Generalized Rough Sets (Preclusivity Fuzzy-Intuitionistic (BZ) Lattices)

The standard Pawlak approach to rough set theory, as an approximation space consisting of a universe U and an equivalence (indiscernibility) relation R U x U, can be equivalently described by the induced preclusivity ("discernibility") relation U x U \ R, which is irreflexive and symmetric.We generalize the notion of approximation space as a pair consisting of a universe U and a discernibility or preclusivity (irreflexive and symmetric) relation, not necessarily induced from an equivalence relation. In this case the "elementary" sets are not mutually disjoint, but all the theory of generalized rough sets can be developed in analogy with the standard Pawlak approach. On the power set of the universe, the algebraic structure of the quasi fuzzy-intuitionistic "classical" (BZ) lattice is introduced and the sets of all "closed" and of all "open" definable sets with the associated complete (in general nondistributive) ortholattice structures are singled out.The rough approximation of any fixed subset of the universe is the pair consisting of the best "open" approximation from the bottom and the best "closed" approximation from the top. The properties of this generalized rough approximation mapping are studied in the context of quasi-BZ lattice structures of "closed-open" ordered pairs (the "algebraic logic" of generalized rough set theory), comparing the results with the standard Pawlak approach. A particular weak form of rough representation is also studied.     

Fuzzy Logic and Arithmetical Hierarchy,II

A very simple many-valued predicate calculus is presented; a completeness theorem is proved and the arithmetical complexity of some notions concerning provability is determined.     

A Logic for Reasoning about Relative Similarity

A similarity relation is a reflexive and symmetric binary relation between objects. Similarity is relative: it depends on the set of properties of objects used in determining their similarity or dissimilarity. A multi-modal logical language for reasoning about relative similarities is presented. The modalities correspond semantically to the upper and lower approximations of a set of objects by similarity relations corresponding to all subsets of a given set of properties of objects. A complete deduction system for the language is presented.     

飞行错觉水平评定方法的初步研究

本文从飞行错觉发生的类型和频率出发,尝试对飞行员在实际飞行中所产生的错觉水平或空间定向能力做出基本的评定。本研究采用区间模糊统计的方法,对飞行错觉的类型（倾斜错觉、俯仰错觉、方位错觉、反旋转错觉和倒飞错觉）和频率（从无、偶尔、时有、经常、总是）做了经验的赋值工作,分别计算出了这些模糊概念（除从无外）的心理量表值和模糊度。同时我们认为,飞行错觉水平是频率与类型的函数,其数学表达式为：,并将这一表达式作为评定飞行员飞行错觉水平的基本模型。     

Choosing what to choose from: Preference for inclusion over exclusion when constructing consideration sets from large choice sets

Decision making is a two‐stage process, consisting of, first, consideration set construction and then final choice. Decision makers can form a consideration set from a choice set using one of two strategies: including the options they wish to further consider or excluding those they do not wish to further consider. The authors propose that decision makers have a relative preference for an inclusion (vs. exclusion) strategy when choosing from large choice sets and that this preference is driven primarily by a lay belief that inclusion requires less effort than exclusion, particularly in large choice sets. Study 1 demonstrates that decision makers prefer using an inclusion (vs. exclusion) strategy when faced with large choice sets. Study 2 replicates the effect of choice set size on preference for consideration set construction strategy and demonstrates that the belief that exclusion is more effortful mediates the relative preference for inclusion in large choice sets. Studies 3 and 4 further support the importance of perceived effort, demonstrating a greater preference for inclusion in large choice sets when decision makers are primed to think about effort (vs. accuracy; Study 3) and when the choice set is perceived as requiring more effort because of more information being presented about each alternative (vs. more alternatives in the choice set; Study 4). Finally, Study 5 manipulates consideration set construction strategy, showing that using inclusion (vs. exclusion) in large choice sets leads to smaller consideration sets, greater confidence in the decision process, and a higher quality consideration set.     

On Revising Fuzzy Belief Bases

We look at the problem of revising fuzzy belief bases, i.e., belief base revision in which both formulas in the base as well as revision-input formulas can come attached with varying degrees. Working within a very general framework for fuzzy logic which is able to capture certain types of uncertainty calculi as well as truth-functional fuzzy logics, we show how the idea of rational change from “crisp” base revision, as embodied by the idea of partial meet (base) revision, can be faithfully extended to revising fuzzy belief bases. We present and axiomatise an operation of partial meet fuzzy base revision and illustrate how the operation works in several important special instances of the framework. We also axiomatise the related operation of partial meet fuzzy base contraction.This paper is an extended version of a paper presented at the Nineteenth Conference on Uncertainty in Arti.cial Intelligence (UAI’03).     

Algebras of Intervals and a Logic of Conditional Assertions

Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable mathematical entities to bear conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an algebraic completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, ukasiewicz's three-valued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Körner's logic of inexact predicates.     

On Modal μ-Calculus and Non-Well-Founded Set Theory

A finitary characterization for non-well-founded sets with finite transitive closure is established in terms of a greatest fixpoint formula of the modal -calculus. This generalizes the standard result in the literature where a finitary modal characterization is provided only for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new interlinks between automata theory and non-well-founded sets.