How does the input data relate to a query output.

Types of Provenance

Why Provenance (Lineage)

What's the smallest fragment of my input needed to produce some row

Why-Not Provenance

What's the least I can add to my input to get a desired row

How Provenance

An execution trace of the result; How were the tuples combined?

Where Provenance

Which cell(s) was a given output value taken from

Taint

Was the output affected by any "tainted" input cell/row

Why is $\left<1\right>$ in $\pi_A (R \bowtie S)$?

$\left\{ R_1, S_1 \right\}$, $\left\{ R_1, S_2 \right\}$, $\left\{ R_2, S_3 \right\}$

Witness: Any subset of the original database that still produces the same result.
(Generally we only want 'minimal' witnesses)

How is $\left<1\right>$ derived in $\pi_A (R \bowtie S)$?

$(R_1 \bowtie S_1) \oplus (R_1 \bowtie S_2) \oplus (R_2 \bowtie S_3)$

Stop thinking about relations as collections of records, and instead think of them as collections of facts

The fact $R(1, 2)$ is true.

The fact $R(2, 1)$ is false (or unknown).

A table contains all facts that are provably true.

$$Q(A) :-~~ R(A, B), S(B, C)$$

• there is some $B$ and $C$ for which...
• the fact $R(A, B)$ is true, and...
• the fact $S(B, C)$ is true.

$\forall A : \big( \exists B, C : R(A, B) \wedge S(B, C) \big) \rightarrow Q(A)$

$$Q(A) :-~~ R(A, B), S(B, C)$$ $$Q(A) :-~~ R(A, B), R(B, C)$$

Treat multiple rules as a disjunction.
($Q(A)$ is true if any rule is satisfied)

As powerful as Set-RA

Projection

$Q := \pi_A(R)$

$Q(A) :-~~ R(A, \ldots)$

Union

$Q := R \cup S$

$Q(\ldots) :-~~ R(\ldots)$

$Q(\ldots) :-~~ S(\ldots)$

Join

$Q := R \bowtie S$

$Q(\ldots) :-~~ R(\ldots), S(\ldots)$

Selection (Equality)

$Q := \sigma_{R.A = R.B}(R)$

$Q(A) :-~~ R(A, A)$

Selection (Equality')

$Q := \sigma_{R.A = 1}(R)$

$Q(B) :-~~ R(1, B)$

Selection (Other)

$Q := \sigma_{A > B}(R)$

$Q(A,B) :-~~ R(A, B), [[ A > B ]]$

Relations are Sets of Facts. We can have a relation consisting of all pairs $A, B$ where $A$ is bigger.

A Finite Relation

... declares a finite number of true facts

An Infinite Relation

... declares an infinite number of true facts

Safety Property: Every variable must appear in at least one finite relation in a rule body.

Recursion

Recursive datalog: The body can reference the head atom

(~Dijkstra's algorithm)

... is like a very large number of queries with no head variables

The fact $Q(1, 1)$ is true if $\exists B : R(1, B) \wedge S(B, 1)$

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

Every row not explicitly listed is mapped to False