11 research outputs found
The Complexity of Reasoning for Fragments of Default Logic
Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified
the complexity of the extension existence problem for propositional default
logic as \SigmaPtwo-complete, and the complexity of the credulous and
skeptical reasoning problem as SigmaP2-complete, resp. PiP2-complete.
Additionally, he investigated restrictions on the default rules, i.e.,
semi-normal default rules. Selman made in 1992 a similar approach with
disjunction-free and unary default rules. In this paper we systematically
restrict the set of allowed propositional connectives. We give a complete
complexity classification for all sets of Boolean functions in the meaning of
Post's lattice for all three common decision problems for propositional default
logic. We show that the complexity is a hexachotomy (SigmaP2-, DeltaP2-, NP-,
P-, NL-complete, trivial) for the extension existence problem, while for the
credulous and skeptical reasoning problem we obtain similar classifications
without trivial cases.Comment: Corrected versio
The complexity of reasoning for fragments of default logic
Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified the complexity of the extension existence problem for propositional default logic as Σ -complete, and the complexity of the credulous and skeptical reasoning problem as Σ -complete, respectively Π-complete. Additionally, he investigated restrictions on the default rules, i.e. semi-normal default rules. Selman used in 1992 a similar approach with disjunction-free and unary default rules. In this article, we systematically restrict the set of allowed propositional connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice for all three common decision problems for propositional default logic. We show that the complexity is a hexachotomy (Σ -, Δ -, NP-, P-, NL-complete, trivial) for the extension existence problem, while for the credulous and skeptical reasoning problem we obtain similar classifications without trivial cases
Complexity of Non-Monotonic Logics
Over the past few decades, non-monotonic reasoning has developed to be one of
the most important topics in computational logic and artificial intelligence.
Different ways to introduce non-monotonic aspects to classical logic have been
considered, e.g., extension with default rules, extension with modal belief
operators, or modification of the semantics. In this survey we consider a
logical formalism from each of the above possibilities, namely Reiter's default
logic, Moore's autoepistemic logic and McCarthy's circumscription.
Additionally, we consider abduction, where one is not interested in inferences
from a given knowledge base but in computing possible explanations for an
observation with respect to a given knowledge base.
Complexity results for different reasoning tasks for propositional variants
of these logics have been studied already in the nineties. In recent years,
however, a renewed interest in complexity issues can be observed. One current
focal approach is to consider parameterized problems and identify reasonable
parameters that allow for FPT algorithms. In another approach, the emphasis
lies on identifying fragments, i.e., restriction of the logical language, that
allow more efficient algorithms for the most important reasoning tasks. In this
survey we focus on this second aspect. We describe complexity results for
fragments of logical languages obtained by either restricting the allowed set
of operators (e.g., forbidding negations one might consider only monotone
formulae) or by considering only formulae in conjunctive normal form but with
generalized clause types.
The algorithmic problems we consider are suitable variants of satisfiability
and implication in each of the logics, but also counting problems, where one is
not only interested in the existence of certain objects (e.g., models of a
formula) but asks for their number.Comment: To appear in Bulletin of the EATC
Deductive Systems in Traditional and Modern Logic
The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic