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1 4. Datalog Queries Datalog query – a finite set of rules of the form: R 0 (x 1,…,x k ) :– R 1 (x 1,1,…, x 1,k 1 ),..., R n (x n,1,…, x n,k n ) where each R i 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 – R 0 body of the rule – R 1,…,R n

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2 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 (x 0,y 0 ). Reach(n) :– Street(n, x 0, y 0 ). 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, t 2 ), Go(z, t 2, y, t).

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3 Example: Find town points covered by a radio station Covered(x 2, y 2 ) :– Broadcast(n, x, y), Town(t, x 2, y 2 ), Parameters(n, s, blat, blong), Parameters(t, s 2, tlat, tlong), x 2 = x + (tlat – blat), y 2 = y + (tlong – blong).

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4 4.2 Datalog with Sets Example: Hamiltonian Cycle Input: Vertices(S) where S is a set of vertices Edge ({c 1 }, {c 2 }) if there is an edge from c 1 to c 2 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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5 Base case – Path is a single vertex. All vertices except the start vertex is unvisited. Path(X 1, B) :– Vertices(A), Start(X 1 ), B = A \ X 1. Recursion – a path to X 1 with B unvisited exists if there is Path(X 1, B) : – Path(X 2, A), a path to X 2 with A unvisited Edge(X 2, X 1 ), and an edge from X 2 to X 1, X 1 A, which is unvisited, and B = A \ X 1. B is A minus X 1

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6 If there is a path from start to X 2 that visits all vertices and an edge from X 2 to start, then there is a Hamiltonian cycle. Hamiltonian(X 1 ) :– Path(X 2, ), Edge(X 2, X 1 ), Start(X 1 ).

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7 4.4 Datalog with Abstract Data Types Example: Streets(Name, Extent) where extent is a set of 2D points. Let (x 0, y 0 ) be a start location. Express the reach relation: Reach(n) :– Street(n, Extent), {( x 0,y 0 )} Extent. Reach(n) :– Reach(m), Street(m, S 1 ), Street(n, S 2 ), S 1 S 2

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8 4.5 Semantics Rule instantiation – substitution of variables by constants ⊢ Q,I R(a 1,…..a k ) – R(a 1,….a k ) has a proof using query Q and input database I, iff R represents input relation r and (a 1,….a k ) r, or There is some rule and instantiation R(a 1,…,a k ):–R 1 (a 1,1,…,a 1,k 1 ),…, R n (a n,1,…, a n, k n ). where ⊢ Q,I R i (a i,1,…,a i,k i ) for each 1 i n.

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9 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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10 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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12 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

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