1. Solving the Resolution-Free SAT Problem by Hyper-Unit Propagation in Linear Time
Gábor Kusper
 Research Institute for Symbolic Computation (RISC-Linz)
Johannes Kepler University, Linz, Austria

2. Outline Motivation Hyper-Unit Propagation Resolution-Free SAT Problem
Unicorn-SAT Algorithm
Example
General-Unicorn-SAT Algorithm
Conclusion

3. Motivation Theorema, Prof. Buchberger http://www.theorema.org/
A1, ..., An G
A1, ..., An, G is unsatisfiable

4. Unit Propagation a  b  c a  b a  c a  b  c a  b  c
by a

5. Hyper-Unit Propagation
a  b  c
a  b
a  c
a  b  c
a  b  c
a  b
a  c
a  b  c
HUP
by a  b
Hyper-Unit Prop. (HUP) by assignment A on S is denoted by S[A].

6. Resolution Resolution (tautologies are not allowed): a  b a  b
a  c
Examples
Resolution OK:
b  c a
b  b  c
a  b
a  c
a  b  c
a  b  c
Examples
Resolution-Free:
a  a  c
a  b  c
a  b  c
a  b  c
a  b  c
Tautologies!
Example
Resolution-Free
Clause Set:

7. Definition
A clause set is resolution-free iff resolution cannot be performed on any two clauses of the clause set.

8. L(C): Negation of a Resolution-Mate
a  b  c
Let C be:
negate a literal
A Resolution-Mate of C:
a  b  c
a  b  c
a  b  c
These are clauses!
L(C): negate all but one literal
negate
A Sub-Model generated from C:
a  b  c
a  b  c
a  b  c
These are
assignments!

9. Definition
We obtain a resolution-mate of a clause by negating one literal of it.
A sub-model generated from clause C is the negation of a resolution-mate of C. It is denoted by L(C). By definition L() = .
L(C) is an assignment, L(C) is a clause.
L(C) is a resolution-mate of clause C.

10. Res.-Mate is not in S
If S is resolution-free, S, CS and B is a resolution-mate of C. Then BS.
Proof: C resolves with B!
a  b  c
a  b  c
a  b  c
a  b  c
a  b  c
a  b  c

11.  is not in S after Sub-Model Propagation
If S is resolution-free, S, CS and C is minimal (B = L(C) ,i.e., B is a resolution-mate of C). Then S[L(C)].
Proof: Since C is minimal it suffices to show that B  S. It is known from the previous slide.
a resolution- mate:
S is resolution-free:
a sub-model:
a  b
a  b  c
a  b  c
minimal:
subsumed:
not subsumed:
a  b
a  b
a  b  c
a  b  c
a  b  c
a  b  c

12. In Picture START input: S S[A] S[A] HUP S is L[C] = A := a  b by A
resolution-free
and S
HUP
by A
L[C] = A := a  b
Let A negation of B
B := a  b
a  b
if S[A] =  then
M := A
Let B a res.-mate of C
Let C be a min.
clause from S
C:= a  b
END

13. HUP Preserves Resolution-Free-ness
If S is resolution-free and A is an assignment. Then S[A] is resolution-free.
Proof: Assume S[A] is not resolution-free. We show that this contradict with S is resolution-free. S:
S[A]:
... y ... x ...
...
... y ... x ...

14. Unicorn-SAT Algorithm
START
input: S
i := 1
S[Ai] is
res.-free
S[Ai]
S := S[Ai], i := i+1
S is
resolution-free
and S
HUP
by Ai
Ai := a  b
Let Ai negation of B
B := a  b
a  b
if S[Ai] =  then
M := A1A2...Ai
Let B a res.-mate of C
Let C be a min.
clause from S
C:= a  b
END

15. Termination If S  , S and CS. Then S[L(C)]
has fewer clauses than S and
contains fewer variables than S.
Proof: Hyper-Unit propagation inherits this property from unit propagation.

16. Example sub-model a  b  c a  b  c a  b  d  e a  b  c  d
a  d  e
a  b  c  d  e
a  b  c  d  e
a  b  c  d  e
b  c  d  e
sub-model
a  b  c
HUP
sub-mo.
d  e
d  e
d  e
HUP
a  b  c  d  e

17. Time Complexity # variables : n # clauses : m O(nm) Argument:
HUP by Ai can be simulated by length(Ai) UP.
If we need k HUP then Summa(length(Ai), i:=1..k) = n.
So we can simulate k HUP by n UP.
Since UP is O(m) time method Unicorn-SAT is O(nm).

18. General-Unicorn-SAT Algorithm
if i = 0 then unsatisfiable
satisfiable
if Si
i := i-1
i := i+1
if Si+1 = 
if Si+1
Si := Sires(C,L(C))
Correct the guess! Si
Si+1
if Si then
Ai := L(C), where
C is a min. clause form Si.
Guess a solution!
if Si+1
i := i+1
i := i-1
START: i := 1, S0 := S, A0 := , Si := Si-1[Ai-1]

19. Conclusion DPLL like algorithms General-Unicorn-SAT
use unit propagation
incrementally construct a solution
General-Unicorn-SAT
uses hyper-unit propagation
guesses a sub-model & correct it if necessary

20. Thank you for your attention!