11.%20Certification

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11. Certification

• Certification algorithms/methods to verify that
  
• a program is safe.
• Used when restriction does not work because we
  use a combination of queries and constraint types
  that are not always safe.
• Advantage more powerful queries than with
  restriction
• Disadvantage harder to use than restriction

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• Certification methods
• Constant propagation some attributes of input
  relations are
• known to be always constants and this
  information is recursively propagated to other
  relations/rules in the query.
• Example
• Textbook example 4.1.4 has Go2(x,y,c) where c is
  a constant.
• Therefore in the second rule the constraint t
  t2 c is a
• half-addition constraint. Since Datalog with
  half-addition
• constraints is safe by restriction the Travel
  query is also safe.
• Note the reduction to restriction in the above.

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• Variable independence knowing the value of a
  variable does not improve our knowledge of the
  value of another variable.
• Example
• x1 and x2 are independent
  variables in
• x1 30, x2 50
• they are dependent variables in
• x1 x2 gt 0
• If a constraint tuple contains only equality,
  inequality, lower
• and upper bound constraints, then the variables
  in it are
• independent.

4

• Example
• Not_Fastest(x, y, t2) Travel(x, y, t1),
  Travel(x, y, t2), t1 lt t2.
• Fastest(x, y, t) Travel(x, y, t), not
  Not_Fastest(x, y, t).
• Suppose we know that the variables are
  independent in
• each constraint tuple of Travel because it
  contains only
• equality, inequality, lower and upper bound
  constraints.
• Then they are also independent in Not_Fastest,
  which
• also only contains those types of constraints.
• Hence by restriction its negation has a
  closed-form,
• and Fastest can be evaluated.

5

• Monotonicity an attribute value is monotone
  increasing
• Example Apply monotonicity to the Subsetsum
  query
• in the Textbook.
• Acyclicity evaluation follows the edges in a
• directed acyclic graph.
• Example Apply acyclicity to the polluted river
  example
• in the Textbook.

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• Tightened bounds the bound of a negative
  constraint increases in each rule application,
  while the variables remain bounded from below by
  some constant.
• Tightened coefficients coefficients in a
  negative
• linear constraint growing faster than the
  bound.

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• Vector addition system a constraint automaton
  where each transition only adds constants to each
  state variable, which are always non-negative.
• Example Tourist constraint automaton.
• Positive stochastic matrix multiplication a
• constraint automaton where each transition
• only multiplies by a positive matrix the state
  
• variables to get the new state.
• Example The water cycle constraint automaton.