SYSTEMS OF CONDITIONAL BELIEFS IN DEMPSTER-SHAFER THEORY AND EXPERT SYSTEMS
Various notions of soundness for systems of conditional beliefs are introduced and interrelated (the term “belief” being used in the sense of Dempster-Shafer theory). An application to the analysis of the use of Dempster-Shafer theory in classical (rule-based, MYCIN-like) expert systems is sketched. Some open problems are formulated.

title: A simple dynamic logic
abstract: A system of dynamic predicate logic is presented. It allows nonstandard modelling of time, which makes recursive axiomatization possible, and is essentially one-sorted (the same elements code data and time), which makes the system reasonably simple and transparent. The system is subjected to usual mathematical considerations; in particular, it is shown to be equivalent to the three-sorted system due to Andreka, Nemeti and Sain (1982). A conservation result for the latter system is obtained as a by-product.

title: GETTING BELIEF FUNCTIONS FROM KRIPKE MODELS

title: Trakhtenbrot Theorem and Fuzzy Logic
abstract: Trakhtenbrot theorem is shown to be valid for the three main fuzzy logics – Łukasiewicz, Godel and product logic.

title: Metamathematics of Fuzzy Logic
abstract: Preface. 1. Preliminaries. 2. Many-Valued Propositional Calculi. 3. Lukasiewicz Propositional Logic. 4. Product Logic, Godel Logic. 5. Many-Valued Predicate Logics. 6. Complexity and Undecidability. 7. On Approximate Inference. 8. Generalized Quantifiers and Modalities. 9. Miscellanea. 10. Historical Remarks.

title: Fuzzy Logic and Arithmetical Hierarchy, II
abstract: 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.

title: Analytic Renormings of C(K) Spaces
abstract: A ladder system is disposed between two spaced storage shelves. The ladder may move longitudinally along the shelves, and may also be moved laterally between the shelves. Further, the ladder may pivot about a lateral axis such that the ladder may be picked up over obstructions in an aisle between the shelves. In this way, one can quickly and easily position the ladder at any position along or between the two storage shelves for accessing items stored on the shelves. This ladder system is particularly useful in the crowded warehouse-type retail outlets which are now gaining popularity.

title: Polynomial algebras on classical banach spaces
abstract: For the classical Banach spacesX = ℓp,C(K) we identify alln such that every polynomial of degreen + 1 onX is uniformly approximable on the unit ball by elements of the algebra generated by all polynomials of degree up ton onX.

title: Dual renormings of Banach spaces
abstract: We prove that a Banach space admitting an equivalent WUR norm is an Asplund space. Some related dual renormings are also presented.

title: Smooth norms that depend locally on finitely many coordinates
abstract: We characterize separable normed spaces that admit equivalent COO-smooth norms depending locally on finitely many coordinates. It follows, in particular, that such norms exist on any normed space with countable algebraic basis. We use the method of Talagrand operators developed by Haydon in  and certain integral convolution techniques [2], [13] to characterize separable normed spaces that admit C??-smooth norms depending locally on finitely many coordinates as spaces that admit norms with countable boundaries. As corollaries we obtain improvements on some results of Fonf  and Vanderwerff  and a new simple proof of a result of Haydon on Cw-smooth renormings of spaces of continuous functions on countable compact sets. Our results should be compared to the result of Godun, Lin, and Troyanski in  where it is shown that every separable Banach space can be given an equivalent norm, the unit ball of which contains countably many strongly extreme points. From this result and Theorem 1 below it follows that strongly extreme points need not necessarily form a boundary of Bx* . We will use the notation standard in Banach space theory. In particular, Bx and Sx will denote respectively the unit ball and the unit sphere of a Banach space X. We say that 11 11 depends locally on finitely many coordinates if for each x E Sx there exist an open neighbourhood 0 of x, a finite set {x, ...,Xk*} C X*, and a function f: Rk -+ R such that IIYII =f(X1*(y), ...,xk*(y)) fory EO. If (X, 11) is a normed space, the set B c Sx* is called a boundary of (X, * 11) if for each x E Sx there exists x* E B such that x*(x) = 1. Theorem 1. Let (X, 11 * 11) be a normed space.  (i) X admits an equivalent norm having a countable boundary. (ii) X admits an equivalent norm with a boundary B, such that there is a sequence {Kn}nEN of norm compact sets in X* satisfing B C UnEN KnX Received by the editors October 13, 1993 and, in revised form, June 15, 1994.