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

When Conditional Logic and Belief Revision Meet Substructural Logics

International audienceTwo threads of research have been pursued in parallel in logic and artificial intelligence. On the one hand, in artificial intelligence, logic-based theories have been developed to study and formalize belief change and the so-called "common sense reasoning" , i.e. the actual reasoning of humans. On the other hand, in logic, substructural logics, i.e. logics lacking some of the structural rules of classical logic, have been studied in depth from a theoretical point of view. However, the powerful (proof-theoretical) techniques and methods developed in logic have not yet been applied to artificial intelligence. Conditional logic and belief revision theory are prominent theories in artificial intelligence dealing with common sense reasoning. We show in this article that they can both be embedded within the framework of substructural logics and can both be seen as extensions of the Lambek calculus. This allows us to compare and relate them to each other systematically, via a natural formalization of the Ramsey test

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

Kiel Declarative Programming Days 2013

This report contains the papers presented at the Kiel Declarative Programming Days 2013, held in Kiel (Germany) during September 11-13, 2013. The Kiel Declarative Programming Days 2013 unified the following events: * 20th International Conference on Applications of Declarative Programming and Knowledge Management (INAP 2013) * 22nd International Workshop on Functional and (Constraint) Logic Programming (WFLP 2013) * 27th Workshop on Logic Programming (WLP 2013) All these events are centered around declarative programming, an advanced paradigm for the modeling and solving of complex problems. These specification and implementation methods attracted increasing attention over the last decades, e.g., in the domains of databases and natural language processing, for modeling and processing combinatorial problems, and for high-level programming of complex, in particular, knowledge-based systems

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