A new definition of program-size complexity is made. H(A;B=C;D) is defined to be the size in bits of the shortest self-delimiting program for calculating strings A and B if one is given a minimal-size selfdelimiting program for calculating strings C and D. This differs from previous definitions: (1) programs are required to be self-delimiting, i.e. no program is a prefix of another, and (2) instead of being given C and D directly, one is given a program for calculating them that is minimal in size. Unlike previous definitions, this one has precisely the formal 2 G. J. Chaitin properties of the entropy concept of information theory. For example, H(A;B) = H(A) + H(B=A) + O(1). Also, if a program of length k is assigned measure 2 \Gammak , then H(A) = \Gamma log 2 (the probability that the standard universal computer will calculate A) +O(1). Key Words and Phrases: computational complexity, entropy, information theory, instantaneous code, Kraft inequality, minimal program, probab...

0 =h 0 in the diagram. The sawtooth line reflects the fact that even when the principle of indifference can be applied, there may be arguments whose strength can be bounded no more precisely than by an adjacent pair of indifference arguments. Note that a=h in the diagram is bounded numerically only by 0.0 and the strength of a 00 =h 00 . Keynes' ideas were taken up by B. O. Koopman [14, 15, 16], who provided an axiomatization for Keynes' probability values. The axioms are qualitative, and reflect what Keynes said about probability judgment. (It should be remembered that for Keynes probability judgment was intended to be objective in the sense that logic is objective. Although different people may accept different premises, whether or not a conclusion follows logically from a given set of premises is objective. Though Ramsey [26] attacked this aspect of Keynes' theory, it can be argued

This paper is concerned with establishing broadly-based system-theoretic foundations and practical techniques for the problem of system identification that are rigorous, intuitively clear and conceptually powerful. A general formulation is first given in which two order relations are postulated on a class of models: a constant one of complexity; and a variable one of approximation induced by an observed behaviour. An admissible model is such that any less complex model is a worse approximation. The general problem of identification is that of finding the admissible subspace of models induced by a given behaviour. It is proved under very general assumptions that, if deterministic models are required then nearly all behaviours require models of nearly maximum complexity. A general theory of approximation between models and behaviour is then developed based on subjective probability concepts and semantic information theory The role of structural constraints such as causality, locality, finite memory, etc., are then discussed as rules of the game. These concepts and results are applied to the specific problem or stochastic automaton, or grammar, inference. Computational results are given to demonstrate that the theory is complete and fully operational. Finally the formulation of identification proposed in this paper is analysed in terms of Klir’s epistemological hierarchy and both are discussed in terms of the rich philosophical literature on the acquisition of knowledge. 1

A new approach to high-order-conditional probability density estimation is developed, based on a partitioning of conditioning space via decision trees. The technique is applied to image compression, image restoration, and texture synthesis, and the results compared with those obtained by standard mixture density and linear regression models. By applying the technique to subband-domain processing, some evidence is provided to support the following statement: the appropriate tradeoff between spatial and spectral localization in linear preprocessing shifts towards greater spatial localization when subbands are processed in a way that exploits interdependence.

In this paper, we introduce the notion of interval structures in an attempt to establish a unified framework for representing uncertain information. Two views are suggested for the interpretation of an interval structure. A typical example using the compatibility view is the roughset model in which the lower and upper approximations form an interval structure. Incidence calculus adopts the allocation view in which an interval structure is defined by the tightest lower and upper incidence bounds. The relationship between interval structures and interval-based numeric belief and plausibility functions is also examined. As an application of the proposed model, an algorithm is developed for computing the tightest incidence bounds.

The Principle of Maximum Entropy is discussed and two classic probabilistic models of information retrieval, the Binary Independence Model of Robertson and Sparck Jones and the Combination Match Model of Croft and Harper are derived using the maximum entropy approach. The assumptions on which the classical models are based are not made. In their place, the probability distribution of maximum entropy consistent with a set of constraints is determined. It is argued that this subjectivist approach is more philosophically coherent than the frequentist conceptualization of probability that is often assumed as the basis of probabilistic modeling and that this philosophical stance has important practical consequences with respect to the realization of information retrieval research.

In this paper we analyse the reasons why teaching probability is difficult for mathematics teachers, we describe the contents needed in the didactical preparation of teachers to teach probability and we present examples of activities to carry out this training. Nowadays probability and statistics is part of the mathematics curricula for primary and secondary school in many countries. The reasons to include this teaching have been repeatedly highlighted over the past 20 years (e.g. Holmes, 1980; Hawkins et al., 1991; Vere-Jones, 1995), and include the usefulness of statistics and probability for daily life, its instrumental role in other disciplines, the need for a basic stochastic knowledge in many professions and its role in developing a critical reasoning. However, teaching probability and statistics is not easy for mathematics teachers. Primary and secondary level mathematics teachers frequently lack specific preparation in statistics education. For example, in Spain, prospective secondary teachers with a major in Mathematics do not receive specific training in statistics education. The situation is even worse for primary teachers, most of whom have not had basic training in statistics and this could be extended to many countries. There can be little support from textbooks and curriculum documents prepared for primary and secondary teachers, because

A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization.

After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its set-theoretical realization in terms of Kolmogorov probability spaces. Since the axioms of mathematical probability theory say nothing about the conceptual meaning of “randomness ” one considers probability as property of the generating conditions of a process so that one can relate randomness with predictability (or retrodictability). In the measure-theoretical codification of stochastic processes genuine chance processes can be defined rigorously as so-called regular processes which do not allow a long-term prediction. We stress that stochastic processes are equivalence classes of individual point functions so that they do not refer to individual processes but only to an ensemble of statistically equivalent individual processes. Less popular but conceptually more important than statistical descriptions are individual descriptions which refer to individual chaotic processes. First, we review the individual description based on the generalized harmonic analysis by Norbert Wiener. It allows the definition of individual purely chaotic processes which can be interpreted as trajectories of regular statistical stochastic processes.

The Archimedean axiom in fuzzy set theory is critically discussed. The axiom is brought into perspective within a measurement theoretic framework and then its validity for fuzzy set theory is questioned. The discussion sheds light into what type of vagueness fuzzy set theory models. 1 Introduction It is one of the basic tenets of fuzzy set theory to take into account the continuous degrees of membership. In that way, fuzzy set theory is distinguished from other many-valued logics. Continuous membership functions and continuous Archimedean triangular norms and conorms together with a negation operator describe an algebraic structure that defines fuzzy set theory. We have investigated the semantic issues that such an algebraic structure raises elsewhere [1, 4, 2, 3] from a measurement point of view. In this paper, we concentrate on the Archimedean axiom and discuss its relevance to fuzzy set theory. In particular we are trying to answer the following questions: 1. Is the Archimedean...