This paper will describe a voyage of discovery — the discovery of Algorithmic Probability. But before I describe that voyage —a few words about motivation. Motivation in science is roughly of two kinds: In one, the motivation is discovery itself — the joy of “going where no one has gone before ” — the

Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid three-dimensional objects, (iii) in the theory of effective computability (Church-Turhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.

We examine the longstanding problem of introducing a time observable in Quantum Mechanics; using the formalism of positive-operatorvalued measures we show how to define such an observable in a natural way and we discuss some consequences. Pacs: 03.65.Bz- Quantum theory; UTF-390.

Empirical equations are an important class of regularities that can be discovered in databases. We concentrate on the role of equations as definitions of attribute values. Such definitions can be used in many ways in a single database and for transfer of knowledge between databases. We present a quest for equations that can be used as definitions of an attribute in a given database. That quest triggers a discovery mechanism that specializes in searching recursively a system of databases and returns a set of partial definitions. We introduce the notion of shared operational semantics. It is founded on an equation-based system of partial definitions and it gives necessary foundations for designing local query answering systems in a distributed two-layered information system (D²LIS). The knowledge exchange between two sites of D²LIS may only improve an equation-based system of partial definitions at each of these sites. At the same time the shared operational ...

Empirical equations and rules are important classes of regularities that can be discovered in databases. We concentrate on their role as definitions of attribute values. Such definitions can be used in many ways in a single database and for transfer of knowledge between databases. We analyze quests for definitions of an attribute in a given database. A quest triggers a discovery mechanism that specializes in searching recursively a system of databases and returns a set of partial definitions. We introduce the notion of shared operational semantics founded on an equation-based and rule-based system of partial de nitions. It gives necessary foundations for designing local query answering systems in a distributed knowledge system (DKS).