114 research outputs found
Generalizing Consistency and other Constraint Properties to Quantified Constraints
Quantified constraints and Quantified Boolean Formulae are typically much
more difficult to reason with than classical constraints, because quantifier
alternation makes the usual notion of solution inappropriate. As a consequence,
basic properties of Constraint Satisfaction Problems (CSP), such as consistency
or substitutability, are not completely understood in the quantified case.
These properties are important because they are the basis of most of the
reasoning methods used to solve classical (existentially quantified)
constraints, and one would like to benefit from similar reasoning methods in
the resolution of quantified constraints. In this paper, we show that most of
the properties that are used by solvers for CSP can be generalized to
quantified CSP. This requires a re-thinking of a number of basic concepts; in
particular, we propose a notion of outcome that generalizes the classical
notion of solution and on which all definitions are based. We propose a
systematic study of the relations which hold between these properties, as well
as complexity results regarding the decision of these properties. Finally, and
since these problems are typically intractable, we generalize the approach used
in CSP and propose weaker, easier to check notions based on locality, which
allow to detect these properties incompletely but in polynomial time
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
Circumscribing datalog: Expressive power and complexity
AbstractIn this paper we study a generalization of datalog, the language of function-free definite clauses. It is known that standard datalog semantics (i.e., least Herbrand model semantics) can be obtained by regarding programs as theories to be circumscribed with all predicates to be minimized. The extension proposed here, called datalogcirc, consists in considering the general form of circumscription, where some predicates are minimized, some predicates are fixed, and some vary. We study the complexity and the expressive power of the language thus obtained. We show that this language (and, actually, its non-recursive fragment) is capable of expressing all the queries in DB-co-NP and, as such, is much more powerful than standard datalog, whose expressive power is limited to a strict subset of PTIME queries. Both data and combined complexities of answering datalogcirc queries are studied. Data complexity is proved to be co-NP-complete. Combined complexity is shown to be in general hard for co-NE and complete for co-NE in the case of Herbrand bases containing k distinct constant symbols, where k is bounded
Combining Relational Algebra, SQL, Constraint Modelling, and Local Search
The goal of this paper is to provide a strong integration between constraint
modelling and relational DBMSs. To this end we propose extensions of standard
query languages such as relational algebra and SQL, by adding constraint
modelling capabilities to them. In particular, we propose non-deterministic
extensions of both languages, which are specially suited for combinatorial
problems. Non-determinism is introduced by means of a guessing operator, which
declares a set of relations to have an arbitrary extension. This new operator
results in languages with higher expressive power, able to express all problems
in the complexity class NP. Some syntactical restrictions which make data
complexity polynomial are shown. The effectiveness of both extensions is
demonstrated by means of several examples. The current implementation, written
in Java using local search techniques, is described. To appear in Theory and
Practice of Logic Programming (TPLP)Comment: 30 pages, 5 figure
Compilability of Abduction
Abduction is one of the most important forms of reasoning; it has been
successfully applied to several practical problems such as diagnosis. In this
paper we investigate whether the computational complexity of abduction can be
reduced by an appropriate use of preprocessing. This is motivated by the fact
that part of the data of the problem (namely, the set of all possible
assumptions and the theory relating assumptions and manifestations) are often
known before the rest of the problem. In this paper, we show some complexity
results about abduction when compilation is allowed
On the Complexity of Finding Second-Best Abductive Explanations
While looking for abductive explanations of a given set of manifestations, an
ordering between possible solutions is often assumed. The complexity of
finding/verifying optimal solutions is already known. In this paper we consider
the computational complexity of finding second-best solutions. We consider
different orderings, and consider also different possible definitions of what a
second-best solution is
The Complexity of Model Checking for Circumscriptive Formulae
this paper we carry out a detailed analysis of the computational complexity of model checking for propositional circumscriptive formulae. Following Schaefer's approach [6], we classify propositional formulae according to the logical relations which are used to represent them. Using the results reported in that paper we prove polynomial as well as co-NP-hardness results, thus providing a detailed picture of the tractability threshold of the problem. In particular we answer negatively to Kolaitis and Papadimitriou's question, showing that the model checking problem is co-NP-complete for a subclass of the propositional formulae. The structure of the paper is as follows: In Section 2 we recall the basic notions on circumscription and formally define our problem, and in Section 3 we present the results of the complexity analysis. 2 Preliminarie
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