Title: CSPs: Adding Structure to SAT
1CSPs Adding Structure to SAT
- George Katsirelos Fahiem Bacchus
- University of Toronto
2Introduction
- 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.
3BackgroundThe SAT and CSP Formalisms
4Formalism
SAT CSPs
Boolean Variables Multi-Valued variables
0,1 values for each variable. 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
5Formalism
- 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.
6Constraints
- 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.
7Extensionally 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 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
8Extensionally 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 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
9Extensionally vs Intensionally Represented
Constraints
- Or as a set of falsifying assignment tuples
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
10Extensionally 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.
11Extensionally vs Intensionally Represented
Constraints
- Constraint could also be represented
intensionally as an algorithm for computing the
Boolean function.
12Extensionally 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
13Extensionally 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.
14Translating 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.
15SAT ? 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 (x0,y0,z1) to
0, all other assignments of x,y,z to 1.
16CSP ? SAT
- The other direction requires two steps
- Converting the multi-valued variables into a set
of Boolean assignment variables. - Converting the constraints into clauses over the
assignment variables.
17CSP ? SATConverting 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 Xdi.
-
18CSP ? 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) X must have a value
- (-xi, -xj) for all (i ? j) X has a unique value
19CSP ? SAT Converting the Constraints into Clauses
- Now we convert the constraints to clauses.
- Each falsifying assignment tuple in the
constraints extensional representation is
equivalent to a clause. - So a constraint becomes a set of clauses, one for
each falsifying assignment.
20CSP ? SAT
- Each falsifying tuple is a set of assignment
variables that cannot be simultaneously true. -
- E.g.. (x1 y2 z1)
- Pushing the negation in we get a clause(-x1 _
-y2 _ -z1)
X Y Z X Y Z 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
21CSP ? SAT
- There are various optimizations that can be
applied to this basic translation. - Specific constraints admit more compact encodings.
22Modeling with CSPs
23Modeling 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.
24N-Queens
Q
Q
Q
Q
- Place N queens on an NxN chess board so that no
queen can attack any other queen.
25N-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
26N-Queens
Q11 Q21
Q22 Q42
Q23
Q24 Q34
- Place N queens on an NxN chess board so that
queen can attack any other queen.
Q1 Q2 Q3 Q4
- N values for each variable
- The row we place that columns queen on.
27N-Queens
- Constraints
- AllDiff(Q1, , QN) each Queen has a unique value
(cant be in the same row) - Cij(Qi,Qj) Qi Qj ? i-j (for each i ? j)
- cant be on same diagonal
28Modeling with CSPs
- A SAT encoding of N-Queens more complex to
specify. - SAT encodings almost impossible to generate by
hand.
29Modeling 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.
30Solving CSPs
31Backtracking 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
32Translation 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.
33Disadvantages(a) Clauses for Multi-valued
Variables
34Disadvantages(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.
35Disadvantages(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 m4 - x1 _ x2 x3 x4
- x3 -x1 -x2 -x4
X1 X2 X3 X4
X1 X2 X3 X4
36Disadvantages(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)
37Disadvantages(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).
38Disadvantages(a) Clauses for Multi-valued
Variables
- Sets of clauses can be replaced by a single
clause. - Dx Dy 1, 2, 3, 4(R, -x1, -y1) (R, -x1,
-y2) (R, -x2, -y1) (R, -x2, -y2) - (R, -x1, -y1) (R, -x1, -y2) (R, -x2, -y1) (R,
-x2, -y2)Equivalent to single clause(R, x3,
x4, y3 , y4).
39Disadvantages(a) Clauses for Multi-valued
Variables
- (R, x3, x4, y3 , y4) (R, (-x2 -x1),
(-y2 -y1)) - Multiply this out and you get 8 clauses.
40Disadvantages(b) Heuristics
41Disadvantages (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).
42Disadvantages (b) Heuristics
- With multi-valued variables we always have
- xi -x1 ? xi-1 xi1 ? -xm
- Hence conflicts arise only from refuting all
values from some CSP variables domain - -x1 ? ? -xm
43Disadvantages (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 shouldnt choose to branch on
xi without considering the status of other
associated assignment variables.
Ansótegui1 et al 2003.
44Disadvantages(c) Propagation
45Disadvantages (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.
46Disadvantages (c) Propagation
- (-x1 _ -y2 _ -z1)
- (-x1 _ y3 _ z1)
X Y Z X Y Z 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
47Disadvantages (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 X1, y2 Y2
48Disadvantages (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)X1 Y3 ?
Z ? 1 Z ? 2
49Disadvantages (c) Propagation
- However, in practice, FC does not perform
particularly well. - A superior form of propagation is GAC.
50GAC (Macworth Freuder 1977-79)
- Given a constraint C(X1,X2, , Xk)
- di 2 DXi is supported (in C) if there exists a
set of assignmentsX1 d1, , Xi di ,, Xk
dk that satisfies CC(X1d1, , Xk dk) 1. - This set is called a support for di.
51GAC (Macworth Freuder 1977-79)
- Supports for X 1 (x1)
- Supports for x3If 3 is removed from the
domain of Z, i.e., -z3 becomes true, x3 will
loose its only support.
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
52GAC (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.
53GAC (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.
54GAC (Macworth Freuder 1977-79)
- During search we make all constraints GAC at the
root. - The assignment X1 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.
55GAC 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
56GAC 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
- T ² c
- T ? c for any c that is a subset of c
57GAC 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 Cs 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.
58GAC 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.
-
59GAC 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.
60Translation 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.
61CSP Solvers
62Using 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 doesnt
integrate well with GAC propagation.
63Learning
- 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.
64Learning
- In CSPs learning from failed subtrees has a long
tradition. Learning is typically called nogood
recording.
65NoGoods
- A NoGood in CSPs is a set of assignments that
cannot be extended to a solution. - (X3, Y2, Z1)
- Translating this to SAT we get -(x3 y2 z1)
(-x3, -y2, -z1) - ? Nogoods are negative clauses
- (clauses containing only negative literals).
66Negative Resolution
- Restricting learning to NoGoods (negative
clauses) restricts the solvers 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 2003Katsirelos PhD thesis
67Negative 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.
68Integrating 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.
69Clause Learning in CSP solvers
X 1 ? CHOICEX ? 2 ? (X ? 1, X ? 2) (variable
can only have one value)Y ? 1 ? (Y ? 1, X2)
(Non-negative clause reason from GAC)Z ? 1 ? (Z
? 1, X2) A 2 ? CHOICEA ? 1 ? A ? 3 ? X ?
1 ? (X ? 1, A ? 2) (conflict clause
from constraint over X, and A)
70Clause 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.
71Computing 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.
72Computing Clauses from GAC
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
- E.g., (X1 Y1 Z1). This support can be
lost if 1 is pruned from the domain of Y or from
Z. (Y?1, or Z?1)
73Computing Clauses from GAC
- If GAC on this constraint prunes X1, a reason
for this pruning is a set of currently true
non-assignments that hits all of X1s supports.
74Computing Clauses from GAC
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
- E.g.,Y?1, Z?2, Z?3 covers all of X1s supports
in this constraint - Y?1Æ Z?2 Æ Z?3!X?1 (Y1, Z2, Z3, X?1)
75Computing Clauses from GAC
- We put this implication on the trailX?1 ? (X?1,
Y1, Z2, Z3) - 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.
76Computing 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.
77Example 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
78All 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.
79The 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.
80Clausal 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.
81Clausal Reasons from AllDiff
- DX 1, 2, 3,4, DY 1, 2, 3,4, DZ 1, 2,
3, DW 1, 4, 5
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
Initially all values are supported
82Clausal Reasons from AllDiff
DX 1, 2, 3,4, DY 1, 2, 3,4, DZ 1, 2,
3, DW 1, 4, 5
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)
W cannot be assigned 1 since that value is
consumed.
83Clausal Reasons from AllDiff
DX 1, 2, 3,4, DY 1, 2, 3,4, DZ 1, 2,
3, DW 1, 4, 5
- X?4 Æ Y?4 ! W?1
- Clause reason for W?1
- (W?1,X4,Y4)
W cannot be assigned 1 since that value is
consumed.
84Clausal 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.
85Computing 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.
86Computing 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.
87Solving CSP via CSP solvers
- More work still needs to be done
- There are many more propagators that we dont yet
know how to get clausal reasons from. - Heuristics remain poorly understood, and still
need improvement.
88Some Empirical Results
89Logistics (AI Planning)
- From Katsirelos Bacchus, Generalized NoGoods
in CSPs
Problem GAC GACG
10-11 gt20,000 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 gt20,000 1899.0
28-12 gt20,000 326.6
30-11 45.77 4.7
90Social Golfer
w,g,s GAC GACS GACG
2-7-5 1586.0s 218.0s 4.4s
2-8-5 gt2000.0s 1211.9s 5.5s
3-6-4 gt2000.0s 869.7s 5.0s
3-7-4 gt2000.0s 549.6s 1.6s
4-7-3 843.4s 91.5s 0.3s
91Other 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.
92Other 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.
93Other 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.
94Conclusions
- 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.