Bertrand I-Peng Lin

Possibilistic reasoning—a mini-survey and uniform semantics

Abstract In this paper, we survey some quantitative and qualitative approaches to uncertainty management based on possibility theory and present a logical framework to integrate them. The semantics of the logic is based on the Dempsters rule of conditioning for possibility theory. It is then shown that classical modal logic, conditional logic, possibilistic logic, quantitative modal logic and qualitative possibilistic logic are all sublogics of the presented logical framework. In this way, we can formalize and generalize some well-known results about possibilistic reasoning in a uniform semantics. Moreover, our uniform framework is applicable to nonmonotonic reasoning, approximate consequence relation formulation, and partial consistency handling.

Proof methods for reasoning about possibility and necessity

Abstract Possibilistic logic is a quantitative method for uncertainty reasoning that is closely related to Zadehs fuzzy set theory. In this paper, we formulate it as a kind of multimodal logic and develop some proof methods for it, including tableau method and two styles of natural deduction methods. The completeness and soundness of these methods are proved. Finally, some potential applications and the possible research directions are pointed out.

Reasoning about Higher Order Uncertainty in Possiblistic Logic

Possibilistic logic is an important approach for reasoning about possibility and necessity. By formulating possibilistic reasoning as a kind of modal logic, we can represent and reason about higher order uncertainty to any nested degrees. In this paper, we present a system with built-in axioms for reasoning about higher order uncertainty. Some intuition behind the system is discussed and the soundness and completness of it with respect to the class of finite transitive and serial (in fuzzy set-theoretic sense) possible-world models are proved.

A theoretical investigation into quantitative modal logic

Quantitative modal logic (QML) is a multi-modal formulation of possibilistic reasoning with the capacity of representing and reasoning about higher order uncertainty. In this paper, some results about QML and possibility theory are investigated. First, the concept of filtration in classical modal logic is introduced into QML and the fundamental filtration theorem is proved. Second, as a corollary, the finite model property of QML is provided. Finally, the theoretical and practical implications of finite model property are discussed.

FUZZY LOGIC WITH EQUALITY

The concept of fuzzy equality and its related contents to the first order predicate calculus are discussed. It is proved that, in the viewpoint of computational logic, resolution and paramodulation mechanisms are complete and sound for fuzzy logic with equality. Term rewriting system, that is the set of left to right directional equations, provides an essential computational paradigm for word problems in universal algebra. We embody the fuzzy equality to the theory of this computation system and give an algorithmic solution to the word problems in fuzzy algebra.

H-extension of ring

A ring R is called an H -ring if for every x ∈ R there exists an interger n = n ( x )> 1 such that x n - x ∈ C , where C is center of R. I. N. Herstein proved that H -rings must be commutative [See 3 pp. 220–221]. We now introduce the following definition.

Gentzen Sequent Calculus for Possibilistic Reasoning

Possibilistic logic is an important uncertainty reasoning mechanism based on Zadehs possibility theory and classical logic. Its inference rules are derived from the classical resolution rule by attaching possibility or necessity weights to ordinary clauses. However, since not all possibility-valued formulae can be converted into equivalent possibilistic clauses, these inference rules are somewhat restricted. In this paper, we develop Gentzen sequent calculus for possibilistic reasoning to lift this restriction. This is done by first formulating possibilistic reasoning ⇒ a kind of modal logic. Then the Gentzen method for modal logics generalized to cover possibilistic logic. Finally, some properties of possibilistic logic, such as Craigs interpolation lemma and Beths definability theorem are discussed in the context of Gentzen methods.

Fuzzy term-rewriting system

Abstract This paper provides an approach to treat fuzzy equality in first order logic, specially in a theory in which only equality occurs as a predicate symbol (namely equational theory). Fuzzy term rewriting systems (FRS), that is the set of left to right directional equations, are also discussed. Finally, a fuzzy recognition algorithm is presented by applying FRS, which shows its potential use.

Nonmonotonic reasoning based on incomplete logic

ABSTRACT What characterizes human reasoning is the ability of dealing with incomplete information. Incomplete logic is developed for modeling incomplete knowledge. The most distinctive feature of incomplete logic is its semantics. This is an alternative presentation of partial semantics. In this paper, we will introduce the general notion of incomplete logic (ICL), compare it with partial logic, and give the resolution method for it. We will also show how ICL can be applied to nonmonotonic reasoning. We define nonmonotonic derivation as monotonic derivation in ICL from the database and some consistent assumptions. The mechanism of ICL makes it easy to assert the consistency of an assumption without asserting the assumption itself.

Abstract minimality and circumscription

Abstract In this paper, we present an alternative approach to the generalization of circumscription. Traditionally, the generalization of circumscription involves the change of ordering among models, while in the present study we only try to generalize the minimality criteria of models. We define the notion of abstractly minimal (or ( P , Z )-minimal) models by isomorphism. Under this generalization we come up with the fact that some theories which are unsatisfiable in the original circumscription will be satisfiable now. Moreover, we prove that this generalization is completely coincident with the original circumscription in the case of well-founded theories.