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

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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.

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3 Background The SAT and CSP Formalisms

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4 Formalism SATCSPs Boolean VariablesMulti-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

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5 Formalism SAT = h V, Ci V = {V 1, V 2, …, V n } is a set of Boolean variables C = {c 1, c 2, …, c k } a set of clauses. CSP = h V, D, C i V = {V 1, V 2, …, V n } is a set of multi-valued variables D = {D 1, D 2, …, D n } is a set of value domains, with D i being the domain of values for variable V i C = {C 1, C 2, …, C k } 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.

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6 Constraints A constraint C(X 1,X 2, …, X k ) over the variables X 1, …, X k is a Boolean function It maps assignments to these variables to {0,1} C(X 1,X 2, …, X k ) : D X 1 £ £ D X k {0,1} If a tuple of assignments maps to 1, then these assignments satisfy the constraint, otherwise these assignments the falsify the constraint.

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7 Extensionally vs Intensionally Represented Constraints We can specify the constraint with a table C(X,Y,Z) with D X = D Y = D Z = {1, 2, 3} XYZC(X,Y,Z)XYZ XYZ

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8 Extensionally vs Intensionally Represented Constraints Thus we can represent the constraint as a set of satisfying assignment tuples XYZC(X,Y,Z)XYZ XYZ

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9 Extensionally vs Intensionally Represented Constraints Or as a set of falsifying assignment tuples XYZC(X,Y,Z)XYZ XYZ

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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.

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11 Extensionally vs Intensionally Represented Constraints Constraint could also be represented intensionally as an algorithm for computing the Boolean function.

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12 Extensionally vs Intensionally Represented Constraints XYZ C(X,Y,Z) XYZ XYZ = X · Y · Z

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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.

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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.

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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 (x=0,y=0,z=1) to 0, all other assignments of x,y,z to 1.

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16 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.

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17 CSP SAT Converting the Multi-Valued Variables Let X be a CSP variable with D x = {d 1, …, d m } We create m Boolean assignment variables x 1, x 2, …, x m these have the the interpretation x i is true iff X=d i.

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18 CSP SAT The CSP variable X must have a value and it must have a unique value. Hence the Boolean assignment variables x 1, x 2, …, x m associated with a particular CSP variable are mutually exclusive and exhaustive. This is captured by adding the clauses (x 1, x 2, …, x m ) X must have a value (-x i, -x j ) for all (i j) X has a unique value

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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.

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20 CSP SAT Each falsifying tuple is a set of assignment variables that cannot be simultaneously true. E.g.. –(x 1 ^ y 2 ^ z 1 ) Pushing the negation in we get a clause (-x 1 _ -y 2 _ -z 1 ) XYZXYZXYZ

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

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22 Modeling with CSPs

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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.

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24 N-Queens Place N queens on an NxN chess board so that no queen can attack any other queen. Q Q Q Q

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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 Q1Q1 Q2Q2 Q3Q3 Q4Q4

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26 N-Queens Place N queens on an NxN chess board so that queen can attack any other queen. Q 1 =1Q 2 =1 Q 2 =2Q 4 =2 Q 2 =3 Q 2 =4Q 3 =4 Q1Q1 Q2Q2 Q3Q3 Q4Q4 N values for each variable: The row we place that columns queen on.

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27 N-Queens Constraints AllDiff(Q 1, …, Q N ) each Queen has a unique value (cant be in the same row) C ij (Q i,Q j ): |Q i – Q j | |i-j| (for each i j) cant be on same diagonal

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28 Modeling with CSPs A SAT encoding of N-Queens more complex to specify. SAT encodings almost impossible to generate by hand.

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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.

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30 Solving CSPs

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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

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32 Translation to SAT ü The clause learning in SAT solvers can be exploited. û The mutually exclusive and exhaustive clauses for the multi-valued variables are not fully exploited. û Branching heuristics insensitive to CSP structure. û Unit propagation weaker than propagation methods employed in CSP solvers.

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33 Disadvantages: (a) Clauses for Multi-valued Variables

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34 Disadvantages: (a) Clauses for Multi-valued Variables These clauses impose a useful structure on the assignment variables. (x 1, x 2, …, x m ) X must have a value (-x i, -x j ) for all (i j) X has a unique value

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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 m=4 x 1 _ x 2 ´ –x 3 ^ –x 4 x 3 ´ -x 1 ^ -x 2 ^ -x 4 X=1X=2X=3X=4 X=1X=2X=3X=4

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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 (y 1, y 2, -x 1, -x 2, -z 3 ) subsumed by (-x 1,-x 2 )

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37 Disadvantages: (a) Clauses for Multi-valued Variables Negative assignment literal remove all positive literals from same variable. (y 1, y 2, -y 3, x 1, -x 2, -z 3 ) Resolve with (-y 1, -y 3 ) and (-y 2, -y 3 ) to obtain subsuming clause (-y 3, -x 2, -z 3 ).

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38 Disadvantages: (a) Clauses for Multi-valued Variables Sets of clauses can be replaced by a single clause. D x = D y = {1, 2, 3, 4} (R, -x 1, -y 1 ) (R, -x 1, -y 2 ) (R, -x 2, -y 1 ) (R, -x 2, -y 2 ) (R, -x 1, -y 1 ) (R, -x 1, -y 2 ) (R, -x 2, -y 1 ) (R, -x 2, -y 2 ) Equivalent to single clause (R, x 3, x 4, y 3, y 4 ).

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39 Disadvantages: (a) Clauses for Multi-valued Variables (R, x 3, x 4, y 3, y 4 ) ´ (R, (-x 2 ^ -x 1 ), (-y 2 ^ -y 1 )) Multiply this out and you get 8 clauses.

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40 Disadvantages: (b) Heuristics

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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).

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42 Disadvantages: (b) Heuristics With multi-valued variables we always have x i ´ -x 1 ^ –x i-1 ^ –x i+1 ^ -x m Hence conflicts arise only from refuting all values from some CSP variables domain -x 1 ^ ^ -x m

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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 x i without considering the status of other associated assignment variables. Ansótegui1 et al 2003.

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44 Disadvantages: (c) Propagation

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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.

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46 Disadvantages: (c) Propagation (-x 1 _ -y 2 _ -z 1 ) (-x 1 _ –y 3 _ –z 1 ) … XYZXYZXYZ

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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 x 1 ´ X=1, y 2 ´ Y=2

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48 Disadvantages: (c) Propagation Then unit propagation will falsify all assignments to the remaining unassigned CSP variable that would violate the constraint (-x 1 _ –y 3 _ –z 1 ), (-x 1 _ –y 3 _ –z 2 ) X=1 ^ Y=3 Z 1 & Z 2

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49 Disadvantages: (c) Propagation However, in practice, FC does not perform particularly well. A superior form of propagation is GAC.

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50 GAC (Macworth & Freuder ) Given a constraint C(X 1,X 2, …, X k ) d i 2 D X i is supported (in C) if there exists a set of assignments {X 1 = d 1, …, X i = d i,…, X k = d k } that satisfies C: C(X 1 =d 1, …, X k = d k ) = 1. This set is called a support for d i.

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51 GAC (Macworth & Freuder ) Supports for X = 1 (x 1 ) Supports for x 3 If 3 is removed from the domain of Z, i.e., -z 3 becomes true, x 3 will loose its only support. XYZ C(X,Y, Z) XYZ XYZ

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52 GAC (Macworth & Freuder ) The constraint C(X 1,X 2, …, X k ) is said to be GAC if for all of its variables X i every value in their domain is supported (in C). We can make C(X 1,X 2, …, X k ) GAC by removing all unsupported values from the domains of its variables.

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53 GAC (Macworth & Freuder ) GAC propagation is the dynamic process of making all of the constraints GAC. If d 2 D X is pruned from the domain of X while making C 1 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.

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54 GAC (Macworth & Freuder ) 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.

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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

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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

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57 GAC Propagation in the SAT encoding Let T C 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 T C by the prime implicates of T C, PI c Theorem: Unit prop over PI c achieves precisely GAC propagation on the constraint C.

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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.

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59 GAC Propagation in the SAT encoding Achieving GAC over a generic constraint C(X 1,X 2, …, X k ) 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.

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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.

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61 CSP Solvers

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62 Using a CSP solver ü Constraints can be represented intensionally and propagators can be exploited. ü The multi-valued variable structure, and information about the constraints can be exploited for branching decisions. û Learning in CSP solvers is much weaker than clause learning in Sat solvers, and it doesnt integrate well with GAC propagation.

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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.

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

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65 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 -(x 3 ^ y 2 ^ z 1 ) ´ (-x 3, -y 2, -z 1 ) Nogoods are negative clauses (clauses containing only negative literals).

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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 2003 Katsirelos PhD thesis

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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.

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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.

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69 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)

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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.

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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.

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72 Computing Clauses from GAC Supports for X = 1 XYZ C(X,Y, Z) XYZ XYZ 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)

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73 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=1s supports.

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74 Computing Clauses from GAC Supports for X = 1 XYZ C(X,Y, Z) XYZ XYZ E.g.,Y 1, Z 2, Z 3 covers all of X=1s supports in this constraint: Y 1 Æ Z 2 Æ Z 3 ! X 1 ´ (Y=1, Z=2, Z=3, X 1)

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75 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.

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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.

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77 Example: All Different AllDiff(X 1, …, X n ) is satisfied only by tuples of assignments to the X i that are all different, i.e., i j ! X i X j

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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.

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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(P 1, …, P n ) 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.

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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.

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81 Clausal Reasons from AllDiff D X = {1, 2, 3,4}, D Y = {1, 2, 3,4}, D Z = {1, 2, 3}, D W = {1, 4, 5} Initially all values are supported D X = {1, 2, 3,4}, D Y = {1, 2, 3,4}, D Z = {1, 2, 3}, D W = {1, 4, 5} Prune 4 from domain of X and Y

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82 Clausal Reasons from AllDiff D X = {1, 2, 3,4}, D Y = {1, 2, 3,4}, D Z = {1, 2, 3}, D W = {1, 4, 5} 1, 2, 3 are consumed by X, Y, Z. (Hall interval) D X = {1, 2, 3,4}, D Y = {1, 2, 3,4}, D Z = {1, 2, 3}, D W = {1, 4, 5} W cannot be assigned 1 since that value is consumed.

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83 Clausal Reasons from AllDiff D X = {1, 2, 3,4}, D Y = {1, 2, 3,4}, D Z = {1, 2, 3}, D W = {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)

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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.

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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.

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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.

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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.

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88 Some Empirical Results

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89 Logistics (AI Planning) From Katsirelos & Bacchus, Generalized NoGoods in CSPs ProblemGACGAC+G 10-11>20,0003, >20, >20,

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90 Social Golfer w,g,sGACGAC+SGAC+G s218.0s4.4s 2-8-5>2000.0s1211.9s5.5s 3-6-4>2000.0s869.7s5.0s 3-7-4>2000.0s549.6s1.6s s91.5s0.3s

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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.

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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.

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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.

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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.

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