Constraint Networks. Constraint networks. Definition. Normalized. Constraint Networks. Deduction. Constraint. Networks and Graphs. Solving.

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1 1 Satisfaction Problems Albert-Ludwigs-Universität Freiburg networks networks and Stefan Wölfl, Christian Becker-Asano, and Bernhard Nebel October 27, 2014 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 3 / 27 networks networks A constraint network is a triple If we assume some ordering of the variables in V, we can write networks more compactly: where: N = V,dom,C V is a non-empty and finite set of variables; dom is a function that assigns to each variable v V a non-empty set dom(v) (dom(v) is called the domain of v, elements of dom(v) are called values); C is a set of relations over variables of V (called constraints), i.e., each constraint is a relation R x1,...,x m over some scheme S = (x 1,...,x m ) of variables in V. networks and A constraint network is a triple N = V,D,C where: V = {,...,v n } is a non-empty and finite orde set of variables (assume order (,...,v n )); D = (D 1,...,D n ) is a sequence of domains for V (with D i the domain of variable v i and D N = D 1 D n is the domain of N); C is a set of constraints (x,r) where x = (v i1,...,v im ) is a scheme of variables in V and R D i1 D im. networks and The set of constraint schemes {S 1,...S t } is called network scheme. Notice: any ordering of the variables suffices. network that differ only in the variable ordering are conside equal. October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 4 / 27 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 5 / 27

2 networks Example: 4-queens problem Note that we consider finitely many variables only, but (e.g., for theoretical studies) this could be relaxed. In the 2nd definition we assumed that the relations of the constraint are embedded in the domain of the network: R D i1 D im. Such networks are called embedded networks (see Bessiere, 2006). The definition does not require that constraint relations are given explicitly (in extension, i.e. by a set of its tuples; table constraint). A constraint relation R could be specified by any Boolean function f R : a tuple satisfies the constraint R iff f R applied on the tuple gives 1. If not stated otherwise, we will always assume that the domains of the variables are given in extension. networks and The 4-queens problem can be represented as a constraint network. For example, consider variables,...,v 4 (each associated to a column of the 4 4-chess board). Each variable v i has as its domain D i = {1,...,4} (conceived of as the row positions of a queen in column i) v 2 v 4 Define then binary constraints (thus encoding non-attacking queen positions ): R v1,v 2 := {(1,3),(1,4),(2,4),(3,1), (4,1),(4,2)} R v1, := {(1,2),(1,4),(2,1),(2,3), (3,2),(3,4),(4,1),(4,3)}... networks and October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 6 / 27 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 7 / 27 Example: colorability of a constraint network k-colorability of a graph G can be represented as a constraint network of the following form: v 2 V = {v i : v i is a vertex in G} D i = {1,...,k} (v i V) C = {((v i,v j ),) : {v i,v j } is an edge of G} Binary constraint networks can be represented by a directed labeled graph (or even: by an undirected graph if all constraints are symmetric). October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 8 / 27 networks and A solution of a constraint network N = V,D,C is a (variable) assignment a: V such that: (a) a(v i ) D i, for each v i V, i : v i V (b) (a(x 1 ),...,a(x m )) R for each constraint R x1,...,x m in C. N is called satisfiable if N has a solution. sol(n) denotes the set of all solutions of N. sol(n) can also be written as: sol(n) = {(d 1,...,d n ) D 1 D n : the assignment D i d 1,...,v n d n defines a solution} Notice: October 27, 2014 sol(n) Wölfl, is anebel relation and Becker-Asano... Satisfaction Problems 9 / 27 networks and

3 Instantiation, partial solution Instantiation, solution Let N = V,D,C be a constraint network. (a) An instantiation of a subset V V is an assignment a : V i : v i V D i with a(v i ) D i. (b) An instantiation a of V is called partial solution if a satisfies each constraint R S in C with S V. In this case a is called locally consistent. (c) Shortcut notation: for an instantiation a of V = {x 1,...,x m } and constraint R S with scope S V, set networks and Note: (a) An instantiation of V V, a, is a partial solution (locally consistent) iff a[s] R, for each constraint R with scope S V. (b) Not every partial solution is part of a (full) solution, i.e., there may be partial solutions of a constraint network that cannot be extended to a solution. For the 4-queens problem, for example: 4 q 3 q networks and a[s] := (a(x 1 ),...,a(x m )). 2 Hence, a solution is an instantiation of all variables in V that is locally consistent. 1 q October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 10 / 27 v 2 v 4 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 11 / 27 Nogoods constraint network Let N = V,D,C be a constraint network. Let N = V,D,C be a constraint network. Due to our definition it is possible that C contains constraints networks R vi1,...,v ik and S vj1,...,v jk networks An instantiation a of subset V V is called a nogood (of N) if a cannot be extended to a (full) solution of N, i.e., there exists no solution a: V i D i such that a V = a. Instantiations that are no nogoods are sometimes called consistent or globally consistent (to emphasize the difference to locally consistent assignments). Later, we will also introduce the notion of globally consistent networks. and where (j 1,...,j k ) is just a permutation of (i 1,...,i k ). Without changing the set of solutions, we can simplify the network by deleting S vj1,...,v jk from C and rewriting R vi1,...,v ik as follows: R vi1,...,v ik R vi1,...,v ik π vi1,...,v ik (S vj1,...,v jk ). Given a fixed order on the set of variables V, we can systematically delete-and-refine constraints. This results in a constraint network that contains at most one constraint for each subset of variables. Such a network is called a normalized constraint network. and October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 12 / 27 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 13 / 27

4 Equivalence Tightness Let N and N be constraint networks on the same set of variables and on the same domains for each variable. N and N are called equivalent if they have the same set of solutions. Example: networks and Let N and N be (normalized) constraint networks on the same set of variables and on the same domains for each variable. N is as tight as N if for each constraint R S of N, (a) N has no constraint with the same scope as R S, or (b) R π S (R S ), where R S is the constraint of N with the same scope as R S. networks and < v 2 v 2 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 14 / 27 = v 2 v 2 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 15 / 27 Intersection of networks The intersection of N and N, N N, is the network defined by intersecting for each scope the constraints R S C and R S C with the same scope, i.e., modulo a suitable permutation of the constraint schemes, R S := R S R S. If for a scope S only one of the networks contains a constraint, then we set: R S := R S (or := R S, resp.) networks and Clearly, if N is as tight as N, then sol(n ) sol(n). tightness has a large influence on the efficiency of constraint satisfaction. Warning: Different concepts of tightness can be found in the literature Here: Tightness does not account for comparing constraints with different arities 2 and Primal Dual and Primal Dual Lemma If N and N are equivalent networks, then N N is equivalent to both networks and as tight as both networks. October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 16 / 27 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 18 / 27

5 Primal constraint graphs Primal constraint graph: Example Let N = V,D,C be a (normalized) constraint network. The primal constraint graph of a network N = V,D,C is the undirected graph G N := V,E N where and Primal Dual Consider a constraint network with variables,...,v 5 and two ternary constraints R v1,v 2, and S v3,v 4,v 5. Then the primal constraint graph of the network has the form: v 2 and Primal Dual {u,v} E N {u,v} is a subset of the scope of some constraint in N. v 4 v 5 Absence of an edge between two variables/nodes means that there is no explicit constraint in which both variables participate. October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 19 / 27 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 20 / 27 Dual constraint graphs hypergraph The dual constraint graph of a constraint network N = V,D,C is the labeled graph with D N := V,E N,l X V X is the scope of some constraint in N {X,Y} E N X Y /0 l : E N 2 V, {X,Y} X Y In the example above, the dual constraint graph is: and Primal Dual The constraint hypergraph of a constraint network N = V,D,C is the hypergraph H N := V,E N with X E N X is the scope of some constraint in N. In the example above (constraint network with variables,...,v 5 and two ternary constraints R v1,v 2, and S v3,v 4,v 5 ) the hypeges of the constraint hypergraph are: and Primal Dual,v 2,,v 4,v 5 E N = {{,v 2, },{,v 4,v 5 }}. October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 21 / 27 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 22 / 27

6 3 Simple solution strategy: Backtracking search and Backtracking: search systematically for locally consistent partial instantiations in a depth-first manner: forward phase: extend the current partial solution by assigning a consistent value to some new variable (if possible) backward phase: if no consistent instantiation for the current variable exists, return to the previous variable. and October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 24 / 27 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 25 / 27 Backtracking algorithm Literature Backtracking(N, a): Input: a constraint network N = V, D, C and a partial assignment a of N (e.g., the empty instantiation a = {}) Output: a solution of N or inconsistent if a is not locally consistent with N: return inconsistent if a is defined for all variables in V: return a select some variable v i for which a is not defined for each value x from D i : a := a {v i x} a Backtracking(N,a ) if a is not inconsistent : return a return inconsistent and Rina Dechter. Processing, Chapter 2, Morgan Kaufmann, 2003 and October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 26 / 27 October 27, 2014 Wölfl, Nebel and Becker-Asano Satisfaction Problems 27 / 27