A Default Logic Patch for Default Logic

International audienceThis paper is about the fusion of multiple information sources represented using default logic. More precisely, the focus is on solving the problem that occurs when the standard-logic knowledge parts of the sources are contradictory, as default theories trivialize in this case. To overcome this problem, it is shown that replacing each formula belonging to Minimally Unsatisfiable Subformulas by a corresponding supernormal default allows appealing features. Moreover, it is investigated how these additional defaults interact with the initial defaults of the theory. Interestingly, this approach allows us to handle the problem of default theories containing inconsistent standard-logic knowledge, using the default logic framework itself

Propositional semantics for default logic

We present new semantics for propositional default logic based on the notion of meta-interpretations - truth functions that assign truth values to clauses rather than letters. This leads to a propositional characterization of default theories: for each such finite 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 an extension and answering questions 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 which is a superset of network default theories and disjunction-free default theories. This leads to the observation that coherence and membership for the class 2-DT is NP-complete and entailment is co-NP-complete.Since propositional satisfiability can be regarded as a constraint satisfaction problem (CSP), this work also paves the way for applying CSP techniques to default reasoning. In particular, we use the taxonomy of tractable CSP to identify new tractable subsets for Reiter's default logic. Our procedures allow also for computing stable models of extended logic programs

Where Fail-Safe Default Logics Fail

Reiter's original definition of default logic allows for the application of a default that contradicts a previously applied one. We call failure this condition. The possibility of generating failures has been in the past considered as a semantical problem, and variants have been proposed to solve it. We show that it is instead a computational feature that is needed to encode some domains into default logic

Default Logic in a Coherent Setting

In this talk - based on the results of a forthcoming paper (Coletti, Scozzafava and Vantaggi 2002), presented also by one of us at the Conference on "Non Classical Logic, Approximate Reasoning and Soft-Computing" (Anacapri, Italy, 2001) - we discuss the problem of representing default rules by means of a suitable coherent conditional probability, defined on a family of conditional events. An event is singled-out (in our approach) by a proposition, that is a statement that can be either true or false; a conditional event is consequently defined by means of two propositions and is a 3-valued entity, the third value being (in this context) a conditional probability

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