Kolmogorov Complexity in perspective. Part II: Classification, Information Processing and Duality

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts published in a same volume. Part II is dedicated to the relation between logic and information system, within the scope of Kolmogorov algorithmic information theory. We present a recent application of Kolmogorov complexity: classification using compression, an idea with provocative implementation by authors such as Bennett, Vitanyi and Cilibrasi. This stresses how Kolmogorov complexity, besides being a foundation to randomness, is also related to classification. Another approach to classification is also considered: the so-called "Google classification". It uses another original and attractive idea which is connected to the classification using compression and to Kolmogorov complexity from a conceptual point of view. We present and unify these different approaches to classification in terms of Bottom-Up versus Top-Down operational modes, of which we point the fundamental principles and the underlying duality. We look at the way these two dual modes are used in different approaches to information system, particularly the relational model for database introduced by Codd in the 70's. This allows to point out diverse forms of a fundamental duality. These operational modes are also reinterpreted in the context of the comprehension schema of axiomatic set theory ZF. This leads us to develop how Kolmogorov's complexity is linked to intensionality, abstraction, classification and information system.Comment: 43 page

Logic and operator algebras

The most recent wave of applications of logic to operator algebras is a young and rapidly developing field. This is a snapshot of the current state of the art.Comment: A minor chang

Hyperset Approach to Semi-structured Databases and the Experimental Implementation of the Query Language Delta

This thesis presents practical suggestions towards the implementation of the hyperset approach to semi-structured databases and the associated query language Delta. This work can be characterised as part of a top-down approach to semi-structured databases, from theory to practice. The main original part of this work consisted in implementation of the hyperset Delta query language to semi-structured databases, including worked example queries. In fact, the goal was to demonstrate the practical details of this approach and language. The required development of an extended, practical version of the language based on the existing theoretical version, and the corresponding operational semantics. Here we present detailed description of the most essential steps of the implementation. Another crucial problem for this approach was to demonstrate how to deal in reality with the concept of the equality relation between (hyper)sets, which is computationally realised by the bisimulation relation. In fact, this expensive procedure, especially in the case of distributed semi-structured data, required some additional theoretical considerations and practical suggestions for efficient implementation. To this end the 'local/global' strategy for computing the bisimulation relation over distributed semi-structured data was developed and its efficiency was experimentally confirmed.Comment: Technical Report (PhD thesis), University of Liverpool, Englan

A proof of strong normalisation using domain theory

Ulrich Berger presented a powerful proof of strong normalisation using domains, in particular it simplifies significantly Tait's proof of strong normalisation of Spector's bar recursion. The main contribution of this paper is to show that, using ideas from intersection types and Martin-Lof's domain interpretation of type theory one can in turn simplify further U. Berger's argument. We build a domain model for an untyped programming language where U. Berger has an interpretation only for typed terms or alternatively has an interpretation for untyped terms but need an extra condition to deduce strong normalisation. As a main application, we show that Martin-L\"{o}f dependent type theory extended with a program for Spector double negation shift.Comment: 16 page

Quantum Cognition based on an Ambiguous Representation Derived from a Rough Set Approximation

Over the last years, in a series papers by Arrechi and others, a model for the cognitive processes involved in decision making has been proposed and investigated. The key element of this model is the expression of apprehension and judgement, basic cognitive process of decision making, as an inverse Bayes inference classifying the information content of neuron spike trains. For successive plural stimuli, it has been shown that this inference, equipped with basic non-algorithmic jumps, is affected by quantum-like characteristics. We show here that such a decision making process is related consistently with ambiguous representation by an observer within a universe of discourse. In our work ambiguous representation of an object or a stimuli is defined by a pair of maps from objects of a set to their representations, where these two maps are interrelated in a particular structure. The a priori and a posteriori hypotheses in Bayes inference are replaced by the upper and lower approximation, correspondingly, for the initial data sets each derived with respect to a map. We show further that due to the particular structural relation between the two maps, the logical structure of such combined approximations can only be expressed as an orthomodular lattice and therefore can be represented by a quantum rather than a Boolean logic. To our knowledge, this is the first investigation aiming to reveal the concrete logic structure of inverse Bayes inference in cognitive processes.Comment: 23 pages, 8 figures, original research pape