CSPs: Adding Structure to SAT

1
CSPs Adding Structure to SAT

• George Katsirelos Fahiem Bacchus
• University of Toronto

2
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.

3
BackgroundThe SAT and CSP Formalisms
4
Formalism
5
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.

6
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.

7
Extensionally vs Intensionally Represented
Constraints

• We can specify the constraint with a table
• C(X,Y,Z) with DX DY DZ 1, 2, 3

8
Extensionally vs Intensionally Represented
Constraints

• Thus we can represent the constraint as a set of
  satisfying assignment tuples

9
Extensionally vs Intensionally Represented
Constraints

• Or as a set of falsifying assignment tuples

10
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.

11
Extensionally vs Intensionally Represented
Constraints

• Constraint could also be represented
  intensionally as an algorithm for computing the
  Boolean function.

12
Extensionally vs Intensionally Represented
Constraints

X Y Z
13
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.

14
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.

15
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 (x0,y0,z1) to
  0, all other assignments of x,y,z to 1.

16
CSP ? 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.

17
CSP ? 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.

18
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) X must have a value
• (-xi, -xj) for all (i ? j) X has a unique value

19
CSP ? 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.

20
CSP ? 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)

21
CSP ? SAT

• There are various optimizations that can be
  applied to this basic translation.
• Specific constraints admit more compact encodings.

22
Modeling with CSPs
23
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.

24
N-Queens

• Place N queens on an NxN chess board so that no
  queen can attack any other queen.

25
N-Queens

• Place N queens on an NxN chess board so that
  queen can attack any other queen.

• N, Queen variables, one for each column

26
N-Queens

• Place N queens on an NxN chess board so that
  queen can attack any other queen.

• N values for each variable
• The row we place that columns queen on.

27
N-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

28
Modeling with CSPs

• A SAT encoding of N-Queens more complex to
  specify.
• SAT encodings almost impossible to generate by
  hand.

29
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.

30
Solving CSPs
31
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

32
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.

33
Disadvantages(a) Clauses for Multi-valued
Variables
34
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.

35
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 m4
• x1 _ x2 x3 x4
• x3 -x1 -x2 -x4

36
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)

37
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).

38
Disadvantages(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).

39
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.

40
Disadvantages(b) Heuristics
41
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).

42
Disadvantages (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

43
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 shouldnt choose to branch on
  xi without considering the status of other
  associated assignment variables.

Ansótegui1 et al 2003.
44
Disadvantages(c) Propagation
45
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.

46
Disadvantages (c) Propagation

• (-x1 _ -y2 _ -z1)
• (-x1 _ y3 _ z1)

47
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 X1, y2 Y2

48
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)X1 Y3 ?
  Z ? 1 Z ? 2

49
Disadvantages (c) Propagation

• However, in practice, FC does not perform
  particularly well.
• A superior form of propagation is GAC.

50
GAC (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.

51
GAC (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.

52
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.

53
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.

54
GAC (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.

55
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
56
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
• T ² c
• T ? c for any c that is a subset of c

57
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 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.

58
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.

59
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.

60
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.

61
CSP Solvers
62
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 doesnt
  integrate well with GAC propagation.

63
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.

64
Learning

• In CSPs learning from failed subtrees has a long
  tradition. Learning is typically called nogood
  recording.

65
NoGoods

• 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).

66
Negative 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
67
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.

68
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.

69
Clause 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)
70
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.

71
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.

72
Computing Clauses from GAC

• Supports for X 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)

73
Computing 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.

74
Computing Clauses from GAC

• Supports for X 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)

75
Computing 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.

76
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.

77
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

78
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.

79
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.

80
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.

81
Clausal 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
82
Clausal 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.
83
Clausal 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.
84
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.

85
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.

86
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.

87
Solving 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.

88
Some Empirical Results
89
Logistics (AI Planning)

• From Katsirelos Bacchus, Generalized NoGoods
  in CSPs

90
Social Golfer
91
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.

92
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.

93
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.

94
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.