106 research outputs found
Decidability and definability with circumscription
AbstractWe consider McCarthy's notions of predicate circumscription and formula circumscription. We show that the decision problems “does θ have a countably infinite minimal model” and “does φ hold in every countably infinite minimal model of θ” are complete Σ12 and complete π12 over the integers, for both forms of circumscription. The set of structures definable (up to isomorphism) as first order definable subsets of countably infinite minimal models is the set of structures which are Δ12 over the integers, for both forms of circumscription. Thus, restricted to countably infinite structures, predicate and formula circumscription define the same sets and have equally difficult decision problems. With general formula circumscription we can define several infinite cardinals, so the decidability problems are dependent upon the axioms of set theory
Space Efficiency of Propositional Knowledge Representation Formalisms
We investigate the space efficiency of a Propositional Knowledge
Representation (PKR) formalism. Intuitively, the space efficiency of a
formalism F in representing a certain piece of knowledge A, is the size of the
shortest formula of F that represents A. In this paper we assume that knowledge
is either a set of propositional interpretations (models) or a set of
propositional formulae (theorems). We provide a formal way of talking about the
relative ability of PKR formalisms to compactly represent a set of models or a
set of theorems. We introduce two new compactness measures, the corresponding
classes, and show that the relative space efficiency of a PKR formalism in
representing models/theorems is directly related to such classes. In
particular, we consider formalisms for nonmonotonic reasoning, such as
circumscription and default logic, as well as belief revision operators and the
stable model semantics for logic programs with negation. One interesting result
is that formalisms with the same time complexity do not necessarily belong to
the same space efficiency class
Complexity of Nested Circumscription and Nested Abnormality Theories
The need for a circumscriptive formalism that allows for simple yet elegant
modular problem representation has led Lifschitz (AIJ, 1995) to introduce
nested abnormality theories (NATs) as a tool for modular knowledge
representation, tailored for applying circumscription to minimize exceptional
circumstances. Abstracting from this particular objective, we propose L_{CIRC},
which is an extension of generic propositional circumscription by allowing
propositional combinations and nesting of circumscriptive theories. As shown,
NATs are naturally embedded into this language, and are in fact of equal
expressive capability. We then analyze the complexity of L_{CIRC} and NATs, and
in particular the effect of nesting. The latter is found to be a source of
complexity, which climbs the Polynomial Hierarchy as the nesting depth
increases and reaches PSPACE-completeness in the general case. We also identify
meaningful syntactic fragments of NATs which have lower complexity. In
particular, we show that the generalization of Horn circumscription in the NAT
framework remains CONP-complete, and that Horn NATs without fixed letters can
be efficiently transformed into an equivalent Horn CNF, which implies
polynomial solvability of principal reasoning tasks. Finally, we also study
extensions of NATs and briefly address the complexity in the first-order case.
Our results give insight into the ``cost'' of using L_{CIRC} (resp. NATs) as a
host language for expressing other formalisms such as action theories,
narratives, or spatial theories.Comment: A preliminary abstract of this paper appeared in Proc. Seventeenth
International Joint Conference on Artificial Intelligence (IJCAI-01), pages
169--174. Morgan Kaufmann, 200
Preferential and Preferential-discriminative Consequence relations
The present paper investigates consequence relations that are both
non-monotonic and paraconsistent. More precisely, we put the focus on
preferential consequence relations, i.e. those relations that can be defined by
a binary preference relation on states labelled by valuations. We worked with a
general notion of valuation that covers e.g. the classical valuations as well
as certain kinds of many-valued valuations. In the many-valued cases,
preferential consequence relations are paraconsistant (in addition to be
non-monotonic), i.e. they are capable of drawing reasonable conclusions which
contain contradictions. The first purpose of this paper is to provide in our
general framework syntactic characterizations of several families of
preferential relations. The second and main purpose is to provide, again in our
general framework, characterizations of several families of preferential
discriminative consequence relations. They are defined exactly as the plain
version, but any conclusion such that its negation is also a conclusion is
rejected (these relations bring something new essentially in the many-valued
cases).Comment: team Logic and Complexity, written in 2004-200
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