Abduction is an important form of nonmonotonic reasoning allowing one to find explanations for certain symptoms or manifestations. When the application domain is described by a logical theory, we speak about logic-based abduction. Candidates for abductive explanations are usually subjected to minimality criteria such as subsetminimality, minimal cardinality, minimal weight, or minimality under prioritization of individual hypotheses. This paper presents a comprehensive complexity analysis of relevant decision and search problems related to abduction on propositional theories. Our results indicate that abduction is harder than deduction. In particular, we show that with the most basic forms of abduction the relevant decision problems are complete for complexity classes at the second level of the polynomial hierarchy, while the use of prioritization raises the complexity to the third level in certain cases.

This paper addresses complexity issues for important problems arising with disjunctive logic programming. In particular, the complexity of deciding whether a disjunctive logic program is consistent is investigated for a variety of well-known semantics, as well as the complexity of deciding whether a propositional formula is satised by all models according to a given semantics. We concentrate on nite propositional disjunctive programs with as wells as without integrity constraints, i.e., clauses with empty heads; the problems are located in appropriate slots of the polynomial hierarchy. In particular, we show that the consistency check is P 2 -complete for the disjunctive stable model semantics (in the total as well as partial version), the iterated closed world assumption, and the perfect model semantics, and we show that the inference problem for these semantics is P 2 -complete; analogous results are derived for the an

| Research in non-monotonic reasoning has focused largely on the idea of representing knowledge about the world via rules that are generally true but can be defeated. Even if relational databases are nowadays the main tool for storing very large sets of data, the approach of using non-monotonic AI formalisms as relational database query languages has been investigated to a much smaller extent. In this work we propose a novel application of Reiter's default logic by introducing a default query language (DQL) for nite relational databases, which is based on default rules. The main result of this paper is that DQL is as expressive as SO 98 , the existential-universal fragment of secondorder logic. This result is not only of theoretical importance: We exhibit queries {which are useful in practice{ that can be expressed with DQL and can not with other query languages based on non-monotonic logics such as DATALOG with negation under the stable model semantics. In particular, we show that DQ...

Default logic was proposed by Reiter as a knowledge representation tool. In this paper, we present our work on the Default Reasoning System, DeReS, the first comprehensive and optimized implementation of default logic. While knowledge representation remains the main application area for default logic, as a source of large-scale problems needed for experimentation and as a source of intuitions needed for a systematic methodology of encoding problems as default theories we use here the domain of combinatorial problems. To experimentally study the performance of DeReS we developed a benchmarking system, the TheoryBase. The TheoryBase is designed to support experimental investigations of nonmonotonic reasoning systems based on the language of default logic or logic programming. It allows the user to create parameterized collections of default theories having similar properties and growing sizes and, consequently, to study the asymptotic performance of nonmonotonic systems under i...

In this paper, we describe a system, called TheoryBase, whose goal is to facilitate experimental studies of nonmonotonic reasoning systems. TheoryBase generates test default theories and logic programs. It has an identification system for generated theories, which allows us to reconstruct a logic program or a default theory from its identifier. Hence, exchanging test cases requires only exchanging identifiers. TheoryBase can generate a large variety of examples of default theories and logic programs. We believe that its universal adoption may significantly advance experimental studies of nonmonotonic reasoning systems.

Although the computational perspective on cognitive tasks has always played a major role in Artificial Intelligence, the interest in the precise determination of the computational costs that are required for solving typical AI problems has grown only recently. In this paper, we will describe what insights a computational complexity analysis can provide and what methods are available to deal with the complexity problem. This work was partially supported by the European Commission as part of DRUMS-II, the ESPRIT Basic Research Project P6156. 1 Introduction It is well-known that typical AI problems, such as natural language understanding, scene interpretation, planning, configuration, or diagnosis are computationally difficult. Hence, it seems to be worthless to analyze the computational complexity of these problems. In fact, some people believe that all AI problems are NP-hard or even undecidable. Conceiving AI as a scientific field that has as its goal the analysis and synthesis of...

Bondarenko et al. have recently proposed an abstract framework for default reasoning. Besides capturing most existing formalisms and proving that their standard semantics all coincide, the framework extends these formalisms by generalising the semantics of admissible and preferred arguments, originally proposed for logic programming only.

Conditional knowledge bases have been proposed as belief bases that include defeasible rules (also called defaults) of the form " ! ", which informally read as "generally, if then ." Such rules may have exceptions, which can be handled in different ways. A number of entailment semantics for conditional knowledge bases have been proposed in the literature. However, while the semantic properties and interrelationships of these formalisms are quite well understood, about their computational properties only partial results are known so far. In this paper, we fill these gaps and first draw a precise picture of the complexity of default reasoning from conditional knowledge bases: Given a conditional knowledge base KB and a default ! , does KB entail ! ? We classify the complexity of this problem for a number of well-known approaches (including Goldszmidt et al.'s maximum entropy approach and Geffner's conditional entailment), where we consider the general propositional case as well as natural syntactic restrictions (in particular, to Horn and literal-Horn conditional knowledge bases). As we show, the more sophisticated semantics for conditional knowledge bases are plagued with intractability in all these fragments. We thus explore cases in which these semantics are tractable, and find that most of them enjoy this property on feedback-free Horn conditional knowledge bases, which constitute a new, meaningful class of conditional knowledge bases. Furthermore, we generalize previous tractability results from Horn to q-Horn conditional knowledge bases, which allow for a limited use of disjunction. Our results complement and extend previous results, and contribute in refining the tractability/intractability frontier of default reasoning from conditional know...

In this paper we show how propositional default theories can be characterized by classical propositional theories: for each finite default theory, we show a classical propositional theory such that there is a one-to-one correspondence between models for the latter and extensions of the former. This means that computing extensions and answering queries about coherence, set-membership and set-entailment are reducible to propositional satisfiability. The general transformation is exponential but tractable for a subset which we call 2-DT --- a superset of network default theories and disjunction-free default theories. Consequently, coherence and set-membership for the class 2-DT is NP-complete and set-entailment is co-NP-complete. This work paves the way for the application of decades of research on efficient algorithms for the satisfiability problem to default reasoning. For example, since propositional satisfiability can be regarded as a constraint satisfaction problem (CSP...

Abductive reasoning (roughly speaking, find an explanation for observations out of hypotheses), has been recognized as an important principle of common-sense reasoning. Since logical knowledge representation is commonly based on nonclassical formalisms like default logic, autoepistemic logic, or circumscription, it is necessary to perform abductive reasoning from theories (i.e. knowledge bases) of nonclassical logics. In this paper, we investigate how abduction can be performed from theories in default logic. In particular, we present a basic model of abduction from default theories. Different modes of abduction are plausible, based on credulous and skeptical default reasoning; they appear useful for different applications such as diagnosis and planning. Moreover, we thoroughly analyze the complexity of the main abductive reasoning tasks, namely finding an explanation, deciding relevance of a hypothesis, and deciding necessity of a hypothesis. These problems are intractable even...