CS240A: Databases and Knowledge Bases Fixpoint Semantics of Datalog

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CS240A Databases and Knowledge BasesFixpoint
Semantics of Datalog

• Carlo Zaniolo
• Department of Computer Science
• University of California, Los Angeles
• WINTER 2002

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Three Equivalent Semantics

• Model Theoretic describing what facts hold true
  because of the given facts and rules.
• Proof Theoretic using SLD-resolution
• Fixpoint Semantics. Defined as the least
  fixpoint of the transformation defined by the
  rules.
• For positive programs (I.e., programs without
  negated goals) all three semantics are-well
  defined and equivalent. Here we will cover only
  3.
• For programs with negated goals things are more
  complex, different semantics are possible and
  equivalence might not hold any longer. Here we
  will only consider stratified negation.

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Example

• anc(X, Y) parent(X, Y).
  anc(X, Z) anc(X,Y),
  parent(Y,Z).
• parent(X, Y) father(X, Y).
  parent(X, Y) mother(X, Y).
• mother(anne, silvia).
  mother(silvia, marc).
• A bottom-up derivation to derive facts
  that are true.
• Step1 mother(anne, silvia), mother(silvia,
  marc).
• Step2 mother(anne, silvia), mother(silvia,
  marc), parent(anne, silvia),
  parent(silvia, marc).
• Step3 those in step 2 anc(anne, silvia),
  anc(silvia, marc).
• Step4 those in step 3 anc(anne, marc).
• Step4 those in step 4 only---we have reached
  fixpoint.

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Herbrand Universe and Ground Instanceof program P

• The Herbrand Universe for P, denoted UP, is
  defined as the set containing (1) the constants
  in P and (2) the terms recursively constructed
  by letting the arguments of functions be elements
  in UP.
• For the example program UP anne, marc,
  silvia.
• Ground version a rule r ground(r). This is the
  set of ground instances of r --i.e., all the
  rules obtained by assigning to the variables in
  r, values from the Herbrand universe UP.
• E.g. from the previous program take parent(X,
  Y) mother(X, Y). Since there are 2 variables in
  this rule and UP 3, then ground(r) consists of
  3 3 rules
• parent(anne, anne) mother(anne, anne).
• parent(anne, marc) mother(anne, marc).
• ...
• parent(silvia, silvia) mother(silvia,
  silvia).

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Semantics of a positive program P as the least
fixpoint of its Immediate Consequence Operator TP

• The ground version of a program P, denoted
  ground(P), is the set of the ground instances of
  its rules
• ground(P) ground( r)    r Î P
• The Immediate Consequence Operator TP
• TP(I) A Î BP r A A1,...,A nÎ
  ground(P), A1,...,An Í I
• The TP operator can be implemented using the
  relational algebra equivalent of the relational
  rules.
• The fixpoint equation I TP(I)
• The solutions of this equations are called
  fixpoint for TP
• When we have several solutions we can find the
  minimal ones and the least ones (according to Í
  )
• For a positive program P the least fixpoint of
  TP exists and is denoted lfp(TP)
• lfp(TP) defines the meaning of P.

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Computation of TP

• Powers of TP TP 0 ( I )   I
• ¼
• TP n1 ( I )   TP ( TP n (I) )
• Moreover, with w denoting the first limit
  ordinal, we define
• TPw (I)     È T n (I)    n ³ 0
• Powers of TP starting from the empty set I.e.
  from I Æ
• Theorem If P is a positive program then lfp(TP)
  TPw (Æ)
• NB the monotonicity of TP is critical for the
  existence of lfp(TP) and its efficient
  computation (next slide).

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Computing TPw (Æ)

• Theorem The successive powers of TP, form an
  ascendingchain I.e. TPn (Æ) Í TP n1 (Æ).
• Proof by induction
• Base Æ TP0 (Æ) Í TP1(Æ),
• Induction If TPn-1 (Æ) Í TPn (Æ) then TPn
  (Æ) Í TPn1 (Æ)
• Indeed, TPn(Æ) TP ( TPn-1(Æ)) and TPn1(Æ)
  TP ( TPn(Æ))
• And the conclusion follows from the monotonicity
  of TP
• Corollaries TPk(Æ)    Èn k TPn (Æ)
• if TPk1(Æ) TPK (Æ) then TPK (Æ) TP
  (Æ)w
• Computation start from the empty set, and repeat
  the application of TP until no new atoms are
  obtainedI.e., the n1-th power is identical to
  the n-th one (if such condition never occurs then
  we have an infinite computation).