[PPT] - Exercise 8: Datalog Database Theory 2020-06-08 Maximilian Marx, PowerPoint Presentation

SLIDE 1

Exercise 8: Datalog

Database Theory 2020-06-08 Maximilian Marx, David Carral

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SLIDE 2

Exercise 1

Exercise. A graph is planar if it can be drawn on the plane without intersections of edges. For example, the following

graph A is planar, while graph B is not:

Figure: A Figure: B

1. Can the graphs A and B be distinguished by a FO query?
2. Show that planarity is not FO-definable by using locality.

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SLIDE 3

Exercise 1

Exercise. A graph is planar if it can be drawn on the plane without intersections of edges. For example, the following

graph A is planar, while graph B is not:

Figure: A Figure: B

1. Can the graphs A and B be distinguished by a FO query?
2. Show that planarity is not FO-definable by using locality.

Solution.

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SLIDE 4

Exercise 1

Exercise. A graph is planar if it can be drawn on the plane without intersections of edges. For example, the following

graph A is planar, while graph B is not:

Figure: A Figure: B

1. Can the graphs A and B be distinguished by a FO query?
2. Show that planarity is not FO-definable by using locality.

Solution.

1. This query matches B but not A:

∃x, y, z, w, v. E(x, y) ∧ E(y, z) ∧ E(z, w) ∧ E(w, x) ∧ E(x, v) ∧ E(y, v) ∧ E(z, v) ∧ E(w, v) ∧ E(x, z) ∧ E(y, w)

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SLIDE 5

Exercise 1

Exercise. A graph is planar if it can be drawn on the plane without intersections of edges. For example, the following

graph A is planar, while graph B is not:

Figure: A Figure: B

1. Can the graphs A and B be distinguished by a FO query?
2. Show that planarity is not FO-definable by using locality.

Solution.

1. This query matches B but not A:

∃x, y, z, w, v. E(x, y) ∧ E(y, z) ∧ E(z, w) ∧ E(w, x) ∧ E(x, v) ∧ E(y, v) ∧ E(z, v) ∧ E(w, v) ∧ E(x, z) ∧ E(y, w)

2. For ϕ with quantifier rank r, consider counterexamples of size d = 3r :

. . . . . . . . . . . . . . . . . . . . . . . . d d d d d d . . . . . . . . . . . . . . . . . . . . . . . . d d d d d d

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SLIDE 6

Exercise 2

Exercise. Consider the example Datalog program from the lecture:

(a) Father(alice,bob) (b) Mother(alice,carla) (c) Mother(evan,carla) (d) Father(carla,david)

Parent(x,y) ← Father(x,y)

(1)

Parent(x,y) ← Mother(x,y)

(2)

Ancestor(x,y) ← Parent(x,y)

(3)

Ancestor(x,z) ← Parent(x,y) ∧ Ancestor(y,z)

(4)

SameGeneration(x,x) ←

(5)

SameGeneration(x,y) ← Parent(x,v) ∧ Parent(y,w) ∧ SameGeneration(v,w)

(6)

1. Give a proof tree for SameGeneration(evan, alice).
2. Compute the sets T 0

P, T 1 P, T 2 P, . . . When is the fixed point reached?

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SLIDE 7

Exercise 2

Exercise. Consider the example Datalog program from the lecture:

(a) Father(alice,bob) (b) Mother(alice,carla) (c) Mother(evan,carla) (d) Father(carla,david)

Parent(x,y) ← Father(x,y)

(1)

Parent(x,y) ← Mother(x,y)

(2)

Ancestor(x,y) ← Parent(x,y)

(3)

Ancestor(x,z) ← Parent(x,y) ∧ Ancestor(y,z)

(4)

SameGeneration(x,x) ←

(5)

SameGeneration(x,y) ← Parent(x,v) ∧ Parent(y,w) ∧ SameGeneration(v,w)

(6)

1. Give a proof tree for SameGeneration(evan, alice).
2. Compute the sets T 0

P, T 1 P, T 2 P, . . . When is the fixed point reached?

Solution.

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SLIDE 8

Exercise 2

Exercise. Consider the example Datalog program from the lecture:

(a) Father(alice,bob) (b) Mother(alice,carla) (c) Mother(evan,carla) (d) Father(carla,david)

Parent(x,y) ← Father(x,y)

(1)

Parent(x,y) ← Mother(x,y)

(2)

Ancestor(x,y) ← Parent(x,y)

(3)

Ancestor(x,z) ← Parent(x,y) ∧ Ancestor(y,z)

(4)

SameGeneration(x,x) ←

(5)

SameGeneration(x,y) ← Parent(x,v) ∧ Parent(y,w) ∧ SameGeneration(v,w)

(6)

1. Give a proof tree for SameGeneration(evan, alice).
2. Compute the sets T 0

P, T 1 P, T 2 P, . . . When is the fixed point reached?

Solution. 1. SameGeneration(evan, alice) Parent(evan, carla) Parent(alice, carla) SameGeneration(carla, carla) Mother(evan, carla) Mother(alice, carla) (2); { x → evan, y → carla } (2); { x → alice, y → carla } (6); { x → evan, v → carla, y → alice, w → carla } (c) (b) (5); { x → carla }

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Exercise 2

Exercise. Consider the example Datalog program from the lecture:

(a) Father(alice,bob) (b) Mother(alice,carla) (c) Mother(evan,carla) (d) Father(carla,david)

Parent(x,y) ← Father(x,y)

(1)

Parent(x,y) ← Mother(x,y)

(2)

Ancestor(x,y) ← Parent(x,y)

(3)

Ancestor(x,z) ← Parent(x,y) ∧ Ancestor(y,z)

(4)

SameGeneration(x,x) ←

(5)

SameGeneration(x,y) ← Parent(x,v) ∧ Parent(y,w) ∧ SameGeneration(v,w)

(6)

1. Give a proof tree for SameGeneration(evan, alice).
2. Compute the sets T 0

P, T 1 P, T 2 P, . . . When is the fixed point reached?

Solution. 2. T 0

P = ∅

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SLIDE 10

Exercise 2

Exercise. Consider the example Datalog program from the lecture:

(a) Father(alice,bob) (b) Mother(alice,carla) (c) Mother(evan,carla) (d) Father(carla,david)

Parent(x,y) ← Father(x,y)

(1)

Parent(x,y) ← Mother(x,y)

(2)

Ancestor(x,y) ← Parent(x,y)

(3)

Ancestor(x,z) ← Parent(x,y) ∧ Ancestor(y,z)

(4)

SameGeneration(x,x) ←

(5)

SameGeneration(x,y) ← Parent(x,v) ∧ Parent(y,w) ∧ SameGeneration(v,w)

(6)

1. Give a proof tree for SameGeneration(evan, alice).
2. Compute the sets T 0

P, T 1 P, T 2 P, . . . When is the fixed point reached?

Solution. 2. T 0

P = ∅

T 1

P = { Father(alice,bob), Mother(alice,carla), Mother(evan,carla), Father(carla,david) }

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SLIDE 11

Exercise 2

Exercise. Consider the example Datalog program from the lecture:

(a) Father(alice,bob) (b) Mother(alice,carla) (c) Mother(evan,carla) (d) Father(carla,david)

Parent(x,y) ← Father(x,y)

(1)

Parent(x,y) ← Mother(x,y)

(2)

Ancestor(x,y) ← Parent(x,y)

(3)

Ancestor(x,z) ← Parent(x,y) ∧ Ancestor(y,z)

(4)

SameGeneration(x,x) ←

(5)

SameGeneration(x,y) ← Parent(x,v) ∧ Parent(y,w) ∧ SameGeneration(v,w)

(6)

1. Give a proof tree for SameGeneration(evan, alice).
2. Compute the sets T 0

P, T 1 P, T 2 P, . . . When is the fixed point reached?

Solution. 2. T 0

P = ∅

T 1

P = { Father(alice,bob), Mother(alice,carla), Mother(evan,carla), Father(carla,david) }

T 2

P = T 1 P ∪ { Parent(alice,bob), Parent(alice,carla), Parent(evan,carla), Parent(carla,david), SameGeneration(alice,alice), SameGeneration(bob,bob), SameGeneration(carla,carla), SameGeneration(david,david), SameGeneration(evan,evan) }

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SLIDE 12

Exercise 2

Exercise. Consider the example Datalog program from the lecture:

(a) Father(alice,bob) (b) Mother(alice,carla) (c) Mother(evan,carla) (d) Father(carla,david)

Parent(x,y) ← Father(x,y)

(1)

Parent(x,y) ← Mother(x,y)

(2)

Ancestor(x,y) ← Parent(x,y)

(3)

Ancestor(x,z) ← Parent(x,y) ∧ Ancestor(y,z)

(4)

SameGeneration(x,x) ←

(5)

SameGeneration(x,y) ← Parent(x,v) ∧ Parent(y,w) ∧ SameGeneration(v,w)

(6)

1. Give a proof tree for SameGeneration(evan, alice).
2. Compute the sets T 0

P, T 1 P, T 2 P, . . . When is the fixed point reached?

Solution. 2. T 0

P = ∅

T 1

P = { Father(alice,bob), Mother(alice,carla), Mother(evan,carla), Father(carla,david) }

T 2

P = T 1 P ∪ { Parent(alice,bob), Parent(alice,carla), Parent(evan,carla), Parent(carla,david), SameGeneration(alice,alice), SameGeneration(bob,bob), SameGeneration(carla,carla), SameGeneration(david,david), SameGeneration(evan,evan) }

T 3

P = T 2 P ∪ { Ancestor(alice,bob), Ancestor(alice,carla), Ancestor(evan,carla), Ancestor(carla,david), SameGeneration(alice,evan), SameGeneration(evan,alice) }

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SLIDE 13

Exercise 2

Exercise. Consider the example Datalog program from the lecture:

(a) Father(alice,bob) (b) Mother(alice,carla) (c) Mother(evan,carla) (d) Father(carla,david)

Parent(x,y) ← Father(x,y)

(1)

Parent(x,y) ← Mother(x,y)

(2)

Ancestor(x,y) ← Parent(x,y)

(3)

Ancestor(x,z) ← Parent(x,y) ∧ Ancestor(y,z)

(4)

SameGeneration(x,x) ←

(5)

SameGeneration(x,y) ← Parent(x,v) ∧ Parent(y,w) ∧ SameGeneration(v,w)

(6)

1. Give a proof tree for SameGeneration(evan, alice).
2. Compute the sets T 0

P, T 1 P, T 2 P, . . . When is the fixed point reached?

Solution. 2. T 0

P = ∅

T 1

P = { Father(alice,bob), Mother(alice,carla), Mother(evan,carla), Father(carla,david) }

T 2

P = T 1 P ∪ { Parent(alice,bob), Parent(alice,carla), Parent(evan,carla), Parent(carla,david), SameGeneration(alice,alice), SameGeneration(bob,bob), SameGeneration(carla,carla), SameGeneration(david,david), SameGeneration(evan,evan) }

T 3

P = T 2 P ∪ { Ancestor(alice,bob), Ancestor(alice,carla), Ancestor(evan,carla), Ancestor(carla,david), SameGeneration(alice,evan), SameGeneration(evan,alice) }

T 4

P = T 3 P ∪ { Ancestor(alice,david), Ancestor(evan,david) } = T 5 P

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Exercise 2

Exercise. Consider the example Datalog program from the lecture:

(a) Father(alice,bob) (b) Mother(alice,carla) (c) Mother(evan,carla) (d) Father(carla,david)

Parent(x,y) ← Father(x,y)

(1)

Parent(x,y) ← Mother(x,y)

(2)

Ancestor(x,y) ← Parent(x,y)

(3)

Ancestor(x,z) ← Parent(x,y) ∧ Ancestor(y,z)

(4)

SameGeneration(x,x) ←

(5)

SameGeneration(x,y) ← Parent(x,v) ∧ Parent(y,w) ∧ SameGeneration(v,w)

(6)

1. Give a proof tree for SameGeneration(evan, alice).
2. Compute the sets T 0

P, T 1 P, T 2 P, . . . When is the fixed point reached?

Solution. 2. T 0

P = ∅

T 1

P = { Father(alice,bob), Mother(alice,carla), Mother(evan,carla), Father(carla,david) }

T 2

P = T 1 P ∪ { Parent(alice,bob), Parent(alice,carla), Parent(evan,carla), Parent(carla,david), SameGeneration(alice,alice), SameGeneration(bob,bob), SameGeneration(carla,carla), SameGeneration(david,david), SameGeneration(evan,evan) }

T 3

P = T 2 P ∪ { Ancestor(alice,bob), Ancestor(alice,carla), Ancestor(evan,carla), Ancestor(carla,david), SameGeneration(alice,evan), SameGeneration(evan,alice) }

T 4

P = T 3 P ∪ { Ancestor(alice,david), Ancestor(evan,david) } = T 5 P = T ∞ P

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SLIDE 15

Exercise 3

Exercise. Consider databases that encode a labelled, directed graph by means of a ternary EDB predicate e (“edge”).

The first two parameters are the source and target nodes of the edge, while the third parameter is its label. For example, the edge m

a

− → n would be represented by the fact e(m, n, a). Moreover, assume that only constants a and b are used as labels. Can you express the following queries using Datalog?

1. “Which nodes in the graph are reachable from the node n?”
2. “Are all nodes of the graph reachable from the node n?”
3. “Does the graph have a directed cycle?”
4. “Does the graph have a path that is labelled by a palindrome?”

(a palindrome is a word that reads the same forwards and backwards)

5. “Is the connected component that contains the node n 2-colourable?”
6. “Is the graph 2-colourable?”
7. “Which pairs of nodes are connected by a path with an even number of a labels?”
8. “Which pairs of nodes are connected by a path with the same number of a and b labels?”
9. “Is there a pair of nodes that is connected by two distinct paths?”

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SLIDE 16

Exercise 3

Exercise. Consider databases that encode a labelled, directed graph by means of a ternary EDB predicate e (“edge”).

The first two parameters are the source and target nodes of the edge, while the third parameter is its label. For example, the edge m

a

− → n would be represented by the fact e(m, n, a). Moreover, assume that only constants a and b are used as labels. Can you express the following queries using Datalog?

1. “Which nodes in the graph are reachable from the node n?”
2. “Are all nodes of the graph reachable from the node n?”
3. “Does the graph have a directed cycle?”
4. “Does the graph have a path that is labelled by a palindrome?”

(a palindrome is a word that reads the same forwards and backwards)

5. “Is the connected component that contains the node n 2-colourable?”
6. “Is the graph 2-colourable?”
7. “Which pairs of nodes are connected by a path with an even number of a labels?”
8. “Which pairs of nodes are connected by a path with the same number of a and b labels?”
9. “Is there a pair of nodes that is connected by two distinct paths?”

Solution.

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SLIDE 17

Exercise 3

Exercise. Consider databases that encode a labelled, directed graph by means of a ternary EDB predicate e (“edge”).

The first two parameters are the source and target nodes of the edge, while the third parameter is its label. For example, the edge m

a

− → n would be represented by the fact e(m, n, a). Moreover, assume that only constants a and b are used as labels. Can you express the following queries using Datalog?

1. “Which nodes in the graph are reachable from the node n?”
2. “Are all nodes of the graph reachable from the node n?”
3. “Does the graph have a directed cycle?”
4. “Does the graph have a path that is labelled by a palindrome?”

(a palindrome is a word that reads the same forwards and backwards)

5. “Is the connected component that contains the node n 2-colourable?”
6. “Is the graph 2-colourable?”
7. “Which pairs of nodes are connected by a path with an even number of a labels?”
8. “Which pairs of nodes are connected by a path with the same number of a and b labels?”
9. “Is there a pair of nodes that is connected by two distinct paths?”

Solution. 1. Reachable(x, y) ← e(x, y, v) Reachable(x, z) ← e(x, y, v) ∧ Reachable(y, z) Ans(x) ← Reachable(n, x)

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SLIDE 18

Exercise 3

Exercise. Consider databases that encode a labelled, directed graph by means of a ternary EDB predicate e (“edge”).

The first two parameters are the source and target nodes of the edge, while the third parameter is its label. For example, the edge m

a

− → n would be represented by the fact e(m, n, a). Moreover, assume that only constants a and b are used as labels. Can you express the following queries using Datalog?

1. “Which nodes in the graph are reachable from the node n?”
2. “Are all nodes of the graph reachable from the node n?”
3. “Does the graph have a directed cycle?”
4. “Does the graph have a path that is labelled by a palindrome?”

(a palindrome is a word that reads the same forwards and backwards)

5. “Is the connected component that contains the node n 2-colourable?”
6. “Is the graph 2-colourable?”
7. “Which pairs of nodes are connected by a path with an even number of a labels?”
8. “Which pairs of nodes are connected by a path with the same number of a and b labels?”
9. “Is there a pair of nodes that is connected by two distinct paths?”

Solution.

2. Not expressible, since Datalog is monotone: any query that is true for some set of ground facts I is also true for

every set of ground facts J ⊇ I, but the query is true on I = { e(n, n, a) }, but not on J = I ∪ { e(m, m, b) }.

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SLIDE 19

Exercise 3

Exercise. Consider databases that encode a labelled, directed graph by means of a ternary EDB predicate e (“edge”).

The first two parameters are the source and target nodes of the edge, while the third parameter is its label. For example, the edge m

a

− → n would be represented by the fact e(m, n, a). Moreover, assume that only constants a and b are used as labels. Can you express the following queries using Datalog?

1. “Which nodes in the graph are reachable from the node n?”
2. “Are all nodes of the graph reachable from the node n?”
3. “Does the graph have a directed cycle?”
4. “Does the graph have a path that is labelled by a palindrome?”

(a palindrome is a word that reads the same forwards and backwards)

5. “Is the connected component that contains the node n 2-colourable?”
6. “Is the graph 2-colourable?”
7. “Which pairs of nodes are connected by a path with an even number of a labels?”
8. “Which pairs of nodes are connected by a path with the same number of a and b labels?”
9. “Is there a pair of nodes that is connected by two distinct paths?”

Solution. 3. Reachable(x, y) ← e(x, y, v) Reachable(x, z) ← e(x, y, v) ∧ Reachable(y, z) Ans() ← Reachable(x, x)

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SLIDE 20

Exercise 3

Exercise. Consider databases that encode a labelled, directed graph by means of a ternary EDB predicate e (“edge”).

The first two parameters are the source and target nodes of the edge, while the third parameter is its label. For example, the edge m

a

− → n would be represented by the fact e(m, n, a). Moreover, assume that only constants a and b are used as labels. Can you express the following queries using Datalog?

1. “Which nodes in the graph are reachable from the node n?”
2. “Are all nodes of the graph reachable from the node n?”
3. “Does the graph have a directed cycle?”
4. “Does the graph have a path that is labelled by a palindrome?”

(a palindrome is a word that reads the same forwards and backwards)

5. “Is the connected component that contains the node n 2-colourable?”
6. “Is the graph 2-colourable?”
7. “Which pairs of nodes are connected by a path with an even number of a labels?”
8. “Which pairs of nodes are connected by a path with the same number of a and b labels?”
9. “Is there a pair of nodes that is connected by two distinct paths?”

Solution. 4. Reachable(x, y) ← e(x, y, v) Reachable(x, z) ← e(x, y, a), Reachable(y, w), e(w, z, a) Reachable(x, z) ← e(x, y, b), Reachable(y, w), e(w, z, b) Ans() ← Reachable(x, y)

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SLIDE 21

Exercise 3

Exercise. Consider databases that encode a labelled, directed graph by means of a ternary EDB predicate e (“edge”).

The first two parameters are the source and target nodes of the edge, while the third parameter is its label. For example, the edge m

a

− → n would be represented by the fact e(m, n, a). Moreover, assume that only constants a and b are used as labels. Can you express the following queries using Datalog?

1. “Which nodes in the graph are reachable from the node n?”
2. “Are all nodes of the graph reachable from the node n?”
3. “Does the graph have a directed cycle?”
4. “Does the graph have a path that is labelled by a palindrome?”

(a palindrome is a word that reads the same forwards and backwards)

5. “Is the connected component that contains the node n 2-colourable?”
6. “Is the graph 2-colourable?”
7. “Which pairs of nodes are connected by a path with an even number of a labels?”
8. “Which pairs of nodes are connected by a path with the same number of a and b labels?”
9. “Is there a pair of nodes that is connected by two distinct paths?”

Solution.

5. Not expressible; consider I = { e(n, 1, a), e(1, 2, a) } and J = I ∪ { e(2, n, a) }.

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SLIDE 22

Exercise 3

Exercise. Consider databases that encode a labelled, directed graph by means of a ternary EDB predicate e (“edge”).

The first two parameters are the source and target nodes of the edge, while the third parameter is its label. For example, the edge m

a

− → n would be represented by the fact e(m, n, a). Moreover, assume that only constants a and b are used as labels. Can you express the following queries using Datalog?

1. “Which nodes in the graph are reachable from the node n?”
2. “Are all nodes of the graph reachable from the node n?”
3. “Does the graph have a directed cycle?”
4. “Does the graph have a path that is labelled by a palindrome?”

(a palindrome is a word that reads the same forwards and backwards)

5. “Is the connected component that contains the node n 2-colourable?”
6. “Is the graph 2-colourable?”
7. “Which pairs of nodes are connected by a path with an even number of a labels?”
8. “Which pairs of nodes are connected by a path with the same number of a and b labels?”
9. “Is there a pair of nodes that is connected by two distinct paths?”

Solution.

6. Not expressible; consider I = { e(n, 1, a), e(1, 2, a) } and J = I ∪ { e(2, n, a) }.

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SLIDE 23

Exercise 3

Exercise. Consider databases that encode a labelled, directed graph by means of a ternary EDB predicate e (“edge”).

The first two parameters are the source and target nodes of the edge, while the third parameter is its label. For example, the edge m

a

− → n would be represented by the fact e(m, n, a). Moreover, assume that only constants a and b are used as labels. Can you express the following queries using Datalog?

1. “Which nodes in the graph are reachable from the node n?”
2. “Are all nodes of the graph reachable from the node n?”
3. “Does the graph have a directed cycle?”
4. “Does the graph have a path that is labelled by a palindrome?”

(a palindrome is a word that reads the same forwards and backwards)

5. “Is the connected component that contains the node n 2-colourable?”
6. “Is the graph 2-colourable?”
7. “Which pairs of nodes are connected by a path with an even number of a labels?”
8. “Which pairs of nodes are connected by a path with the same number of a and b labels?”
9. “Is there a pair of nodes that is connected by two distinct paths?”

Solution. 7. Reachable(x, y) ← e(x, y, b) Reachable(x, z) ← e(x, y, a), e(y, z, a) Reachable(x, z) ← e(x, y, a), Reachable(y, w), e(w, z, a) Reachable(x, z) ← Reachable(x, y), Reachable(y, z) Ans(x, y) ← Reachable(x, y)

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SLIDE 24

Exercise 3

Exercise. Consider databases that encode a labelled, directed graph by means of a ternary EDB predicate e (“edge”).

The first two parameters are the source and target nodes of the edge, while the third parameter is its label. For example, the edge m

a

− → n would be represented by the fact e(m, n, a). Moreover, assume that only constants a and b are used as labels. Can you express the following queries using Datalog?

1. “Which nodes in the graph are reachable from the node n?”
2. “Are all nodes of the graph reachable from the node n?”
3. “Does the graph have a directed cycle?”
4. “Does the graph have a path that is labelled by a palindrome?”

(a palindrome is a word that reads the same forwards and backwards)

5. “Is the connected component that contains the node n 2-colourable?”
6. “Is the graph 2-colourable?”
7. “Which pairs of nodes are connected by a path with an even number of a labels?”
8. “Which pairs of nodes are connected by a path with the same number of a and b labels?”
9. “Is there a pair of nodes that is connected by two distinct paths?”

Solution. 8. Reachable(x, z) ← e(x, y, a), e(y, z, b) Reachable(x, z) ← e(x, y, b), e(y, z, a) Reachable(x, z) ← Reachable(x, y), Reachable(y, z) Ans(x, y) ← Reachable(x, y)

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SLIDE 25

Exercise 3

Exercise. Consider databases that encode a labelled, directed graph by means of a ternary EDB predicate e (“edge”).

The first two parameters are the source and target nodes of the edge, while the third parameter is its label. For example, the edge m

a

− → n would be represented by the fact e(m, n, a). Moreover, assume that only constants a and b are used as labels. Can you express the following queries using Datalog?

1. “Which nodes in the graph are reachable from the node n?”
2. “Are all nodes of the graph reachable from the node n?”
3. “Does the graph have a directed cycle?”
4. “Does the graph have a path that is labelled by a palindrome?”

(a palindrome is a word that reads the same forwards and backwards)

5. “Is the connected component that contains the node n 2-colourable?”
6. “Is the graph 2-colourable?”
7. “Which pairs of nodes are connected by a path with an even number of a labels?”
8. “Which pairs of nodes are connected by a path with the same number of a and b labels?”
9. “Is there a pair of nodes that is connected by two distinct paths?”

Solution.

9. Not expressible, since Datalog is homomorphism-closed; consider I = { e(n, 1, a), e(1, m, a), e(n, 2, a), e(2, m, a) }

and J = { e(n, 1, a), e(1, m, a) } and the homomorphism ϕ : I → J = { 2 → 1 }.

25 / 47

SLIDE 26

Exercise 4

Exercise. Consider a UCQ of the following form

(r11(x) ∧ r12(x)) ∨ . . . ∨ (rℓ1(x) ∧ rℓ2(x)) Find a Datalog query that expresses this UCQ. How many rules and how many additional IDB predicates does your solution use (depending on ℓ)?

26 / 47

SLIDE 27

Exercise 4

Exercise. Consider a UCQ of the following form

(r11(x) ∧ r12(x)) ∨ . . . ∨ (rℓ1(x) ∧ rℓ2(x)) Find a Datalog query that expresses this UCQ. How many rules and how many additional IDB predicates does your solution use (depending on ℓ)? Solution.

27 / 47

SLIDE 28

Exercise 4

Exercise. Consider a UCQ of the following form

(r11(x) ∧ r12(x)) ∨ . . . ∨ (rℓ1(x) ∧ rℓ2(x)) Find a Datalog query that expresses this UCQ. How many rules and how many additional IDB predicates does your solution use (depending on ℓ)? Solution. Ans(x) ← r11(x), r12(x) Ans(x) ← r21(x), r22(x) . . . . . . . . . Ans(x) ← rℓ1(x), rℓ2(x)

28 / 47

SLIDE 29

Exercise 4

Exercise. Consider a UCQ of the following form

(r11(x) ∧ r12(x)) ∨ . . . ∨ (rℓ1(x) ∧ rℓ2(x)) Find a Datalog query that expresses this UCQ. How many rules and how many additional IDB predicates does your solution use (depending on ℓ)? Solution. Ans(x) ← r11(x), r12(x) Ans(x) ← r21(x), r22(x) . . . . . . . . . Ans(x) ← rℓ1(x), rℓ2(x) This solution uses ℓ rules and one additional IDB predicate.

29 / 47

SLIDE 30

Exercise 5

Exercise. Consider a Datalog query of the following form:

A1(x) ← r11(x) . . . Aℓ(x) ← rℓ1(x) A1(x) ← r12(x) . . . Aℓ(x) ← rℓ2(x) Ans(x) ← A1(x), . . . , Aℓ(x) Find a UCQ that expresses this Datalog query. How many CQs does your solution contain (depending on ℓ)?

30 / 47

SLIDE 31

Exercise 5

Exercise. Consider a Datalog query of the following form:

A1(x) ← r11(x) . . . Aℓ(x) ← rℓ1(x) A1(x) ← r12(x) . . . Aℓ(x) ← rℓ2(x) Ans(x) ← A1(x), . . . , Aℓ(x) Find a UCQ that expresses this Datalog query. How many CQs does your solution contain (depending on ℓ)? Solution.

31 / 47

SLIDE 32

Exercise 5

Exercise. Consider a Datalog query of the following form:

A1(x) ← r11(x) . . . Aℓ(x) ← rℓ1(x) A1(x) ← r12(x) . . . Aℓ(x) ← rℓ2(x) Ans(x) ← A1(x), . . . , Aℓ(x) Find a UCQ that expresses this Datalog query. How many CQs does your solution contain (depending on ℓ)? Solution. ϕ11···1(x) = r11(x) ∧ r21(x) ∧ · · · ∧ rℓ1(x) ϕ21···1(x) = r12(x) ∧ r21(x) ∧ · · · ∧ rℓ1(x) ϕ12···1(x) = r11(x) ∧ r22(x) ∧ · · · ∧ rℓ1(x) ϕ22···1(x) = r12(x) ∧ r22(x) ∧ · · · ∧ rℓ1(x) . . . . . . . . . . . . ϕ22···2(x) = r12(x) ∧ r22(x) ∧ · · · ∧ rℓ2(x) ϕ =

i∈{ 11···1,21···1,...,22···2 } ϕi

32 / 47

SLIDE 33

Exercise 5

Exercise. Consider a Datalog query of the following form:

A1(x) ← r11(x) . . . Aℓ(x) ← rℓ1(x) A1(x) ← r12(x) . . . Aℓ(x) ← rℓ2(x) Ans(x) ← A1(x), . . . , Aℓ(x) Find a UCQ that expresses this Datalog query. How many CQs does your solution contain (depending on ℓ)? Solution. ϕ11···1(x) = r11(x) ∧ r21(x) ∧ · · · ∧ rℓ1(x) ϕ21···1(x) = r12(x) ∧ r21(x) ∧ · · · ∧ rℓ1(x) ϕ12···1(x) = r11(x) ∧ r22(x) ∧ · · · ∧ rℓ1(x) ϕ22···1(x) = r12(x) ∧ r22(x) ∧ · · · ∧ rℓ1(x) . . . . . . . . . . . . ϕ22···2(x) = r12(x) ∧ r22(x) ∧ · · · ∧ rℓ2(x) ϕ =

i∈{ 11···1,21···1,...,22···2 } ϕi

This solution uses 2ℓ CQs.

33 / 47

SLIDE 34

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

34 / 47

SLIDE 35

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution.

35 / 47

SLIDE 36

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅).

36 / 47

SLIDE 37

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅). ◮ Assume that I ⊆ TP(I) for some set of ground facts I, and consider a ground fact H ∈ TP(I). Then there is some ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ I. Since I ⊆ TP(I), we have B1, . . . , Bn ∈ TP(I), and hence H ∈ TP(TP(I)).

37 / 47

SLIDE 38

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅). ◮ Assume that I ⊆ TP(I) for some set of ground facts I, and consider a ground fact H ∈ TP(I). Then there is some ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ I. Since I ⊆ TP(I), we have B1, . . . , Bn ∈ TP(I), and hence H ∈ TP(TP(I)). ◮ Thus, we have T i−1

P

⊆ T i

P, and, in particular, TP(T ∞ P ) ⊇ T ∞ P . 38 / 47

SLIDE 39

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅). ◮ Assume that I ⊆ TP(I) for some set of ground facts I, and consider a ground fact H ∈ TP(I). Then there is some ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ I. Since I ⊆ TP(I), we have B1, . . . , Bn ∈ TP(I), and hence H ∈ TP(TP(I)). ◮ Thus, we have T i−1

P

⊆ T i

P, and, in particular, TP(T ∞ P ) ⊇ T ∞ P .

◮ Assume that we have some ground fact H ∈ TP(T ∞

P ), but H T ∞ P . 39 / 47

SLIDE 40

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅). ◮ Assume that I ⊆ TP(I) for some set of ground facts I, and consider a ground fact H ∈ TP(I). Then there is some ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ I. Since I ⊆ TP(I), we have B1, . . . , Bn ∈ TP(I), and hence H ∈ TP(TP(I)). ◮ Thus, we have T i−1

P

⊆ T i

P, and, in particular, TP(T ∞ P ) ⊇ T ∞ P .

◮ Assume that we have some ground fact H ∈ TP(T ∞

P ), but H T ∞ P .

◮ Then there is a ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ T ∞

P . 40 / 47

SLIDE 41

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅). ◮ Assume that I ⊆ TP(I) for some set of ground facts I, and consider a ground fact H ∈ TP(I). Then there is some ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ I. Since I ⊆ TP(I), we have B1, . . . , Bn ∈ TP(I), and hence H ∈ TP(TP(I)). ◮ Thus, we have T i−1

P

⊆ T i

P, and, in particular, TP(T ∞ P ) ⊇ T ∞ P .

◮ Assume that we have some ground fact H ∈ TP(T ∞

P ), but H T ∞ P .

◮ Then there is a ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ T ∞

P .

◮ Since T ∞

P = i≥0 T i P, there are i1, . . . , in with Bij ∈ T ij P , and thus B1, . . . , Bn ∈ T k P with k = max{ i1, . . . , in }. 41 / 47

SLIDE 42

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅). ◮ Assume that I ⊆ TP(I) for some set of ground facts I, and consider a ground fact H ∈ TP(I). Then there is some ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ I. Since I ⊆ TP(I), we have B1, . . . , Bn ∈ TP(I), and hence H ∈ TP(TP(I)). ◮ Thus, we have T i−1

P

⊆ T i

P, and, in particular, TP(T ∞ P ) ⊇ T ∞ P .

◮ Assume that we have some ground fact H ∈ TP(T ∞

P ), but H T ∞ P .

◮ Then there is a ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ T ∞

P .

◮ Since T ∞

P = i≥0 T i P, there are i1, . . . , in with Bij ∈ T ij P , and thus B1, . . . , Bn ∈ T k P with k = max{ i1, . . . , in }.

◮ But then H ∈ TP(T k

P ) = T k+1 P

⊆ T ∞

P , which contradicts H T ∞ P . 42 / 47

SLIDE 43

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅). ◮ Assume that I ⊆ TP(I) for some set of ground facts I, and consider a ground fact H ∈ TP(I). Then there is some ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ I. Since I ⊆ TP(I), we have B1, . . . , Bn ∈ TP(I), and hence H ∈ TP(TP(I)). ◮ Thus, we have T i−1

P

⊆ T i

P, and, in particular, TP(T ∞ P ) ⊇ T ∞ P .

◮ Assume that we have some ground fact H ∈ TP(T ∞

P ), but H T ∞ P .

◮ Then there is a ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ T ∞

P .

◮ Since T ∞

P = i≥0 T i P, there are i1, . . . , in with Bij ∈ T ij P , and thus B1, . . . , Bn ∈ T k P with k = max{ i1, . . . , in }.

◮ But then H ∈ TP(T k

P ) = T k+1 P

⊆ T ∞

P , which contradicts H T ∞ P .

2.

◮ First, note that TP is clearly monotone, i.e., that for sets I ⊆ J of ground facts, we have TP(I) ⊆ TP(J).

43 / 47

SLIDE 44

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅). ◮ Assume that I ⊆ TP(I) for some set of ground facts I, and consider a ground fact H ∈ TP(I). Then there is some ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ I. Since I ⊆ TP(I), we have B1, . . . , Bn ∈ TP(I), and hence H ∈ TP(TP(I)). ◮ Thus, we have T i−1

P

⊆ T i

P, and, in particular, TP(T ∞ P ) ⊇ T ∞ P .

◮ Assume that we have some ground fact H ∈ TP(T ∞

P ), but H T ∞ P .

◮ Then there is a ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ T ∞

P .

◮ Since T ∞

P = i≥0 T i P, there are i1, . . . , in with Bij ∈ T ij P , and thus B1, . . . , Bn ∈ T k P with k = max{ i1, . . . , in }.

◮ But then H ∈ TP(T k

P ) = T k+1 P

⊆ T ∞

P , which contradicts H T ∞ P .

2.

◮ First, note that TP is clearly monotone, i.e., that for sets I ⊆ J of ground facts, we have TP(I) ⊆ TP(J). ◮ Consider some fixed point F of TP. We show T i

P ⊆ F for all i ≥ 0. 44 / 47

SLIDE 45

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅). ◮ Assume that I ⊆ TP(I) for some set of ground facts I, and consider a ground fact H ∈ TP(I). Then there is some ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ I. Since I ⊆ TP(I), we have B1, . . . , Bn ∈ TP(I), and hence H ∈ TP(TP(I)). ◮ Thus, we have T i−1

P

⊆ T i

P, and, in particular, TP(T ∞ P ) ⊇ T ∞ P .

◮ Assume that we have some ground fact H ∈ TP(T ∞

P ), but H T ∞ P .

◮ Then there is a ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ T ∞

P .

◮ Since T ∞

P = i≥0 T i P, there are i1, . . . , in with Bij ∈ T ij P , and thus B1, . . . , Bn ∈ T k P with k = max{ i1, . . . , in }.

◮ But then H ∈ TP(T k

P ) = T k+1 P

⊆ T ∞

P , which contradicts H T ∞ P .

2.

◮ First, note that TP is clearly monotone, i.e., that for sets I ⊆ J of ground facts, we have TP(I) ⊆ TP(J). ◮ Consider some fixed point F of TP. We show T i

P ⊆ F for all i ≥ 0.

◮ Clearly, T 0

P = ∅ ⊆ F. 45 / 47

SLIDE 46

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅). ◮ Assume that I ⊆ TP(I) for some set of ground facts I, and consider a ground fact H ∈ TP(I). Then there is some ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ I. Since I ⊆ TP(I), we have B1, . . . , Bn ∈ TP(I), and hence H ∈ TP(TP(I)). ◮ Thus, we have T i−1

P

⊆ T i

P, and, in particular, TP(T ∞ P ) ⊇ T ∞ P .

◮ Assume that we have some ground fact H ∈ TP(T ∞

P ), but H T ∞ P .

◮ Then there is a ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ T ∞

P .

◮ Since T ∞

P = i≥0 T i P, there are i1, . . . , in with Bij ∈ T ij P , and thus B1, . . . , Bn ∈ T k P with k = max{ i1, . . . , in }.

◮ But then H ∈ TP(T k

P ) = T k+1 P

⊆ T ∞

P , which contradicts H T ∞ P .

2.

◮ First, note that TP is clearly monotone, i.e., that for sets I ⊆ J of ground facts, we have TP(I) ⊆ TP(J). ◮ Consider some fixed point F of TP. We show T i

P ⊆ F for all i ≥ 0.

◮ Clearly, T 0

P = ∅ ⊆ F.

◮ Assume that T i

P ⊆ F for some i ≥ 0. Then T i+1 P

= TP(T i

P) ⊆ TP(F) = F, by monotonicity and since F is a fixed point. 46 / 47

SLIDE 47

Exercise 6

Exercise. Show that T ∞

P is the least fixed point of the TP operator.

1. Show that it is a fixed point, i.e., that TP(T ∞

P ) = T ∞ P .

2. Show that every fixed point of TP must contain every fact in T ∞

P .

Solution. 1.

◮ We first show that TP is extensive, i.e., that I ⊆ TP(I) for any set of ground facts I: Clearly ∅ ⊆ TP(∅). ◮ Assume that I ⊆ TP(I) for some set of ground facts I, and consider a ground fact H ∈ TP(I). Then there is some ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ I. Since I ⊆ TP(I), we have B1, . . . , Bn ∈ TP(I), and hence H ∈ TP(TP(I)). ◮ Thus, we have T i−1

P

⊆ T i

P, and, in particular, TP(T ∞ P ) ⊇ T ∞ P .

◮ Assume that we have some ground fact H ∈ TP(T ∞

P ), but H T ∞ P .

◮ Then there is a ground rule H ← B1, . . . , Bn ∈ ground(P) with B1, . . . , Bn ∈ T ∞

P .

◮ Since T ∞

P = i≥0 T i P, there are i1, . . . , in with Bij ∈ T ij P , and thus B1, . . . , Bn ∈ T k P with k = max{ i1, . . . , in }.

◮ But then H ∈ TP(T k

P ) = T k+1 P

⊆ T ∞

P , which contradicts H T ∞ P .

2.

◮ First, note that TP is clearly monotone, i.e., that for sets I ⊆ J of ground facts, we have TP(I) ⊆ TP(J). ◮ Consider some fixed point F of TP. We show T i

P ⊆ F for all i ≥ 0.

◮ Clearly, T 0

P = ∅ ⊆ F.

◮ Assume that T i

P ⊆ F for some i ≥ 0. Then T i+1 P

= TP(T i

P) ⊆ TP(F) = F, by monotonicity and since F is a fixed point.

◮ But then T i

P ⊆ F for all i ≥ 0, and hence also T ∞ P = i≥0 T i P ⊆ F. 47 / 47