V. Wiktor Marek

A theory of nonmonotonic rule systems

We introduce here the study of generalnonmonotonic rule systems. These deal with situations where a conclusion is drawn from a “system of beliefs”S (and seen to be inS), basedboth on some “premises” being inS and on some “restraints” not being inS. In the monotone systems of traditional logic there are no restraints, conclusions are drawn solely based on premises being inS. Nonmonotonic rule systems capture the essential syntactic, semantic, and algorithmic features of many nonmonotone systems such as default logic, negation as failure, truth maintenance, autoepistemic logic, and also important combinatorial questions from mathematics such as the marriage problem. This reveals semantics and syntax and proof procedures and algorithms for computing belief sets in many cases where none were previously available and entirely uniformly. In particular, we introduce and study deductively closed sets, extensions and weak extensions. Semantics of nonmonotonic rule systems is studied in part II of this paper and extensions to predicate classical, intuitionistic, and modal logics are left to a later paper.

Modal logic for default reasoning

In the paper we introduce a variant of autoepistemic logic that is especially suitable for expressing default reasonings. It is based on the notion of iterative expansion. We show a new way of translating default theories into the language of modal logic under which default extensions correspond exactly to iterative expansions. Iterative expansions have some attractive properties. They are more restrictive than autoepistemic expansions, and, for some classes of theories, than moderately grounded expansions. At the same time iterative expansions avoid several undesirable properties of strongly grounded expansions, for example, they are grounded in the whole set of the agents initial assumptions and do not depend on their syntactic representation.Iterative expansions are defined syntactically. We define a semantics which leads to yet another notion of expansion — weak iterative expansion — and we show that there is an important class of theories, that we call ℐ-programs, for which iterative and weak iterative expansions coincide. Thus, for ℐ-programs, iterative expansions can be equivalently defined by semantic means.

Modal nonmonotonic logics: ranges, characterization, computation

Many nonmonotonic formalism, including default logic, logic programming with stable models, and autoepistemic logic, can be represented faithfully by means of modal nonmonotonic logics in the family proposed by McDermott and Doyle. In this paper properties of logics in this family are thoroughly investigated. We present several results on characterization of expansions. These results are applicable to a wide class of nonmonotonic modal logics. Using these characterization results, algorithms for computing expansions for finite theories are developed. Perhaps the most important finding of this paper is that the structure of the family of modal nonmonotonic logics is much simpler than that of the family of underlying modal (monotonic) logics. Namely, it is often the case that different monotonic modal logics collapse to the same nonmonotonic system. We exhibit four families of logics whose nonmonotonic variants coincide: 5-KD45, TW5-SW5, N-WK, and W5-D4WB. These nonmonotonic logics naturally represent logics related to commonsense reasoning and knowledge representation such as autoepistemic logic, reflexive autoepistemic logic, default logic, and truth maintenance with negation.

The relationship between stable, supported, default and autoepistemic semantics for general logic programs

Abstract We investigate the relationship between various alternative semantics for logic programming, viz. the stable model semantics of Gelfond and Lifschitz (1988), the supported model semantics as developed by Apt, Blair and Walker (1988), autoepistemic translations (cf. Moore (1985)) of general logic programs and default translations of general logic programs, Reiter (1980).

Contributions to the Theory of Rough Sets

We study properties of rough sets, that is, approximations to sets of records in a database or, more formally, to subsets of the universe of an information system. A rough set is a pair 〈L, U〉 such that L, U are definable in the information system and L ⊆ U. In the paper, we introduce a language, called the language of inclusion-exclusion, to describe incomplete specifications of (unknown) sets. We use rough sets in order to define a semantics for theories in the inclusion-exclusion language. We argue that our concept of a rough set is closely related to that introduced by Pawlak. We show that rough sets can be ordered by the knowledge ordering (denoted

Revision Programming, Database Updates and Integrity Constraints

\preceq

The expressiveness of locally stratified programs

kn)- We prove that Pawlaks rough sets are characterized as