##### Question 1: Element3D

Consider a 3D Element constraint \(T[W][X][Y]=Z\) where \(T\) is a 3D array of integers of dimensions \(4 \times 5 \times 10\). Assuming \(D(W)=\{0..3\}, D(X)=\{0..4\}, D(Y)=\{0..9\}, D(Z)=\{100..200\}\), what are the dimensions (number of rows, number of columns) of the table for a Table constraint that encodes this 3D Element constraint? Write your answer in the format "nRows,nColumns", without the quotes, for instance 12,3 .

##### Question 2: Eternity II (number of tuples)

For the Eternity II edge-matching problem, assuming a 10x10 board and a set of 100 pieces that are all different and have no rotation symmetries, how many tuples does the table of each Table constraint posted in the model have?

##### Question 3: Eternity II (number of constraints)

For the Eternity II edge-matching problem, assuming a 10x10 board and a set of 100 pieces that are all different and have no rotation symmetries, how many Table constraints are posted in the model?

##### Question 4: Eternity II (table arity)

For the Eternity II edge-matching problem, assuming a 10x10 board and a set of 100 pieces that are all different and have no rotation symmetries, what is the arity (number of variables in the scope) of each Table constraint posted in the model?

##### Question 5: Regular Constraint

A Regular constraint has the form Regular:math:([x_1,...,x_n],A) where \(A\) is a finite-state automaton:

- \(\Sigma\) is the finite input alphabet;
- \(Q\) is the finite set of states;
- \(\delta\) is the transition function: \(\delta : Q \times \Sigma \rightarrow Q\);
- \(F\) is the set of accepting states and a subset of \(Q\).
- \(q_0\) is the initial state and is an element of \(Q\);

Select all the true statements:

A Regular constraint can be encoded with Table constraints for expressions of the form \(T[S_i][x_i]=S_{i+1}\) where \(T\) represents the transition function.

The number of tuples in the table depends only on the number of transitions in the automaton.

A Regular constraint can be encoded in polynomial time and space with a unique Table constraint of arity \(n\) whose table encodes as tuples all the valid paths in the automaton.

The number of tuples in the table depends only on the number of states in the automaton.

The number of tuples in the table depends only on the number of states in the automaton and the size of the alphabet.

The number of tuples in the table depends only on the number \(n\) and domain sizes of the variables \(x_i\).

##### Question 6: STR2 Filtering Algorithm

In the STR2 filtering algorithm for a Table constraint, how many StateInt are necessary to maintain in a stateful way (on backtrack) the set of valid tuples?

##### Question 7: Table Constraint

Select all the true statements:

To enforce domain consistency on all the variables of a 2D Element constraint, one typically uses a Table constraint.

The Table constraint can be used to enforce domain consistency on a Regular constraint on a sequence of variables.

A Table constraint can encode any kind of constraints: it is the most expressive constraint.

The filtering algorithms for a Table constraint generally enforce bound consistency.

An Alldifferent constraint can be encoded with a Table constraint and the time complexity to filter the constraint would be the same as with Régin's algorithm.