4. Datalog Queries

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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),
• wic 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).

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• 4.2 Datalog with Sets
• Example Hamiltonian Cycle
• Input
• 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