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Ultimate Well-founded and Stable Semantics for
Logic Programs with Aggregates
Marc Denecker
a
Nikolay Pelov
b
Maurice Bruynooghe
b
a
Universit´e Libre de Bruxe lles, Bld du Triomphe CP 212, 1050 Brussels, Belgium
b
K.U.Leuven, Celestijnenlaan 200A, 3001 Heverlee, Belgium
In [3] we investigate the problem of defining a well-founded and stable s e -
mantics for programs with aggregates. Our work is based on Approximation The-
ory [1] which is a general algebraic framework for approximating non-monotone
operators on a complete lattice L by approximating operators o n the bi-lattice L
2
.
The theory identifies basic properties of the approximating operators and gives
a method to construct a stable ope rator and their associated fixpoints - Kripke-
Kleene (KK), well-founded (WF), and set of stable fixpoints. The authors then
show that the different fixpoints enjoy similar properties and relationships a s the
corresponding semantics in standard logic programming. For example the WF
fixpoint is mo re precise than the KK fixpoint and a lso approximates every stable
fixpoint. Moreover, stable fixpoints are always minimal fixpoints.
In a mo re recent paper [2] the authors study the concept of precision of an
approximating oper ator. They show that more precise approximating operators
have mo re precise KK and WF fixpoints and a larger number of stable fixpoints.
Moreover, there exists a most-precise approximating operator which they call the
ultimate approximation. In contrast with other approximating op e rators which
have to be defined explicitly, the construction of the ultimate approximating oper-
ator is entirely algebraic and depends only on the original non-monotone oper ator.
This makes it particularly well-suited for our purposes be cause defining a T
P
oper-
ator for programs with aggregates is relatively straightforward. Another advantage
of the ultimate approximation is that in cases where T
P
is monotone the ultimate
well-founded model will be 2-valued and will coincide with the least fixpoint of
T
P
. This is not the case for the standard well-founded semantics. For example in
the standard well-founded model of the pr ogram:
p ← ¬p.
p ← p.
p is undefined while the associated T
P
operator is monotone and p is true in the
ultimate well-founded model.
One disadvantage of using the ultimate semantics is that it has a higher com-
putational cost even for programs without aggregates. The complexity goes one
level higher in the polynomial hierarchy to ∆
p
2
for the well-founded model and to
Σ
p
2
for a stable model which is also complete for this cla ss [2]. Fortunately, by
adding aggregates the complexity does not increase further.
To g ive an example of a logic program with aggregates we consider the problem
of computing the length of the shortest path between two nodes in a directed graph.
The graph is represented with a set of edge(X, Y, W ) facts meaning that there is
an edge between X and Y with weight W . The program consists of a single rule:
sp(X, Y, L) ← min({C | edge(X, Y, C)∨
∃Z, C
1
, C
2
. (sp(X, Z, C
1
) ∧ edge(Z, Y, C
2
)∧
C = C
1
+ C
2
)}, L).
This program shows the syntax which we use. First of all, we allow arbitrary
first-order fo rmulas in the bodies of rules and in the conditions of set expressions.
Unlike most o f the previous work o n aggregates, we use an explicit notation for set
expressions {X | ϕ[X]} denoting the set of all elements d for which the condition
ϕ[d] is true. Aggregate functions are written in a form of atoms like f(s, t) where
F is an aggregate function, s is a set expression, and t is a term. It will evaluate
to true if the aggregate function f applied to the set denoted by s is equal to the
interpretation of the term t.
Having defined a 2-valued semantics fo r formulas containing aggregate atoms
it is easy to extend the definition of the 2-valued van Emden-Kowalski immediate
consequence operator T
P
for programs with a ggregates. Constructing the ultimate
approximating operator of T
P
we obtain an ultimate well-founded and ultimate
stable semantics for pro grams with aggregates. The two semantics have all the
properties which we discussed earlier. In addition for aggregate stratified programs
the ultimate well-founded model will be 2-valued and will coincide with the unique
ultimate stable model.
Our current research is focused on defining a less precise a pproximating oper-
ator for programs with aggregates which extends the s tandard well-founded and
stable semantics for programs without aggregates. Although the complexity of
this operator will still be the same as the the complexity of the ultimate approx-
imation, if we limit our language to some specific aggregate functions we can stay
in the complexity classes o f the standard semantics - P for the well-founded model
and NP for the stable models.
References
[1] M. Denecker, V. Marek, and M. Truszczy´nski. Approximating operators, stable
operators, well-founded fixpoints and applications in non-monotonic reasoning.
In J . Minker, editor, Logic-based Artificial Intelligence, pages 127–144. Kluwer
Academic P ublishers, 2000.
[2] M. Denecker, V. Marek, and M. Truszczy´nsky. Ultimate approximations in
nonmonotonic knowledge representation systems. In, KR’02, pages 17 7–188.
Morgan Ka ufmann, 2002.
[3] M. Denecker, N. Pelov, and M. Br uy nooghe. Well-founded and stable model
semantics for logic programs with aggregates. In P. Codognet, editor, ICLP’01,
volume 2237 of LNCS, pages 212–226. Springer-Verlag, 2001.