Polynomial algebras on classical banach spaces
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: Remarks on symmetric smooth norms

title: Logics of Knowing and Believing
abstract: What is the relation of intelligence (natural or artificial) to logic? This is surely a fascinating question with many aspects. We shall not discuss what intelligence is; but we shall start by saying that logic is understood as the study of consequence and of deduction as obtaining consequences from (accepted, assumed) premises (axioms). It goes without saying that intelligence has a logical aspect (among various other aspects). Intelligent systems are said to work with various sorts of knowledge; knowledge representation and processing, and even reasoning about knowledge are considered to be important topics in AI. Knowledge may be uncertain; thus one has to work with beliefs. It has turned out that logical aspects of reasoning about knowledge and beliefs are well formalized by means of some systems of modal logic called logic of knowledge and of belief respectively. An excellent survey of propositional (Boolean) logics of knowledge and belief in relation to computational complexity (with full proofs of main results) is given by Halpern (1992). Most basic facts on these logics are surveyed in Sect. 2 of this chapter. Section 3 surveys basic facts on predicate (Boolean) logics of knowledge and belief; Hughess and Cresswell (1984) contains details. The main part is Sect. 4 where we take into consideration the fact that most knowledge in AI systems (not speaking about natural intelligence) is imprecise (vague, fuzzy), which leads us to fuzzy logic.

title: Arithmetical Interpretations of Dynamic Logic
abstract: An arithmetical interpretation of dynamic propositional logic (DPL) is a mapping f satisfying the following: (1) fassociates with each formula A of DPL a sentencef(A) of Peano arithmetic (PA) and with each program a a formula f(a) of PA with one free variable describing formally a supertheory of PA; (2) f commutes with logical connectives; (3) f([aK]A) is the sentence saying that f(A) is provable in the theory f(a); (4) for each axiom A of DPL, f(A) is provable in PA (and consequently, for each A provable in DPL, f(A) is provable in PA). The arithmetical completeness theorem is proved saying that a formula A of DPL is provable in DPL iff for each arithmetical interpretationf,f(A) is provable in PA. Various modifications of this result are considered.  The aim of the present paper is to apply techniques of arithmetical interpretations (or: provability interpretations) of modal propositional logic (as developed and studied by deJongh, Sambin, Solovay, Smoryn'ski, Guaspari and others, see ,  ) to dynamic logic. Dynamic logic (or: logic of programs) is a generalization of modal logic extensively studied in computer science (Pratt, Fischer, Ladner, Segerbergh, Parikh, see , , ). Both various systems of modal logic and systems of dynamic logic have well-behaving Kripke semantics (necessity means validity in all alternative states-worlds) and complete axiomatizations (for modal logic see [6] and , for dynamic logic see ). The idea behind arithmetical interpretations of modal logic is to interpret necessity as formal provability (in arithmetic). Now, dynamic logic is a modal logic with many modalities of necessity (corresponding to programs) so that the question arises whether one could interpret these modalities by formal provability in various theories (and obtain some completeness results). This question will be answered positively. What is the meaning of our results? First, the question just formulated seems to be a very natural question and it seems not to be trivial to answer it. Second, our result could elucidate from an unexpected point of view the saying of Kowalski : "algorithm logic + control". We shall see that programs (algorithms)i.e. their logical components-may be modelled by some formal theories. Third, it has proved to be useful to investigate modal calculi having various modalities, not just the necessity, when studying self-reference in Peano arithmetic ((see [10], ). And we have in dynamic logic an example of such a calculus having both Received October 15, 1981. ? 1983, Association for Symbolic Logic 0022-4812/83/4803-0021 /$02.00