Grigori Schwarz

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.

Minimal knowledge problem: a new approach

Abstract In this paper we propose a new logic of minimal knowledge. Our approach falls into the general scheme of Shohams preference semantics. It stems from an earlier work on logics of minimal knowledge by Halpern and Moses, Lin and Shoham, and Lifschitz. The novelty of our work is in a procedure for minimizing knowledge which we propose in this paper, and which is different from earlier proposals. We show that our logic preserves most desirable properties of earlier formalisms and at the same time avoids some of their drawbacks. In addition to a semantic definition of our system, we provide its equivalent syntactic characterization which relates our logic with the nonmonotonic modal logic S4F and allows us to use in our investigations standard modal logic techniques.

On embedding default logic into Moore's autoepistemic logic

Abstract Recently Gottlob proved [2] that there does not exist a faithful modular translation of default logic into autoepistemic logic, and presented a non-modular translation. Gottlobs translation, however, is indirect (it uses “nonmonotonic logic N” as an intermediate point), quite complex and exploits sophisticated encoding of proof theory in autoepistemic formulas. We provide a simpler and more intuitive (non-modular) direct translation. In addition, our argument is purely model-theoretic.

In search of a “true” logic of knowledge: the nonmonotonic perspective

Abstract Modal logics are currently widely accepted as a suitable tool of knowledge representation, and the question what logics are better suited for representing knowledge is of particular importance. Usually, some axiom list is given, and arguments are presented justifying that suggested axioms agree with intuition. The question why the suggested axioms describe all the desired properties of knowledge remains answered only partially, by showing that the most obvious and popular additional axioms would violate the intuition. We suggest the general paradigm of maximal logics and demonstrate how it can work for nonmonotonic modal logics. Technically, we prove that each of the modal logics KD45, SW5, S4F and S4.2 is the strongest modal logic among the logics generating the same nonmonotonic logic. These logics have already found important applications in knowledge representation, and the obtained results contribute to the explanation of this fact.

Ranges of Strong Modal Nonmonotonic Logics

In this paper we consider the effects of Gelfond-Lifschitz reduct of theories in modal language on the properties of associated notions of expansion. It turns out that, as in the general case without Gelfond-Lifschitz reduct, various monotonic modal logics collapse in this situation to the same nonmonotonic logic.

Nonmonotonic Reasoning is Sometimes Simpler

We establish the complexity of decision problems associated with the nonmonotonic modal logic S4. We prove that the problem of existence of an S4-expansion for a given set A of premises is Σ 2 P -complete. Similarly, we show that for a given formula ϕ and a set A of premises, it is Σ 2 P -complete to decide whether ϕ belongs to at least one S4-expansion for A, and it is Π 2 P -complete to decide whether ϕ belongs to all S4-expansions for A. An interesting aspect of these results is that reasoning (testing satisfiability and provability) in the monotonic modal logic S4 is PSPACE-complete. To the best of our knowledge, the nonmonotonic logic S4 is the first example of a nonmonotonic formalism which is computationally easier than the monotonic logic that underlies it (assuming PSPACE does not collapse to Σ P 2) .