CSPs: Adding Structure to
SAT
George Katsirelos
Fahiem Bacchus
University of Toronto
1
Introduction
● Finite domain Constraint Satisfaction Problems
(CSPs).
● Formally equivalent to SAT
● Important practical differences.
∎ Different algorithmic techniques have been
developed in the two areas.
∎ Understanding these can help cross fertilize both
fields.
2
Background
The SAT and CSP Formalisms
3
Formalism
SAT
Boolean Variables
{0,1} values for each
variable.
CSPs
Multi-Valued variables
Possibly distinct
domain of values for
each variable.
Clauses
restricting the
possible assignments
of values to variables
Constraints
restricting the possible
assignments of values
to variables
4
Formalism
● SAT = h V, Ci
∎ V = {V1, V2, …, Vn} is a set of Boolean variables
∎ C = {c1, c2, …, ck} a set of clauses.
● CSP = h V, D, C i
∎ V = {V1, V2, …, Vn} is a set of multi-valued variables
∎ D = {D1, D2, …, Dn} is a set of value domains, with Di being
the domain of values for variable Vi
∎ C = {C1, C2, …, Ck} is a set of constraints.
● In both CSP and SAT the aim is to find an assignment
of values for all of the variables:
∎ In SAT these values must satisfy the clauses
∎ In CSPs these values must satisfy the constraints.
5
Constraints
● A constraint C(X1,X2, …, Xk) over the variables
X1, …, Xk is a Boolean function
∎ It maps assignments to these variables to {0,1}
C(X1,X2, …, Xk) : DX1 £  £ DXk  {0,1}
● If a tuple of assignments maps to 1, then
these assignments satisfy the constraint,
otherwise these assignments the falsify the
constraint.
6
Extensionally vs Intensionally
Represented Constraints
● We can specify the constraint with a table
∎ C(X,Y,Z) with DX = DY = DZ = {1, 2, 3}
X
Y Z
C(X,Y,Z) X
1
1 1 1
2
1
1 2 1
1
Y Z
C(X,Y,Z)
X
Y Z C(X,Y,Z)
1 1 0
3
1 1 0
2
1 2 0
3
1 2 0
1 3 1
2
1 3 0
3
1 3 0
1
2 1 0
2
2 1 0
3
2 1 0
1
2 2 1
2
2 2 1
3
2 2 0
1
2 3 1
2
2 3 1
3
2 3 0
1
3 1 0
2
3 1 0
3
3 1 0
1
3 2 0
2
3 2 0
3
3 2 0
1
3 3 1
2
3 3 1
3
3 3 1
7
Extensionally vs Intensionally
Represented Constraints
● Thus we can represent the constraint as a set
of satisfying assignment tuples
X
Y Z
C(X,Y,Z) X
1
1 1 1
2
1
1 2 1
1
Y Z
C(X,Y,Z)
X
Y Z C(X,Y,Z)
1 1 0
3
1 1 0
2
1 2 0
3
1 2 0
1 3 1
2
1 3 0
3
1 3 0
1
2 1 0
2
2 1 0
3
2 1 0
1
2 2 1
2
2 2 1
3
2 2 0
1
2 3 1
2
2 3 1
3
2 3 0
1
3 1 0
2
3 1 0
3
3 1 0
1
3 2 0
2
3 2 0
3
3 2 0
1
3 3 1
2
3 3 1
3
3 3 1
8
Extensionally vs Intensionally
Represented Constraints
● Or as a set of falsifying assignment tuples
X
Y Z
C(X,Y,Z) X Y Z
1
1 1 1
1
C(X,Y,Z)
X
Y Z C(X,Y,Z)
2 1 1 0
3
1 1 0
1 2 1
2 1 2 0
3
1 2 0
1
1 3 1
2 1 3 0
3
1 3 0
1
2 1 0
2 2 1 0
3
2 1 0
1
2 2 1
2 2 2 1
3
2 2 0
1
2 3 1
2 2 3 1
3
2 3 0
1
3 1 0
2 3 1 0
3
3 1 0
1
3 2 0
2 3 2 0
3
3 2 0
1
3 3 1
2 3 3 1
3
3 3 1
9
Extensionally vs Intensionally
Represented Constraints
● Extensional representations specify the
constraint as an explicit list of satisfying
assignments (or falsifying assignments).
● Extensional representations were used in the
2005 CSP solver competition. But are almost
never used in practice.
∎ The extensional representation becomes very
large, growing exponentially with the number of
variables the constraint is over.
10
Extensionally vs Intensionally
Represented Constraints
● Constraint could also be represented
intensionally as an algorithm for computing
the Boolean function.
11
Extensionally vs Intensionally
Represented Constraints
X Y Z
C(X,Y,Z)
X Y Z
C(X,Y,Z)
X Y Z
C(X,Y,Z)
1 1 1 1
2 1 1 0
3 1 1 0
1 1 2 1
2 1 2 0
3 1 2 0
1 1 3 1
2 1 3 0
3 1 3 0
1 2 1 0
2 2 1 0
3 2 1 0
1 2 2 1
2 2 2 1
3 2 2 0
1 2 3 1
2 2 3 1
3 2 3 0
1 3 1 0
2 3 1 0
3 3 1 0
1 3 2 0
2 3 2 0
3 3 2 0
1 3 3 1
2 3 3 1
3 3 3 1
=X·Y·Z
12
Extensionally vs Intensionally
Represented Constraints
● Intensional representations are typical in
practice.
● To specify a CSP problem in a CSP solver one
supplies subroutines to implementing the
constraints of the problem.
∎ Commercial CSP solvers supply a large library of
predefined common constraints.
∎ You then simply specify the variables of the CSP,
their domains, and the constraints that are over
them.
13
Translating between SAT and
CSPs
● Further insight into the relation between
SAT and CSPs is provided by looking at
how we can translate between the
formalisms.
14
SAT  CSP
● Translating in this direction is trivial
● Each SAT variable becomes a CSP
variable, with {0,1} as its domain of
values.
● Each clause is equivalent to a Boolean
function from the variables it is over
∎ (x, y, -z)
A function mapping (x=0,y=0,z=1) to 0, all
other assignments of x,y,z to 1.
15
CSP  SAT
● The other direction requires two steps
1. Converting the multi-valued variables into
a set of Boolean assignment variables.
2. Converting the constraints into clauses
over the assignment variables.
16
CSP  SAT
Converting the Multi-Valued Variables
● Let X be a CSP variable with
Dx = {d1, …, dm}
● We create m Boolean assignment
variables x1, x2, …, xm these have the
the interpretation
xi is true iff X=di.
17
CSP  SAT
● The CSP variable X must have a value and
it must have a unique value.
● Hence the Boolean assignment variables
x1, x2, …, xm associated with a particular
CSP variable are mutually exclusive and
exhaustive.
● This is captured by adding the clauses
∎ (x1, x2, …, xm)
∎ (-xi, -xj) for all (i  j)
X must have a value
X has a unique value
18
CSP  SAT
Converting the Constraints into Clauses
● Now we convert the constraints to
clauses.
● Each falsifying assignment tuple in the
constraint’s extensional representation
is equivalent to a clause.
● So a constraint becomes a set of clauses,
one for each falsifying assignment.
19
CSP  SAT
● Each falsifying tuple
X YZ
X Y Z
X YZ
1 11 1
2 1 1 0
3 110
1 12 1
2 1 2 0
3 120
1 13 1
2 1 3 0
3 130
1 21 0
2 2 1 0
3 210
1 22 1
2 2 2 1
3 220
1 23 1
2 2 3 1
3 230
in we get a clause
1 31 0
2 3 1 0
3 310
1 32 0
2 3 2 0
3 320
(-x1 _ -y2 _ -z1)
1 33 1
2 3 3 1
3 331
is a set of
assignment variables
that cannot be
simultaneously true.
● E.g.. –(x1 ^ y2 ^ z1)
● Pushing the negation
20
CSP  SAT
● There are various optimizations that can
be applied to this basic translation.
● Specific constraints admit more compact
encodings.
21
Modeling with CSPs
22
Modeling with CSPs
● CSPs offer
∎ Multi valued variables: more natural for
modeling real problems.
∎ Constraints over groups of variables that
permit a more natural encoding of the
constraints of the problem.
● Industrial applications are much easier
to formalize using CSPs, and the range
of application of CSP technology in
industry far exceeds that of SAT.
23
●
N-Queens
Q
Place N queens on
an NxN chess
board so that no
queen can attack
any other queen.
Q
Q
Q
24
N-Queens
● Place N queens on
an NxN chess
board so that
queen can attack
any other queen.
Q1
Q2
Q3
Q4
● N, Queen variables, one
for each column
25
N-Queens
● Place N queens on
Q1=1 Q2=1
an NxN chess
board so that
queen can attack
any other queen.
Q2=2
Q4=2
Q2=3
Q2=4 Q3=4
Q1
Q2
Q3
Q4
● N values for each variable:
∎ The row we place that column’s
queen on.
26
N-Queens
● Constraints
∎ AllDiff(Q1, …, QN) each Queen has a
unique value (can’t be in the same row)
∎ Cij(Qi,Qj): |Qi – Qj|  |i-j| (for each i  j)
● can’t be on same diagonal
27
Modeling with CSPs
● A SAT encoding of N-Queens more
complex to specify.
● SAT encodings almost impossible to
generate by hand.
28
Modeling with CSPs
● Modeling using the richer language of
CSPs, translate to SAT (automatically),
solve using standard SAT solver.
∎ Understanding the pros and cons of this
approach gives us further insight into the
algorithmic differences between CSP and
SAT solvers.
29
Solving CSPs
30
Backtracking Search
● SAT and CSP backtracking solvers differ
in the three main parts of backtracking
∎ Propagation as we descend the search tree
∎ Learning as we ascend from failed subtrees
∎ Heuristics for guiding the branching
decisions
31
Translation to SAT




The clause learning in SAT solvers can
be exploited.
The mutually exclusive and exhaustive
clauses for the multi-valued variables
are not fully exploited.
Branching heuristics insensitive to CSP
structure.
Unit propagation weaker than
propagation methods employed in CSP
solvers.
32
Disadvantages:
(a) Clauses for Multi-valued
Variables
33
Disadvantages:
(a) Clauses for Multi-valued Variables
● (x1, x2, …, xm)
X must have a value
● (-xi, -xj) for all (i  j)
X has a unique value
● These clauses impose a useful structure
on the assignment variables.
34
Disadvantages:
(a) Clauses for Multi-valued Variables
● In general, the disjunction of any subset
of positive literals is equivalent to the
conjunction of the complimentary set of
negative literals. E.g., if m=4
∎ x1 _ x2 ´ –x3 ^ –x4
X=1
∎ x3
X=1
X=2
X=3
´ -x1 ^ -x2 ^ -x4
X=2
X=3
X=4
X=4
35
Disadvantages:
(a) Clauses for Multi-valued Variables
● This structure could be exploited in
various ways. For example,
● Two negative assignment literals 
clause is redundant
∎ (y1, y2, -x1, -x2, -z3)
subsumed by (-x1,-x2)
36
Disadvantages:
(a) Clauses for Multi-valued Variables
● Negative assignment literal  remove all
positive literals from same variable.
∎ (y1, y2, -y3, x1, -x2, -z3)
∎ Resolve with (-y1, -y3) and (-y2, -y3) to
obtain subsuming clause (-y3, -x2, -z3).
37
Disadvantages:
(a) Clauses for Multi-valued Variables
● Sets of clauses can be replaced by a
single clause.
∎ Dx = Dy = {1, 2, 3, 4}
(R,
(R,
(R,
(R,
-x1, -y1) (R, -x1, -y2)
-x2, -y1) (R, -x2, -y2)
-x1, -y1) (R, -x1, -y2)
-x2, -y1) (R, -x2, -y2)
Equivalent to single clause
(R, x3, x4, y3 , y4).
38
Disadvantages:
(a) Clauses for Multi-valued Variables
● (R, x3, x4, y3 , y4) ´
(R, (-x2 ^ -x1), (-y2 ^ -y1))
● Multiply this out and you get 8 clauses.
39
Disadvantages:
(b) Heuristics
40
Disadvantages: (b) Heuristics
● Under unit propagation contradictions
arise when –x is inferred in a context
where x is already true
∎ This causes some clause to be falsified
(conflict clause).
41
Disadvantages: (b) Heuristics
● With multi-valued variables we always
have
∎ xi ´ -x1  ^ –xi-1 ^ –xi+1  ^ -xm
● Hence conflicts arise only from refuting
all values from some CSP variable’s
domain
∎ -x1  ^  ^ -xm
42
Disadvantages: (b) Heuristics
● In CSP solvers the number of unrefuted
values of a variable is always considered
in the branching heuristic.
● In a SAT solver we shouldn’t choose to
branch on xi without considering the
status of other associated assignment
variables.
Ansótegui1 et al
2003.
43
Disadvantages:
(c) Propagation
44
Disadvantages: (c) Propagation
● Unit Prop in a SAT solver on the clauses
generated by a constraint is equivalent
to Forward Checking in CSPs.
● Forward Checking. Wait until all but one
variable of the constraint is instantiated,
and then prune incompatible values
from the domain of the sole remaining
uninstantiated variable.
45
Disadvantages: (c) Propagation
● (-x1 _ -y2 _ -z1)
● (-x1 _ –y3 _ –z1)
● …
X YZ
X Y Z
X YZ
1 11 1
2 1 1 0
3 110
1 12 1
2 1 2 0
3 120
1 13 1
2 1 3 0
3 130
1 21 0
2 2 1 0
3 210
1 22 1
2 2 2 1
3 220
1 23 1
2 2 3 1
3 230
1 31 0
2 3 1 0
3 310
1 32 0
2 3 2 0
3 320
1 33 1
2 3 3 1
3 331
46
Disadvantages: (c) Propagation
● Each clause contains one negated
assignment literal from each CSP
variable in the constraint.
● To make the clause unit one has to make
all but of these assignment variables
true:
∎ Equivalent to assigning the corresponding
CSP variable
∎ x1 ´ X=1, y2 ´ Y=2
47
Disadvantages: (c) Propagation
● Then unit propagation will falsify all
assignments to the remaining
unassigned CSP variable that would
violate the constraint
∎ (-x1 _ –y3 _ –z1), (-x1 _ –y3 _ –z2)
X=1 ^ Y=3  Z  1 & Z  2
48
Disadvantages: (c) Propagation
● However, in practice, FC does not
perform particularly well.
● A superior form of propagation is GAC.
49
GAC (Macworth & Freuder 1977-79)
● Given a constraint C(X1,X2, …, Xk)
∎ di 2 DX is supported (in C)
i
if there exists a set of assignments
{X1 = d1, …, Xi = di ,…, Xk = dk}
that satisfies C:
C(X1=d1, …, Xk = dk) = 1.
∎ This set is called a support for di.
50
GAC (Macworth & Freuder 1977-79)
● Supports for X = 1 (x1)
● Supports for x3
If 3 is removed from
the domain of Z, i.e., -z3
becomes true, x3 will
loose its only support.
X YZ
C(X,Y,
Z)
X YZ
C(X,Y,
Z)
X YZ
C(X,Y,
Z)
1 11 1
2 11 0
3 110
1 12 1
2 12 0
3 120
1 13 1
2 13 0
3 130
1 21 0
2 21 0
3 210
1 22 1
2 22 1
3 220
1 23 1
2 23 1
3 230
1 31 0
2 31 0
3 310
1 32 0
2 32 0
3 320
1 33 1
2 33 1
3 331
51
GAC (Macworth & Freuder 1977-79)
● The constraint C(X1,X2, …, Xk) is said to
be GAC if for all of its variables Xi every
value in their domain is supported (in C).
● We can make C(X1,X2, …, Xk) GAC by
removing all unsupported values from
the domains of its variables.
52
GAC (Macworth & Freuder 1977-79)
● GAC propagation is the dynamic process
of making all of the constraints GAC.
∎ If d 2 DX is pruned from the domain of X
while making C1 GAC.
∎ Then the other constraints over X must
have GAC reestablished.
∎ This might prune values of other variables,
and their constraints in turn must be made
GAC once again.
53
GAC (Macworth & Freuder 1977-79)
● During search we make all constraints
GAC at the root.
● The assignment X=1 means X  2, X  3,
…
● Thus constraints over X have to have
GAC reestablished by GAC propagation.
● Reestablishing GAC at every node is
called Maintaining GAC.
54
GAC Propagation in the SAT encoding
● By using extra variables in the SAT
encoding we can establish GAC with Unit
Propagation.
● But Unit Propagation on the standard
encoding is less powerful than GAC.
Bessière et al.
2003
55
GAC Propagation in the SAT encoding
● The power of GAC can be characterized
using the notion of prime implicates.
∎ If T is a set of clauses, then the clause c is a
prime implicate if
● c is non-tautological
● T²c
● T ⊭ c’ for any c’ that is a subset of c
56
GAC Propagation in the SAT encoding
● Let TC be the set of clauses of the
constraint C, along with the mutually
exclusive and exhaustive clauses for
each of the C’s variables.
● Now we replace TC by the prime
implicates of TC, PIc
Theorem: Unit prop over PIc achieves
precisely GAC propagation on the
constraint C.
57
GAC Propagation in the SAT encoding
● Note that GAC is local to the constraint.
Communication between constraints
occurs only through unit implicants
(pruned values).
● So GAC is complete local inference for
units.
58
GAC Propagation in the SAT encoding
● Achieving GAC over a generic constraint
C(X1,X2, …, Xk) requires time exponential
in K.
● However, there is however a huge body
of knowledge in the CSP literature on
how to achieve GAC on particular
constraints in time polynomial in K.
∎ These methods (called propagators) exploit
the special structure of the constraint.
59
Translation to SAT Summary
● The fundamental problems
∎ is the size of the encoding
∎ The vast body of knowledge about
propagators for GAC cannot be exploited.
● Exploiting propagators GAC has much in
common with exploiting specialized
theories in SMT.
60
CSP Solvers
61
Using a CSP solver



Constraints can be represented intensionally
and propagators can be exploited.
The multi-valued variable structure, and
information about the constraints can be
exploited for branching decisions.
Learning in CSP solvers is much weaker than
clause learning in Sat solvers, and it doesn’t
integrate well with GAC propagation.
62
Learning
● We can improving learning in CSPs and
achieve a better integration with GAC.
● We can also integrate GAC propagators
with learning using ideas that are
essentially identical to those used in SMT
● These ideas were developed
independently.
63
Learning
● In CSPs learning from failed subtrees has
a long tradition. Learning is typically
called “nogood” recording.
64
NoGoods
● A NoGood in CSPs is a set of
assignments that cannot be extended to
a solution.
∎ (X=3, Y=2, Z=1)
∎ Translating this to SAT we get
-(x3 ^ y2 ^ z1) ´ (-x3, -y2, -z1)
●  Nogoods are negative clauses
∎ (clauses containing only negative literals).
65
Negative Resolution
● Restricting learning to NoGoods (negative
clauses) restricts the solver’s “resolution
power” to Negative Resolution.
∎ Negative resolution: every resolution step involves
a clause negative clause.
● Negative resolution not as powerful as general
resolution
∎ CSP solvers sometimes suffer a super-polynomial
slowdown over SAT solvers running on the SAT
encoding.
Mitchell 2003
Katsirelos PhD thesis
66
Negative Resolution
● As a result of this restriction to learning
negative clauses learning is hardly ever
used CSP solvers in practice.
● Learning negative clauses is also
produces particularly ineffective clauses
from GAC.
67
Integrating SAT style Clause
Learning
● GAC prunes domain values. It forces
negated assignment literals.
● Like SMT all we need to do is to label
those literals with clauses.
68
Clause Learning in CSP solvers
X = 1  CHOICE
X  2  (X  1, X  2) (variable can only have one value)
Y  1  (Y  1, X=2) (Non-negative clause reason from GAC)
Z  1  (Z  1, X=2)
A = 2  CHOICE
A1 …
A3…
X  1  (X  1, A  2) (conflict clause from
constraint over X, and A)
69
Clause Learning in CSP solvers
● We can resolve backwards from a conflict
along the implication trail from a conflict
to learn various types of new clauses
● I.e., we can apply standard SAT clause
learning techniques.
70
Computing Clauses from GAC
● With Unit Prop each literal is implied by a
specific clause that became unit, so the clause
for labeling an implied literal is obvious.
● With GAC a value is pruned (an assignment
variable is made false) as the result of many
different clauses of the constraint.
∎ In particular, a value is pruned by GAC when it
looses all of its supports.
∎ Each support (which is a tuple of assignments to
the variables of the Constraint) is lost when one of
its assignments is made false.
71
Computing Clauses from GAC
● Supports for X = 1
● E.g.,
(X=1 ^ Y=1 ^ Z=1).
This support can
be lost if 1 is
pruned from the
domain of Y or
from Z. (Y1, or
Z1)
X YZ
C(X,Y,
Z)
X YZ
C(X,Y,
Z)
X YZ
C(X,Y,
Z)
1 11 1
2 11 0
3 110
1 12 1
2 12 0
3 120
1 13 1
2 13 0
3 130
1 21 0
2 21 0
3 210
1 22 1
2 22 1
3 220
1 23 1
2 23 1
3 230
1 31 0
2 31 0
3 310
1 32 0
2 32 0
3 320
1 33 1
2 33 1
3 331
72
Computing Clauses from GAC
● If GAC on this constraint prunes X=1, a
reason for this pruning is a set of
currently true non-assignments that hits
all of X=1’s supports.
73
Computing Clauses from GAC
● Supports for X = 1
X YZ
C(X,Y,
Z)
X YZ
C(X,Y,
Z)
X YZ
C(X,Y,
Z)
1 11 1
2 11 0
3 110
1 12 1
2 12 0
3 120
1 13 1
2 13 0
3 130
1 21 0
2 21 0
3 210
1 22 1
2 22 1
3 220
1 23 1
2 23 1
3 230
Y1Æ Z2 Æ Z3!X1
1 31 0
2 31 0
3 310
1 32 0
2 32 0
3 320
(Y=1, Z=2, Z=3, X1)
1 33 1
2 33 1
3 331
E.g.,Y1, Z2, Z3
covers all of X=1’s
supports in this
constraint:
´
74
Computing Clauses from GAC
● We put this implication on the trail
X1  (X1, Y=1, Z=2, Z=3)
● Note that
∎ this is a non-negative clause
∎ We can compute this clause on the fly from
an extensional representation of the
constraint.
● There is no need to precompute and store all
such possible “pruning” clauses.
75
Computing clauses from
Intensional Constraints
● How do we obtain clausal reasons from
GAC propagators?
● We can no longer find a hitting set for
the supports of the value, there is no
explicit representation of these
supports.
76
Example: All Different
● AllDiff(X1, …, Xn) is satisfied only by
tuples of assignments to the Xi that are
all different, i.e.,
∎ ij ! XiXj
77
All Different
● A way of enforcing GAC on AllDiff in
poly-time was the probably the first
propagator developed in the CSP
literature. Regin [1994].
● The method utilizes maximum
matchings in bipartite graphs.
● Since then dozens of propagators have
been developed.
78
The power of propagators
● DPLL must take exponential time on the
pigeon hole problem PHP: this problem is hard
for general resolution.
● PHP can be encoded as a single
AllDiff(P1, …, Pn) each with domain of values
{1, …, n-1}.
● This constraint has no satisfying tuples so
every value will be pruned by GAC.
● A CSP solver can solve this problem in
polynomial time. GAC propagation at the root,
no search.
79
Clausal Reasons from AllDiff
● How do we obtain a clausal reason for a
value pruned by AllDiff?
● For Alldiff a value is pruned from a
variable domain only when that value is
consumed by some other variables. That
is, the value must be used by some other
variable.
80
Clausal Reasons from AllDiff
DX = {1, 2, 3,4},
DY = {1, 2, 3,4},
DZ = {1, 2, 3},
DW = {1, 4, 5}
Initially all values
are supported
DX = {1, 2, 3,4},
DY = {1, 2, 3,4},
DZ = {1, 2, 3},
DW = {1, 4, 5}
Prune 4 from
domain of X and Y
81
Clausal Reasons from AllDiff
DX = {1, 2, 3,4},
DY = {1, 2, 3,4},
DZ = {1, 2, 3},
DW = {1, 4, 5}
1, 2, 3 are
consumed by X, Y,
Z. (Hall interval)
DX = {1, 2, 3,4},
DY = {1, 2, 3,4},
DZ = {1, 2, 3},
DW = {1, 4, 5}
W cannot be
assigned 1 since
that value is
consumed.
82
Clausal Reasons from AllDiff
DX = {1, 2, 3,4},
DY = {1, 2, 3,4},
DZ = {1, 2, 3},
DW = {1, 4, 5}
W cannot be
assigned 1 since
that value is
consumed.
X4 Æ Y4 ! W1
Clause reason for
W1
(W1,X=4,Y=4)
83
Clausal Reasons from All Diff
● If allDiff  Xd
the reason is
∎ First find the set of variables that still have
d in their domain.
∎ These variables must be consuming d.
∎ And the reason is the set of pruned values
in their domains.
84
Computing clauses from
Intensional Constraints
● Katsirelos has found ways to compute
clausal reasons for a variety of known
propagators.
● He has implemented this in a CSP solver
called EFC, which contains
∎ a number of built in intensional constraints
∎ GAC propagators for them.
∎ Clausal reasons supplied by these
propagators.
85
Computing clauses from
Intensional Constraints
● This solver can be thought of as a multi-
valued variable SMT solver, where T
includes a set of constraints known in
the CSP literature.
● Like SMT solvers it can display very
impressive performance.
86
Solving CSP via CSP solvers
● More work still needs to be done
∎ There are many more propagators that we
don’t yet know how to get clausal reasons
from.
∎ Heuristics remain poorly understood, and
still need improvement.
87
Some Empirical Results
88
Logistics (AI Planning)
● From Katsirelos & Bacchus, “Generalized
NoGoods in CSPs”
Problem
GAC
GAC+G
10-11
>20,000
3,906.3
15-15
52.4
2.3
18-11
497.3
89.1
22-11
85.2
13.9
26-12
678.55
19.4
26-13
>20,000
1899.0
28-12
>20,000
326.6
30-11
45.77
4.7
89
Social Golfer
w,g,s
GAC
GAC+S
GAC+G
2-7-5
1586.0s
218.0s
4.4s
2-8-5
>2000.0s
1211.9s
5.5s
3-6-4
>2000.0s
869.7s
5.0s
3-7-4
>2000.0s
549.6s
1.6s
4-7-3
843.4s
91.5s
0.3s
90
Other ideas from CSP solvers for SAT/SMT
● CSP solvers use propagation Queues
∎ Sequences the propagators, as some
propagators are more expensive.
● Flexible ways of specifying the level
of propagation for each constraint.
∎ Only forward checking is preformed
on some, GAC on others, we delay
GAC until all but 3 variables of the
constraint are instantiated etc.
91
Other ideas from CSP solvers for SAT/SMT
● Bounds propagation.
∎ Order the domain values and instead of
pruning all unsupported values, we
maintain upper and lower bounds on
the possible domain values.
∎ Possible to do more efficiently than
GAC in many cases.
● This would correspond to generating only
a subset of the implied literals.
● Once we get to a solution all literals are
set anyway.
92
Other ideas from CSP solvers for SAT/SMT
● Multi-valued variables are very
useful. But not fully exploited in
SAT solvers.
● Huge body of known constraints
and algorithms for propagating
them.
93
Conclusions
● CSPs add structure to SAT.
∎ This structure can be exploited to
make modeling easier.
∎ Can be used to identify groups of
clauses over which
● higher levels of propagation can be
profitable
● Specialized non-resolution based
algorithms can be exploited.
● Connecting these extra kinds of
reasoning with clause learning adds
considerable extra power.
94