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4. Datalog Queries

Datalog query – a finite set of rules of the form:

R0(x1,…,xk) :– R1(x1,1,…, x1,k1),..., Rn(xn,1,…, xn,kn

)

where each Ri is either an input or a defined relation name.

including built-in relations such as +(x,y,z) which means

x + y = z. (We normally use the latter syntax.)

head of the rule – R0

body of the rule – R1,…,Rn

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Example:

Find the SSN and the tax.

Tax_Due(s, t) :– Taxrecord(s, w, i, c), Taxtable(inc, t),

w+i+c = inc.

Find the streets that can be reached from (x0,y0).

Reach(n) :– Street(n, x0, y0).

Reach(n) :– Reach(m), Street(m, x, y), Street(n, x, y).

Find the time to travel from x to y.

Travel(x, y, t) :– Go(x, 0, y, t).

Travel(x, y, t) :– Travel(x, z, t2), Go(z, t2, y, t).

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Example:

Find town points covered by a radio station

Covered(x2, y2) :– Broadcast(n, x, y), Town(t, x2, y2),

Parameters(n, s, blat, blong),

Parameters(t, s2, tlat, tlong),

x2 = x + (tlat – blat),

y2 = y + (tlong – blong). 2S

S

2S

S

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4.2 Datalog with Sets

Example: Hamiltonian CycleInput:

• Vertices(S) where S is a set of vertices

• Edge ({c1}, {c2}) if there is an edge from c1 to c2

• Start({c}) where c is start city name

Output:• Path ({c}, B) if there is a path from c that uses

all vertices except those in B.

• Hamiltonian ({c}) if there is a Hamiltonian path.

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Base case – Path is a single vertex. All vertices except the start vertex is unvisited.

Path(X1, B) :– Vertices(A), Start(X1), B = A \ X1.

Recursion –

a path to X1 with B unvisited

exists if there is

Path(X1, B) : – Path(X2, A), a path to X2 with A unvisited

Edge(X2, X1), and an edge from X2 to X1,

X1 A, which is unvisited, and

B = A \ X1. B is A minus X1

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If there is a path from start to X2 that visits all

vertices and an edge from X2 to start, then there

is a Hamiltonian cycle.

Hamiltonian(X1) :– Path(X2, ),

Edge(X2, X1),

Start(X1).

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4.4 Datalog with Abstract Data Types

Example:

Streets(Name, Extent) where extent is a set of 2D points. Let (x0, y0) be a start location.

Express the reach relation:

Reach(n) :– Street(n, Extent), {(x0,y0)} Extent.

Reach(n) :– Reach(m), Street(m, S1), Street(n, S2),

S1 S2

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4.5 Semantics

Rule instantiation – substitution of variables by constants

⊢Q,I R(a1,…..ak) – R(a1,….ak) has a proof using query Q

and input database I, iff

• R represents input relation r and (a1,….ak) r , or

• There is some rule and instantiation

R(a1,…,ak):–R1(a1,1,…,a1,k1),…, Rn(an,1,…, an, kn

).

where ⊢Q,I Ri(ai,1,…,ai,ki) for each 1 i n .

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Example:

Reach(Vine)

Reach(Vine) :– Street(Vine, 5, 2).

Reach(Bear)

Reach(Bear) :– Reach(Vine), Street(Vine, 5, 12),

Street(Bear, 5, 12).

Reach(Hare) Reach(Hare) :– Reach(Bear), Street(Bear, 8, 13),

Street(Hare, 8, 13).

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Example: By the input database (Figure 1.2):

Go(Omaha, 0, Lincoln, 60)

Go(Lincoln, 60, Kansas_City, 210)

Go(Kansas_City, 210, Des_Moines, 390)

Go(Des_Moines, 390, Chicago, 990)

We also have:

Travel(Omaha, Lincoln, 60)

Travel(Omaha, Lincoln, 60):- Go(Omaha, 0, Lincoln, 60)

Travel(Omaha, Kansas_City, 210)

Travel(Omaha,Kansas_City,210):- Travel(Omaha,Lincoln,60),

Go(Lincoln,60,Kansas_City,210).

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Proof-based semantics – derived relations are the set

of tuples that can be proven.

Fixed point semantics – an interpretation of the derived

relations such that nothing new

can be proven.

Least fixed point semantics – smallest possible FP semantics.

Proof-based semantics = Least fixed point semantics