rf ia
C5: ¬ uf  ia
C6: ¬ ok ¬ rf
C7: ¬ ok ¬ uf
C8: ¬ rf  ¬ uf
C9: ¬ a ¬ ia
BC health: ok, rf, uf
No input cmd: nci
BC activity: a, ia
No output cmd: nco
9/20/2000
copyright Brian Williams

29. LTMS’ Conservative Re-Support Strategy
Delete C10 then add C11
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco pf a
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Performing Unit Propagation
Also show one step example
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
~ 300
C4: ¬ rf ia
C2: ¬ ia  nco
9/20/2000
copyright Brian Williams
Support has ~ 300 consequences

30. LTMS’ Conservative Re-Support Strategy
Delete C10 then add C11
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco pf a
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Performing Unit Propagation
Also show one step example
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
~ 300
C4: ¬ rf ia
C2: ¬ ia  nco
9/20/2000
copyright Brian Williams
Trace support and delete ~ 300 consequences

31. LTMS’ Conservative Re-Support Strategy
Delete C10 then add C11
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco pf a
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Performing Unit Propagation
Also show one step example
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
~ 300
C4: ¬ rf ia
C2: ¬ ia  nco
9/20/2000
copyright Brian Williams
Trace support and delete ~ 300 consequences

32. LTMS’ Conservative Re-Support Strategy
Delete C10 then add C11
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco pf a
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Performing Unit Propagation
Also show one step example
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
~ 300
C4: ¬ rf ia
C2: ¬ ia  nco
9/20/2000
copyright Brian Williams
Trace support and delete ~ 300 consequences

33. LTMS’ Conservative Re-Support Strategy
Delete C10 then add C11
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco pf a
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Performing Unit Propagation
Also show one step example
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
~ 300
C4: ¬ rf ia
C2: ¬ ia  nco
9/20/2000
copyright Brian Williams
Trace support and delete ~ 300 consequences

34. LTMS’ Conservative Re-Support Strategy
Delete C10 then add C11
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco pf a
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Performing Unit Propagation
Also show one step example
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
C4: ¬ rf ia
C2: ¬ ia  nco
9/20/2000
copyright Brian Williams
Trace support and delete ~ 300 consequences

35. LTMS’ Conservative Re-Support Strategy
Delete C10 then add C11
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco pf a
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Problem 2: Blocking by logical inconsistency
Solution ***
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
C4: ¬ rf ia
C2: ¬ ia  nco
C11: rf
9/20/2000
copyright Brian Williams
Add C11 and propagate consequences, ...

36. LTMS’ Conservative Re-Support Strategy
Delete C10 then add C11
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco pf a
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Problem 2: Blocking by logical inconsistency
Solution ***
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
C4: ¬ rf ia
C2: ¬ ia  nco
C11: rf
9/20/2000
copyright Brian Williams
Add C11 and propagate consequences, ...

37. LTMS’ Conservative Re-Support Strategy
Delete C10 then add C11
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco pf a
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Problem 2: Blocking by logical inconsistency
Solution ***
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
C4: ¬ rf ia
C2: ¬ ia  nco
C11: rf
9/20/2000
copyright Brian Williams
Add C11 and propagate consequences, ...

38. LTMS’ Conservative Re-Support Strategy
Delete C10 then add C11
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco pf a
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Problem 2: Blocking by logical inconsistency
Solution ***
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf
Resup
~300 rf
C4: ¬ rf ia
C2: ¬ ia  nco
C11: rf
9/20/2000
copyright Brian Williams
Add C11 and propagate consequences, ~300 unchanged.

39. copyright Brian Williams
Outline
Motivation: Fast Mode Estimation and Reconfiguration
Review of unit propagation
Incremental unit propagation as truth maintenance
LTMS
ITMS – a repair-based approach
9/20/2000
copyright Brian Williams

40. ITMS: Re-Support via Local Cycle Detection
To Avoid Cycles, Avoid Back Edges
Number literals by depth of support.
Back edges point to lower numbers. 4
C1: ¬ nci ¬ a  nco
nci a 2
C3: ¬ ok  a
C10: ok
Performing Unit Propagation
Also show one step example 5 ok 1 ia 3
nco 2
Sup
~300 rf
9/20/2000
C11: rf
copyright Brian Williams
C4: ¬ rf ia
C2: ¬ ia  nco

41. ITMS: Re-Support via Local Cycle Detection
Number literals by depth of support.
Back edges point to lower numbers. 4
C1: ¬ nci ¬ a  nco
nci a 2
C3: ¬ ok  a
C10: ok
Performing Unit Propagation
Also show one step example 5 ok 1 ia 3
nco 2
Sup
~300 rf
9/20/2000
C11: rf
copyright Brian Williams
C4: ¬ rf ia
C2: ¬ ia  nco

42. copyright Brian Williams
Add Before Delete
nci
C1: ¬ nci ¬ a  nco a
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia
Performing Unit Propagation
Also show one step example ok ia
nco
C6: ¬ ok ¬ rf
~300
literals rf
9/20/2000
C11: rf
copyright Brian Williams
C4: ¬ rf ia
C2: ¬ ia  nco

43. Add Before Delete -> Propagation blocked
nci
C1: ¬ nci ¬ a  nco a
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia
Performing Unit Propagation
Also show one step example ok ia
nco
C6: ¬ ok ¬ rf
~300
literals rf
9/20/2000
C11: rf
copyright Brian Williams
C4: ¬ rf ia
C2: ¬ ia  nco

44. Idea: Roll back Consequences of C10 Just Enough to Let C11 Through
nci
C1: ¬ nci ¬ a  nco a
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Performing Unit Propagation
Also show one step example
C6: ¬ ok ¬ rf
~300
literals rf
C4: ¬ rf ia
C2: ¬ ia  nco
C11: rf
9/20/2000
copyright Brian Williams

45. Idea: Roll back Consequences of C10 Just Enough to Let C11 Through
nci
C1: ¬ nci ¬ a  nco a
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Performing Unit Propagation
Also show one step example
C6: ¬ ok ¬ rf
~300
literals rf
C4: ¬ rf ia
C2: ¬ ia  nco
C11: rf
9/20/2000
copyright Brian Williams

46. Idea: Roll back Consequences of C10 Just Enough to Let C11 Through
nci
C1: ¬ nci ¬ a  nco a
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia ok ia
nco
Performing Unit Propagation
Also show one step example
C6: ¬ ok ¬ rf
~300
literals rf
C4: ¬ rf ia
C2: ¬ ia  nco
C11: rf
9/20/2000
copyright Brian Williams

47. ITMS: Roll Back by Repairing Conflicts
Flip rf in conflict C11
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco 4 pf 2 a 2
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia ok 1 ia 5
nco 3
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
C11: rf
C2: ¬ ia  nco 2
C4: ¬ rf ia
9/20/2000
copyright Brian Williams

48. ITMS: Roll Back by Repairing Conflicts
Flip ia in conflict C4
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco 4 pf 2 a 2
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia ok 1 ia 5
nco 3
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
C11: rf
C2: ¬ ia  nco 1
C4: ¬ rf ia
9/20/2000
copyright Brian Williams

49. ITMS: Roll Back by Repairing Conflicts
Ready to resupport nco
Flip ok in conflict C6
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco 4 pf 2 a 2
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia ok 1 ia 5
nco 2
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
C11: rf
C2: ¬ ia  nco 1
C4: ¬ rf ia
9/20/2000
copyright Brian Williams

50. ITMS: Roll Back by Repairing Conflicts
Remove consequences of ok
trying local resupport
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco 4 pf 2 a 2
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia ok 1 ia 5
nco 2
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
C11: rf
C2: ¬ ia  nco 1
C4: ¬ rf ia
9/20/2000
copyright Brian Williams

51. ITMS: Roll Back by Repairing Conflicts
Propagate consequences of ok
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco 4 pf a
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia ok 1 ia 5
nco 2
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
C11: rf
C2: ¬ ia  nco 1
C4: ¬ rf ia
9/20/2000
copyright Brian Williams

52. ITMS: Roll Back by Repairing Conflicts
Finally delete C10
C7: ¬ ok ¬ pf
nci
C1: ¬ nci ¬ a  nco 4 pf 2 a 2
C10: ok
C3: ¬ ok  a
C9: ¬ a ¬ ia ok 1 ia 5
nco 2
C8: ¬ rf  ¬ pf
C6: ¬ ok ¬ rf rf
C11: rf
C2: ¬ ia  nco 2
C4: ¬ rf ia
9/20/2000
copyright Brian Williams

53. ITMS significantly decreases extra label changes
9/20/2000
copyright Brian Williams

54. Comparing the ITMS to the LTMS
9/20/2000
copyright Brian Williams