CSE-4/562 Spring 2019

Think of the relation as a function from potential facts to their truth value.

R	A	B
1	1	2	→ T
2	1	3	→ T
3	2	3	→ T
4	2	4	→ T
5	1	1	→ F
6	...		→ F

Every row not explicitly listed is mapped to False

R	A	B
1	1	2
2	1	3
3	2	3
4	2	4

S	B	C
1	2	1
2	2	2
3	3	3

$Q(1) \equiv ~~~~~F~~~~ \wedge S(1, 1)$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge S(2, 1)$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge S(3, 1)$

...

$~~~~~~~~ \vee ~~ ~~~~~F~~~~ \wedge S(1, 2)$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge S(2, 2)$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge S(3, 2)$

...

R	A	B
1	1	2
2	1	3
3	2	3
4	2	4

S	B	C
1	2	1
2	2	2
3	3	3

$Q(1) \equiv ~~~~~F~~~~ \wedge ~~~~F~~~~~$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge ~~~~T~~~~~$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge ~~~~F~~~~~$

...

$~~~~~~~~ \vee ~~ ~~~~~F~~~~ \wedge ~~~~F~~~~~$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge ~~~~T~~~~~$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge ~~~~F~~~~~$

...

R	A	B
1	1	2
2	1	3
3	2	3
4	2	4

S	B	C
1	2	1
2	2	2
3	3	3

$Q(1) \equiv$ $~~~~~~F~~~~ \wedge ~~~~F~~~~~$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge ~~~~T~~~~~$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge ~~~~F~~~~~$

...

$~~~~~~~~ \vee ~~ ~~~~~F~~~~ \wedge ~~~~F~~~~~$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge ~~~~T~~~~~$

$~~~~~~~~ \vee ~~ ~~~~~T~~~~ \wedge ~~~~F~~~~~$

...

R	A	B
1	1	2
2	1	2
3	1	2
4	1	3
5	2	3
6	2	3
7	2	4

R	A	B
1	1	2	→ 3
2	1	3	→ 1
3	2	3	→ 2
4	2	4	→ 1

R	A	B
1	1	2	→ 3
2	1	3	→ 1
3	2	3	→ 2
4	2	4	→ 1

S	B	C
1	2	1	→ 1
2	2	2	→ 2
3	3	3	→ 3

R	A	B
1	1	2	→ 3
2	1	3	→ 1
3	2	3	→ 2
4	2	4	→ 1

S	B	C
1	2	1	→ 1
2	2	2	→ 2
3	3	3	→ 3

R	A	B
1	1	2	→ 3
2	1	3	→ 1
3	2	3	→ 2
4	2	4	→ 1

S	B	C
1	2	1	→ 1
2	2	2	→ 2
3	3	3	→ 3

R	A	B
1	1	2	→ a
2	1	3	→ b
3	2	3	→ c
4	2	4	→ d
5	...	→ $\mathbf{0}$

S	B	C
1	2	1	→ e
2	2	2	→ f
3	3	3	→ g
4	...	→ $\mathbf{0}$

R	A	B
1	1	2	→ a
2	1	3	→ b
3	2	3	→ c
4	2	4	→ d
5	...	→ $\mathbf{0}$

S	B	C
1	2	1	→ e
2	2	2	→ f
3	3	3	→ g
4	...	→ $\mathbf{0}$

Ground Rules for $\oplus$ , $\otimes$

Commutative, Associative
Must be some $\mathbf{0}$ s.t. $a \oplus \mathbf{0} = a$
Must be some $\mathbf{1}$ s.t. $a \otimes \mathbf{1} = a$
$a \otimes \mathbf{0} = \mathbf{0}$
$a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c)$

Any pair of operators (along with their domain) that follows these rules is called a commutative semiring

Bringing it Back to RA

If a Table is a function, so is a query result!

$[[\pi_A R(A, B)]](x)$

$[[\sigma_\phi R(A, B)]](x, y)$

$[[R(A, B) \cup S(A, B)]](x, y)$

$[[R(A, B) \times S(B, C)]](x, y, z)$

Domain of Tuple IDs: $\mathbb T$

A Set of Tuple IDs: $2^{\mathbb T}$

A Set of Sets of Tuple IDs: $2^{2^{\mathbb T}}$

e.g., $\{ \{t_1, t_5\}, \{t_1, t_6\}, \{t_2, t_7\} \}$

Provenance

CSE-4/562 Spring 2019

Textbook: Readings Only

Ground Rules for $\oplus$ , $\otimes$

Commutative Semirings

Bringing it Back to RA

Mimicking RA

Adding sets of sets

Multiplying sets of sets