The causal ladder and the strength of K-causality. I

A unifying framework for the study of causal relations is presented. The causal relations are regarded as subsets of M x M and the role of the corresponding antisymmetry conditions in the construction of the causal ladder is stressed. The causal hierarchy of spacetime is built from chronology up to K-causality and new characterizations of the distinction and strong causality properties are obtained. The closure of the causal future is not transitive, as a consequence its repeated composition leads to an infinite causal subladder between strong causality and K-causality - the A-causality subladder. A spacetime example is given which proves that K-causality differs from infinite A-causality.Comment: 16 pages, one figure. Old title: ``On the relationship between K-causality and infinite A-causality''. Some typos fixed; small change in the proof of lemma 4.

A prototypical approach to machine learning

This paper presents an overview of a research programme on machine learning which is based on the fundamental process of categorization. This work draws upon the psychological theory of prototypical concepts . This theory is that concepts learnt naturally from interaction with the environment (basic categories) are not structured or defined in logical terms but are clustered in accordance with their similaritry to a central prototype, representing the "most typical" member. A structure of a computer model designed to achieve categorization is outlined and the knowledge representational forms and developmental learning associated with this approach are discussed

A THEORY OF QUALITATIVE SIMILARITY

The central result of this paper establishes an isomorphism between two types of mathematical structures: ""ternary preorders"" and ""convex topologies."" The former are characterized by reflexivity, symmetry and transitivity conditions, and can be interpreted geometrically as ordered betweenness relations; the latter are defined as intersection-closed families of sets satisfying an ""abstract convexity"" property. A large range of examples is given. As corollaries of the main result we obtain a version of Birkhoff''s representation theorem for finite distributive lattices, and a qualitative version of the representation of ultrametric distances by indexed taxonomic hierarchies.