1 CSPs: Adding Structure to SAT George Katsirelos Fahiem Bacchus University of Toronto.

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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.

∎ Understanding these can help cross fertilize both fields.

BackgroundThe SAT and CSP Formalisms

FormalismSAT 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

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.

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

● 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

● 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)

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

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

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

● Or as a set of falsifying assignment tuples

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

● 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.

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

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

● 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

● Further insight into the relation between SAT and CSPs is provided by looking at how we can translate between the formalisms.

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● 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.

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 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● 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 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

CSP SAT

● There are various optimizations that can be applied to this basic translation.

● Specific constraints admit more compact encodings.

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● Place N queens

on an NxN chess board so that no queen can attack any other queen.

on an NxN chess board so that queen can attack any other queen.

● N, Queen variables, one for each column

Q1 Q2 Q3 Q4

on an NxN chess board so that queen can attack any other queen.

Q1=1 Q2=1

Q2=2 Q4=2

Q2=4 Q3=4

Q1 Q2 Q3 Q4

● N values for each variable:∎ The row we place that column’s

queen on.

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 CSPs● A SAT encoding of N-Queens more

complex to specify.● SAT encodings almost impossible to

generate by hand.

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.

Solving CSPs

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 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

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

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

X=1 X=2 X=3 X=4

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)

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).

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)

Equivalent to single clause(R, x3, x4, y3 , y4).

Variables● (R, x3, x4, y3 , y4) ´

(R, (-x2 ^ -x1), (-y2 ^ -y1))

● Multiply this out and you get 8 clauses.

Disadvantages:(b) Heuristics

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).

● 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

● 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) Propagation

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.

● (-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

● 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

● 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

● However, in practice, FC does not perform particularly well.

● A superior form of propagation is GAC.

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 assignments{X1 = d1, …, Xi = di ,…, Xk = dk}

that satisfies C:C(X1=d1, …, Xk = dk) = 1.

∎ This set is called a support for di.

● 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 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

● 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 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.

● During search we make all constraints GAC at the root.

● The assignment X=1 means X 2X 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

● 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

● 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

● 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.

● 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.

● 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

● 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.

CSP Solvers

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

● 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.

Learning

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

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 ● 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 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

● 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

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, 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 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 ● 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 GAC ● Supports for X = 1 X Y Z C(X,Y,

Z)X Y Z C(X,Y,

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., (X=1 ^ Y=1 ^ Z=1). This support can be lost if 1 is pruned from the domain of Y or from Z. (Y1, or Z1)

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 GAC ● Supports for X = 1 X Y Z C(X,Y,

Z)X Y Z C(X,Y,

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.,Y1, Z2, Z3 covers all of X=1’s supports in this constraint:

Y1Æ Z2 Æ Z3!X1´ (Y=1, Z=2, Z=3, X1)

Computing Clauses from GAC

● We put this implication on the trailX1 (X1, 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

● 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

● AllDiff(X1, …, Xn) is satisfied only by tuples of assignments to the Xi that are all different, i.e.,∎ ij ! XiXj

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● 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

● 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.