CSPs and Relational DBs

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• CSPs and Relational DBs
• Foundations of Constraint Processing
• CSCE421/821, Fall 2003
• www.cse.unl.edu/choueiry/F03-421-821/
• Berthe Y. Choueiry (Shu-we-ri)
• Ferguson Hall, Room 104
• choueiry_at_cse.unl.edu
• Tel [phone]

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Background

• Strong historical conceptual connections exist
  between
• Constraint Databases Constraint Logic
  Programming
• Query processing in Relational DBs Solving
  CSPs
• Indeed
• Constraint databases (deductive BD, Datalog) and
  constraint logic programming (CLP) share the
  representation language (restricted forms of FOL)
  
• Relational databases and Constraint Satisfaction
  share computation mechanisms

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Relations

• Binary relation Given two sets Da and Db, a set
  of any 2-tuples
• lt x, ygt with ? Da and y ? Db defines a relation
  Rab
• Ra,b (x, y) ? Da x Db
• Function (special binary relation)
• For any element x in Da there is at most one
  tuple lt x, ? gt ? Rab
• Da is called the domain of the the function
• Db is called the range of the function
• k-ary relation Given k sets D1, D2, , Dk, any
  set of k-tuples
• lt x1, x2, ..., xk gt with x1 ? D1, x2 ? D2, ,
  xk ? Dk defines a
• k-ary relation R1, 2, ..., k (x1, x2, ...,
  xk) ? D1 x D2 x x Dk

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Representation of relations

• Binary arrays
• ? 2-dim binary array (i.e., binary matrix)
• ? more generally, k-dimensional binary arrays
• Tables

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Comparison of terminology
DB terminology CSP terminology
Table, relation Constraint
Relation arity Constraint arity
Attribute CSP variable
Value of an attribute Value of a variable
Domain of an attribute Domain of a variable
Tuple in a table Tuple in a constraint Tuple allowed by an constraint Tuple consistent with a constraint
Constraint relation (in constraint databases) Constraint of linear (in)equality
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Relational Algebra operations on relations

• Database
• Intersection
• Union
• Difference
• Selection
• Projection
• Join (Cartesian product), etc.
• CSP
• The above and composition ( combination of join
  and projection)

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Operators in Relational Algebra

• Selection, projection
• unary operators, defined on one relation
• Intersection, union, difference
• binary operators
• relations must have same scope
• Join
• binary operator
• relations have different scopes

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Intersection

• Input two relations of the same scope
• Output a new more restrictive relation with the
  same scope, made of tuples that are in all the
  input relations (simultaneously)
• Matrix operation logical AND
• R R' R
• R ? R ? R ? R' ?

Okay
Not defined
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Union

• Input two relations of the same scope
• Output a new less restrictive relation with the
  same scope made of tuples that are in any of the
  input relations
• Matrix operation logical OR
• R R' R
• R ? R'? R ? R''?

Not defined
Okay
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Difference

• Input two relations R and R' of the same scope
• Output a new more restrictive relation than R
  made of tuples that are in R but not in R'
• Matrix operation Boolean difference
• R R' R
• R - R'? R - R''?

Not defined
Okay
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Selection

• Input A relation R and some test/predicate on
  attributes of R
• Output A relation R', same scope as R but
  containing only a
• subset of the tuples in R (those that satisfy
  the predicate)
• Relation operation row selection
• R
• Select such that x1gt x2, ?x1gt x2(R)?

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Projection

• Input A relation R and a subset s of the scope
  (attributes)
• Output A relation R' of scope s with the tuples
  rewritten such that
• positions not in s are removed
• Relation operation column elimination
• R
• Project R on x1, x2, ?x1, x2(R)?

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Join (natural join)

• Input Two relations R and R'
• Output A relation R'', whose scope is union of
  scopes of R and R' and tuples satisfy both R and
  R'.
• ? R and R' have no attribute common Cartesian
  product
• ? R and R' have an attribute in common, compute
  all possibilities
• Operation Compute all solutions to a CSP.
• R R"
• Join R and R'', R R''?

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Composition of relations Montanari'74

• Input two binary relations Rab and Rbc with 1
  variable in common.
• Output a new induced relation Rac (to be
  combined by intersection to a pre-existing
  relation between them, if any).
• Matrix operation binary matrix multiplication
• Note - generalization as constraint synthesis
  Freuder, 1978
• - Direct (explicit) vs. induced
  (implicit) relations

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Questions

• Given
• two variables V1 and V2 and
• two constraints C1 and C2 between them
• How do the two expressions C1 ? C2 and C1
  C2 relate?
• Given
• three variables V1, V2, V3 and
• the binary constraints CV1, V2 and CV2, V3
• write the induced CV1, V3, in relational
  algebra
• Given
• three variables V1, V2, V3 and
• the binary constraints CV1, V2, CV1, V3, and CV2,
  V3,
• write the new induced CV1, V3 in
  relational algebra

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Comparison of Terminology
Databases CSPs
(Natural, inner) join Synthesized constraint
Left/right outer join Synthesized constraint including (some) inconsistent tuples
Projection of a join Induced constraint (Composition of two constraints)
Computing r1 r2 ri Finding all solutions to the conjunction of the constraints rk