9. Evaluation of Queries

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9. Evaluation of Queries

• Query evaluation

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• 9.1 Quantifier Elimination and Satisfiability
• Example
• Logical Level r ? ? y1,yn
  ? r
• Constraint level S ? eliminate x
  ? S
• Infinite relation r(x, y1,yn) equivalent to
  constraint relation S.
• Infinite relation r(y1,yn) equivalent to
  constraint relation S.
• Quantifier elimination S ?x S S is
  equivalent to
• 
  eliminating x from S
• Closed quantifier elimination when S and S
  have the
• 
  same type of constraints.

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• QE is closed for conjunctions of
• Infinite domain equality and inequality
• Rational order
• Integer/rational gap-order, lower and upper
  bound
• Integer/rational half-addition, lower and upper
  bound
• Integer/rational gap-order, upper bound, and
  positive linear
• Rational linear inequality
• Real polynomial inequality
• Boolean equality
• Boolean order
• atomless Boolean equality and inequality

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• 9.2 Evaluation of Relational Algebra Queries

points(DB) the logical level infinite
relation/database that is
equivalent to the constraint relation/database
DB. For any relational algebra operator/query O
on infinite databases we can always find a
finitely evaluable relational algebra
operator/query ô on constraint databases such
that O(points(DB))
points(ô(DB))
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Example
A
B
X Y
x y x y 5
X Y
x y x y lt 9
C
X Y
x y x y 5, x y lt 9

C A n B
D
X Y
x y x y 5
X Y x y lt 9

D A ? B
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Example
E
X Y
x y x y 5, x y ? 9

E A B
F
X Y
x y x 7

F ?x E ?y E
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• 9.3 Evaluation of SQL queries
• Max/Min evaluated by linear programming
• if all constraints are linear inequalities.
• Example
• SELECT Min(x)
• FROM E
• will return 7.

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9.4 Evaluation of Datalog Queries

• Constraint instantiation substitution of rule
  body relation
• symbols
  by constraint tuples.
• Constraint proof constraint tuple t0 ? R0 can
  be proven in
• query Q and database instance I,
  written t ?Q,I R0, if
• t ? R0, or
• there is a constraint instantiation
• t0(...) t1(), , tn().
• where ti ?Q,I Ri for 1 i n and
• t0 ? t1(), , tn()
• where are variables that appear only in
  the rule body.

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• Theorem
• The proof-based and the constraint proof-based
  semantics
• are equivalent.
• (a1,,an) ?Q,I R(a1,.,ak)
  points(t) ?Q,I t ? R

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Datalog evaluation algorithm

• Repeat
• do constraint rule instantiation using the
• constraint tuples already in the DB
• evaluate the rule, using variable elimination
• add the derived constraint tuple to the DB,
• if it is at least partially new
• Until any tuple is added to DB
• The goal is to prove termination of the above
  algorithm.

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• Evaluation terminates (in closed-form) for
• Infinite domain equality and inequality
• Rational order
• Integer/rational half-addition, lower and
  upper bound
• Integer/rational gap-order, upper bound,
  positive linear
• Boolean order

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• Approximate evaluation for integer/rational
  addition
• Let l be a fixed negative constant. We
  modify in each
• constraint tuple each constraint with bound
  b lt l by
• changing b to l to get a lower
  bound Q(D)l
• deleting the constraint to get an upper
  bound Q(D)l

The approximate evaluation terminates and
returns correct upper and lower bounds, that is,
Q(D)l ? Q(D) ? Q(D)l also,
for any l1 and l2 such that l1 l2 lt 0
Q(D)l2 ? Q(D) l1 and Q(D)l1 ? Q(D)l2