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CSPs: Adding Structure to SAT George Katsirelos Fahiem Bacchus University of Toronto Introduction Finite domain Constraint Satisfaction Problems (CSPs). Formally equivalent to SAT Important practical differences. Different algorithmic techniques have been developed in the two areas.

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csps adding structure to sat

CSPs: Adding Structure to SAT

George Katsirelos Fahiem Bacchus

University of Toronto

introduction
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.
formalism5
Formalism
  • SAT = hV, Ci
    • V = {V1, V2, …, Vn} is a set of Boolean variables
    • C = {c1, c2, …, ck} a set of clauses.
  • CSP = hV, D, Ci
    • 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.
constraints
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.
extensionally vs intensionally represented constraints
Extensionally vs Intensionally Represented Constraints
  • We can specify the constraint with a table
    • C(X,Y,Z) with DX = DY = DZ = {1, 2, 3}
extensionally vs intensionally represented constraints8
Extensionally vs Intensionally Represented Constraints
  • Thus we can represent the constraint as a set of satisfying assignment tuples
extensionally vs intensionally represented constraints9
Extensionally vs Intensionally Represented Constraints
  • Or as a set of falsifying assignment tuples
extensionally vs intensionally represented constraints10
Extensionally vs Intensionally Represented Constraints
  • Extensionalrepresentations 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.
extensionally vs intensionally represented constraints11
Extensionally vs Intensionally Represented Constraints
  • Constraint could also be represented intensionally as an algorithm for computing the Boolean function.
extensionally vs intensionally represented constraints13
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.
translating between sat and csps
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.
sat csp
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.
csp sat
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.
csp sat converting the multi valued variables
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 X=di.

csp sat18
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
csp sat converting the constraints into clauses
CSP SATConverting 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.
csp sat20
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)
csp sat21
CSP SAT
  • There are various optimizations that can be applied to this basic translation.
  • Specific constraints admit more compact encodings.
modeling with csps23
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.
n queens
N-Queens
  • Place N queens on an NxN chess board so that no queen can attack any other queen.
n queens25
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
n queens26
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 column’s queen on.
n queens27
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
modeling with csps28
Modeling with CSPs
  • A SAT encoding of N-Queens more complex to specify.
  • SAT encodings almost impossible to generate by hand.
modeling with csps29
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.
backtracking search
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
translation to sat
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.
disadvantages a clauses for multi valued variables34
Disadvantages:(a) Clauses for Multi-valued Variables
  • These clauses impose a useful structure on the assignment variables.
  • (x1, x2, …, xm) X must have a value
  • (-xi, -xj) for all (i  j) X has a unique value
disadvantages a clauses for multi valued variables35
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
    • x3´ -x1^ -x2^ -x4
disadvantages a clauses for multi valued variables36
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)
disadvantages a clauses for multi valued variables37
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).
disadvantages a clauses for multi valued variables38
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).

disadvantages a clauses for multi valued variables39
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.
disadvantages b heuristics41
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).
disadvantages b heuristics42
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
disadvantages b heuristics43
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.

disadvantages c propagation45
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.
disadvantages c propagation46
Disadvantages: (c) Propagation
  • (-x1_ -y2_ -z1)
  • (-x1_ –y3_ –z1)
disadvantages c propagation47
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
disadvantages c propagation48
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
disadvantages c propagation49
Disadvantages: (c) Propagation
  • However, in practice, FC does not perform particularly well.
  • A superior form of propagation is GAC.
gac macworth freuder 1977 79
GAC (Macworth & Freuder 1977-79)
  • Given a constraint C(X1,X2, …, Xk)
    • di2 DXi is supported(in C) 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.
gac macworth freuder 1977 7951
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.
gac macworth freuder 1977 7952
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.
gac macworth freuder 1977 7953
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.
gac macworth freuder 1977 7954
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.
gac propagation in the sat encoding
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

gac propagation in the sat encoding56
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
gac propagation in the sat encoding57
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.

gac propagation in the sat encoding58
GAC Propagation in the SAT encoding
  • Note that GAC is localto the constraint. Communication between constraints occurs only through unit implicants (pruned values).
  • So GAC is complete local inference for units.
gac propagation in the sat encoding59
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.
translation to sat summary
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.
using a csp solver
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.
learning
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.
learning64
Learning
  • In CSPs learning from failed subtrees has a long tradition. Learning is typically called “nogood” recording.
nogoods
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).
negative resolution
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 2003Katsirelos PhD thesis

negative resolution67
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.
integrating sat style clause learning
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.
clause learning in csp solvers
Clause Learning in CSP solvers

X = 1  CHOICEX  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  CHOICEA  1  …A  3  …X  1  (X  1, A  2) (conflict clause from constraint over X, and A)

clause learning in csp solvers70
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.
computing clauses from gac
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.
computing clauses from gac72
Supports for X = 1Computing Clauses from GAC
  • 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)
computing clauses from gac73
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.
computing clauses from gac74
Supports for X = 1Computing Clauses from GAC

E.g.,Y1, Z2, Z3 covers all of X=1’s supports in this constraint:

Y1Æ Z2 Æ Z3!X1´(Y=1, Z=2, Z=3, X1)

computing clauses from gac75
Computing Clauses from GAC
  • We put this implication on the trailX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.
computing clauses from intensional constraints
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.
example all different
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
all different
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.
the power of propagators
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.
clausal reasons from alldiff
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 consumedby some other variables. That is, the value must be used by some other variable.
clausal reasons from alldiff81
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

clausal reasons from alldiff82
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.

clausal reasons from alldiff83
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,X=4,Y=4)

W cannot be assigned 1 since that value is consumed.

clausal reasons from all diff
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.
computing clauses from intensional constraints85
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.
computing clauses from intensional constraints86
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.
solving csp via csp solvers
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.
logistics ai planning
Logistics (AI Planning)
  • From Katsirelos & Bacchus, “Generalized NoGoods in CSPs”
other ideas from csp solvers for sat smt
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.
other ideas from csp solvers for sat smt92
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.
other ideas from csp solvers for sat smt93
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.
conclusions
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.
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