CS240A: Databases and Knowledge Bases
Fixpoint Semantics of Datalog
Carlo Zaniolo
Department of Computer Science
University of California, Los Angeles
WINTER 2002
Three Equivalent Semantics
1.
Model Theoretic: describing what facts hold true
because of the given facts and rules.
2.
Proof Theoretic: using SLD-resolution
3.
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.
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.
Herbrand Universe and Ground Instance
of 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).
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  P}

The Immediate Consequence Operator TP:
TP(I) = {A  BP | $r A  A1,...,A n ground(P), { A1,...,An }  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  )

For a positive program P the least fixpoint of TP exists and is denoted
lfp(TP)

lfp(TP) defines the meaning of P.
Computation of TP

TP 0 ( I ) = I
Powers of TP :

TP n+1 ( I ) = TP ( TP n (I) )

Moreover, with w denoting the first limit ordinal, we define:
TPw (I) =

{T
n
(I) | n  0 }
Powers of TP starting from the empty set; I.e. from I = 
Theorem: If P is a positive program; then lfp(TP) = TPw ()
NB: the monotonicity of TP is critical for the existence of
lfp(TP) and its efficient computation (next slide).
Computing TPw ()
Theorem: The successive powers of TP, form an
ascendingchain: I.e. TPn ()  TP n+1 ().
Proof by induction:
Base:  = TP0 ()  TP1(),
Induction: If TPn-1 ()  TPn () then TPn ()  TPn+1 ()
Indeed, TPn() = TP ( TPn-1()) and TPn+1() = TP ( TPn())
And the conclusion follows from the monotonicity of TP
Corollaries: TPk() =
n  k TPn ()
if TPk+1()= TPK () then TPK ()= TP ()w
Computation: start from the empty set, and repeat the
application of TP until no new atoms are obtained—I.e., the
n+1-th power is identical to the n-th one (if such condition
never occurs then we have an infinite computation).