##### Question 1: Consistency AllDifferent

We consider the \(\mathit{AllDifferent}(x_1,x_2,x_3,x_4)\) constraint with domains \(D(x_1)=\{1,3,4\}, D(x_2)=\{1,3\}, D(x_3)=\{1,3\}, D(x_4)=\{1,4,5\}\).

Select all the correct statements:

The constraint is domain consistent.

The constraint is bound consistent.

##### Question 2:

A binary constraint \(C(X,Y)\) is represented visually (feasible solutions inside the green shapes).

The domains \(D(X)\) and \(D(Y)\) are also represented as small green rectangles on the axes.

The image displays the domains AFTER they have been pruned by the constraint.

Select all the true statements given the current represented domains:

The constraint is domain consistent.

The constraint is bound consistent.

The constraint has no solution (it is inconsistent).

##### Question 3: General assertions

Select all the correct statements given a constraint \(C(X,Y,Z)\) with three finite-domain integer variables in its scope:

Assuming the constraint is bound consistent and \(v\) is in \(D(X)\), there exists a solution to this constraint with \(X=v\).

Assuming the constraint is domain consistent, removing a value from \(D(X)\) removes at least one solution to the constraint.

Assuming the constraint is bound consistent, removing a value from \(D(X)\) removes at least one solution to the constraint.

Assuming the constraint is domain consistent and \(v\) is in \(D(X)\), there exists a solution to this constraint with \(X=v\).

##### Question 4: Range consistency

C is range-consistent iff

\(\forall 1 \leq i \leq n, \forall v_i \in D(i):\quad \exists (v_1,\dots,v_{i-1},v_{i+1},\dots,v_n) \in\) \([\min(D_1),\max(D_1)] \times \cdots \times [\min(D_{i-1}),\max(D_{i-1})] \times [\min(D_{i+1}),\max(D_{i+1})] \times \cdots \times [\min(D_n),\max(D_n)]\) \(\text{ such that } C(v_1,\dots,v_i,\ldots,v_n)\)

Select all the correct statements:

Range consistency is weaker (less restrictive) than domain consistency.

Range consistency is stronger (more restrictive) than bound consistency.

Range consistency is stronger (more restrictive) than domain consistency.

Range consistency is weaker (less restrictive) than bound consistency.

##### Question 5: Binary Sum

Consider the constraint \(X+Y=0\) with \(D(X)=[100,110]\) and \(D(Y)=[-1000,120] \cap [-105, 1000]\). Assuming a consistent (does not remove any solution) filtering algorithm for the constraint that achieves bound consistency (after its filtering the constraint is bound consistent), what will be \(D(X)\)?

Write your answer in the format \(\{v_1,v_2,...,v_n\}\) with \(v_i < v_i+1\). For instance, write \(\{6,10,13\}\) but not \(\{13,6,10\}\).