(baby) Bio
• I am Patrick Prosser
• I am a senior lecturer at Glasgow
• I teach AF2 & CP4
• I am a member of
• the algorithms group
• the apes (distributed, not disbanded)
• I am a Glaswegian
This is all that I am allowed to tell you
10 years of conflict-directed backjumping
Plan of the talk
• What’s a csp?
• The simplest algorithm (BT) and its behaviour
• Some improvements (BJ, GBJ)
• A better way (CBJ)
• Improvements to CBJ
• Strange things about CBJ
• k-inconsistencies
• the bridge and the long jump
• value ordering and insolubility
• So, what about CBJ?
• some say its good
• some say it’s a waste of time
• who’s using it
• who’s not using
• Where’s it going
• QBF?
What’s a csp?
<V,D,C>
• a set of variables
• each with a domain of values
• a collection of constraints (I’m going to assume binary)
• assign each variable a value from its domain to satisfy the constraint
Example of a csp
A
B
E
C
D
H
F
G
3 colour me!
Example of a csp
A
B
E
C
D
H
F
G
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Why did it do that?
That was dumb!
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
Example of a csp
A
Variables and Instantiation Order
Checking back
B
Va
E
C
D
H
Vb
Vc
Vd
F
G
Ve
Vf
1 = red
2 = blue
3 = green
Vg
Vh
That was good old fashioned chronological backtracking
• instantiate a variable
• check current against past variables to see if okay
• current? past? Future?
• If not okay try another value
• If no values left go back to previous variable
Example of a csp
A
What would have happened if we had
the E* intermediate variables?
i.e. it falls back on E4, then E3, towards E?
B
E
C
D
H
F
1 = red
2 = blue
3 = green
E1
E2
G
E3
E4
It’s all just depth first search, right?
past variable v[h]
conflict with v[h]
future variable
v[j]
BT Thrashes!
BT
Assume that when we instantiate the current variable
v[i] we check against all past variables from v[1] to v[i-1]
to check if consistent
consistent := false
for x in domain[i] while not(consistent) // find a consistent value
begin
consistent := true
v[i] := x
for h in (1 .. i-1) while consistent
// check backwards
begin
consistent := (check(v[i],v[h])
end
if not(consistent)
then delete(x,domain[i])
end
// did we find a good value?
John Gaschnig’s BJ
Make a small modification to BT. Remember the last variable we check against
consistent := false
maxCheck[i] := 0
for x in domain[i] while not(consistent) // find a consistent value
begin
consistent := true
v[i] := x
for h in (1 .. i-1) while consistent
// check backwards
begin
consistent := (check(v[i],v[h])
maxCheck := max(h,maxCheck[i])
end
if not(consistent)
then delete(x,domain[i])
end
// did we find a good value?
If the loop terminates with consistent false
then jump back to v[h], where h := lastCheck[i]
This is then the deepest variable we failed against
John Gaschnig’s BJ
consistent := false
maxCheck[i] := 0
for x in domain[i] while not(consistent) // find a consistent value
begin
consistent := true
v[i] := x
for h in (1 .. i-1) while consistent
// check backwards
begin
consistent := (check(v[i],v[h])
maxCheck := max(h,maxCheck[i])
end
if not(consistent)
then delete(x,domain[i])
end
// did we find a good value?
If the loop terminates with consistent true ...
lastCheck[i] = i-1
we then step back!
BJ jumps then steps back. It can only jump after it has moved forwards!
BJ reduces thrashing
maxCheck[i]
1
0
2
1
3
2
4
3
5
4
6
4
Jump back to v[4]
then step back
Rina Dechter’s GBJ
Graph-based backjumping, exploits topology of constraint graph
If there are no values remaining for v[i]
Jump back to v[h], where v[h] is the deepest variable
connected to v[i] in the constraint graph
If there are no values remaining for v[h]
Jump back to v[g], where v[g] is the deepest variable
connected to v[h] or v[i] in the constraint graph
If there are no values remaining for v[g]
Jump back to v[f], where v[f] is the deepest variable
connected to v[g] or v[h] or v[i] in the constraint graph
What happens if constraint graph is a clique?
We want something that can jump and jump again,
something that takes into consideration what caused
a dead-end, not something that just looks at the
topology of the constraint graph
Combine BJ and GBJ
CBJ
Remember your conflicts, and when you have used them forget them.
When we instantiate v[i] := x and
check(v[i],v[h]) and
it fails
• v[i] is in conflict with v[h]
• add h to the set confSet[i]
confSet[i] is then the set of past variables that conflict
with values in the domain of v[i]
CBJ
Conflict-directed backjumping, exploits failures within the search process
If there are no values remaining for v[i]
Jump back to v[h], where v[h] is the deepest variable in conflict with v[i]
The hope: re-instantiate v[h] will allow us to find a good value for v[i]
If there are no values remaining for v[h]
Jump back to v[g], where v[g] is the deepest variable in conflict with v[i] or v[h]
The hope: re-instantiate v[g] will allow us to find a good value for v[i] or a
good value for v[h] that will be good for v[i]
If there are no values remaining for v[g]
Jump back to v[f], where v[f] is the deepest variable in conflict with v[i] or v[h] or v[g]
The hope: re-instantiate v[f] will allow us to find a good value for v[i] or a
good value for v[h] that will be good for v[i] or a
good value for v[g] that will be good for v[h] and v[i]
What happens if: constraint graph is dense, tight, or highly consistent?
CBJ
consistent := false
confSet[i] := {0}
for x in domain[i] while not(consistent) // find a consistent value
begin
consistent := true
v[i] := x
for h in (1 .. i-1) while consistent
// check backwards
begin
consistent := (check(v[i],v[h])
if not(consistent) then confSet[i] := confSet[i]  {h}
end
if not(consistent)
then delete(x,domain[i])
end
CBJ
When jumping back from v[i] to v[h] update conflict sets
confSet[h] := confSet[h]  confSet[i] \ {h}
confSet[i] := {0}
That is, when we jump back from v[h] jump back to a
variable that is in conflict with v[h] or with v[i]
Throw away everything you new on v[i]
Reset all variables from v[h+1] to v[i] (i.e. domain and confSet)
CBJ
1
2
conflict set
3
4
{2,0}
5
6
{4,1,0}
CBJ
1
2
conflict set
3
4
5
6
{2,1,0}
CBJ (reduce thrashing)
1
2
3
4
{2,1,0}
5
6
Jump back to deepest past variable
in confSet (call it h) and then
combine confSet[i] with confSet[h]
•History:
•Konkrat and V Beek,
•Gent and Underwood
Forward Checking
1
2
3
4
5
6
7
8
9
NOTE: arrows go forward!
1
2
3
4
5
6
7
8
9
Check Forwards, Jump Back!
1
2
3
4
5
6
7
8
9
•There are no values in cd[6] compatible with v[9]
• get more values into cd[9] (undo v[1]?) OR
• get more values into cd[6] (undo v[4])
• … and if that doesn’t work?
undo v[3] so cd[4] gets value compatible with
cd[6] that is then compatible with cd[9]
CBJ Variants
BM-CBJ, FC-CBJ, MAC-CBJ
CBJ DkC
If we jump from v[i] to v[h] and confSet[i] = {0,h}
then remove value(v[h]) from domain(h)
value(v[h]) is 2-inconsistent wrt v[i]
If we jump from v[h] to v[g] and confSet[h] = {0,g}
then remove value(v[g]) from domain(g)
value(v[g]) is 3-inconsistent wrt v[i] and v[h]
If we jump from v[g] to v[f] and confSet[g] = {0,f}
then remove value(v[f]) from domain(f)
value(v[f]) is 4-inconsistent wrt v[i] and v[h] and v[g]
What happens if the problem is highly consistent?
See JAIR 14 2001, Xinguang Chen & Peter van Beek
CBJ ATMS
If we jump from v[i] to v[g] and confSet[h]  {0 .. g-1}
then do NOT reset domain(h) and
do NOT reset confSet(h)
Consider the past variables as assumptions and confSet[i]
as an explanation
Down side, we have more work to do.
This is a kind of learning (what kind?)
CBJ ~ DB
confSet[x,i] gives the past variable in conflict with v[i] := x
Finer grained:
on jumping back we can deduce better what values to return to domains
Down side, we have more work to do.
This is an algorithm between CBJ and DB
Funny things about cbj
The bridge and the long jump
Funny things about cbj
Value ordering on insoluble problems can have an effect
Problem: V1 to V7, each with domain {A,B}
nogoods {(1A,7A),(3A,7B),(5A,7B),(6A,7A),(6A,7B),(6B,7A),(6B,7B)}
Var
V1
V2
V3
V4
V5
V6
V7
Val confSet
A
A
A
A
A
A
A/B {1,3}
Var
V1
V2
V3
V4
V5
V6
V7
Val confSet
A
A
B
A
A
A
A/B {1,5}
Var
V1
V2
V3
V4
V5
V6
V7
Val confSet
A
A
B
A
B
A
A/B {6}
Finally V6 has no values and cbj jumps to V0
Var
V1
V2
V3
V4
V5
V6
V7
Val confSet
A
A
B
A
B
B
A/B {6}
Funny things about cbj
Value ordering on insoluble problems can have an effect
Problem: V1 to V7, each with domain {A,B}
nogoods {(1A,7A),(3A,7B),(5A,7B),(6A,7A),(6A,7B),(6B,7A),(6B,7B)}
We now order domains and choose B then A!
Var
V1
V2
V3
V4
V5
V6
V7
Val confSet
B
B
B
B
B
B
A/B {6}
Var
V1
V2
V3
V4
V5
V6
V7
Val confSet
B
B
B
B
B
A
A/B {6}
Finally V6 has no values and cbj jumps to V0
Value ordering made a difference to an insoluble problem!
Conflicting claims
Smith & Grant IJCAI95:
CBJ helps minimise occurrence of EHP’s
random problems as evidence
Bessier & Regin CP96:
CBJ is nothing but an overhead
random problems as evidence
Chen & van Beek JAIR 2001:
CBJ is a tiny overhead
When it makes a difference it is a HUGE difference
random & real problems as evidence
New CBJ
I believe all state of the art sat solvers are using cbj
(or have rediscovered cbj but don’t know it)
CBJ for QSAT: see recent AIJ
conflict and solution directed!
Who is not using cbj?
Constraint programming!
We don’t jump and we don’t learn
Is speed everything?
No
How about explanations and retraction?
Why is cbj not in CP?
Need to propagate laterally (see MAC-CBJ tech report)
but this is no big deal
Need to get explanations out of constraints!
Not just writing a good constraint propagator
but a good constraint explainer!
Maybe there is not yet the demand for retraction and explanation
(but I don’t believe that)
So?
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Paper rejected from IJCAI91 (written in 1990)
I was a Lisp programmer at the time (it shows)
I think the experiments were very good (so there!)
Nice study of influence of topological parameter on search cost.
In conclusion I forgot to say CBJ was new … why?
I like BMJ, it is cool (I was smart for 1 day)
I think CBJ is pretty (natural, discovered, not invented)
I like FC-CBJ (I can understand it)
I identify work to be done (researchers love that (why?))
… and I make errors re-dvo’s (researchers love that (why?))
I put my results in perspective (trash them :)
I got encouragement (Nadel) and help (Ole and Peter)
I got a whole load of background (Rina)
But it hurt … why did it take 3 years to get somebody to read it?
It is still alive! We are not done yet.