Recursive input-output models for non-linear multivariate discrete-time systems are derived, and sufficient conditions for their existence are defined. The paper is divided into two parts. The first part introduces and defines concepts such as Nerode realization, multistructural forms and results from differential geometry which are then used to derive a recursive input-output model for multivariable deterministic non-linear systems. The second part introduces several examples, compares the derived model with other representations and extends the results to create prediction error or innovation input-output models for non-linear stochastic systems. These latter models are the generalization of the multivariable ARM AX models for linear systems and are referred to as NARMAX or Non-linear AutoRegressive Moving Average models with exogenous inputs.

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Input-output parametric models for non-linear systems Part II: stochastic non-linear systems

Motivation. Foundations. CATEGORIES, FUNCTORS, AND NATURAL TRANSFORMATIONS. Categories and Functors. Subcategories. Concrete Categories and Concrete Functors. Natural Transformations. OBJECTS AND MORPHISMS. Objects and Morphisms in Abstract Categories. Objects and Morphisms in Concrete Categories. Injective Objects and Essential Embeddings. SOURCES AND SINKS. Sources and Sinks. Limits and Colimits. Completeness and Cocompleteness. Functors and Limits. FACTORIZATION STRUCTURES. Factorization Structures for Morphisms. Factorization Structures for Sources. E-Reflective Subcategories. Factorization Structures for Functors. ADJOINTS AND MONADS. Adjoint Functors. Adjoint Situations. Monads. TOPOLOGICAL AND ALGEBRAIC CATEGORIES. Topological Categories. Topological Structure Theorems. Algebraic Categories. Algebraic Structure Theorems. Topologically Algebraic Categories. Topologically Algebraic Structure Theorems. CARTESIAN CLOSEDNESS AND PARTIAL MORPHISMS. Cartesian Closed Categories. Partial Morphisms, Quasitopoi, and Topological Universes. Bibliography. Tables. Table of Categories. Table of Symbols. Index.

Abstract and concrete categories : the joy of cats

A formal approach to analysing the user complexity of interactive systems or devices is described, based on theoretical results from cognitive psychology. The user's knowledge of how to use a system to accomplish the various tasks is represented in a procedural notation that permits quantification of the amount and complexity of the knowledge required and the cognitive processing load involved in using a system. Making a system more usable can be accomplished by altering its design until the knowledge is adequately simplified. By representing the device behaviour formally as well, it is possible to simulate the user-device interaction to obtain rigorous measures of user complexity.

An approach to the formal analysis of user complexity

What can computers do in principle? What are their inherent theoretical limitations? These are questions to which computer scientists must address themselves. The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function: intuitively a function whose values can be calculated in an effective or automatic way. This book is an introduction to computability theory (or recursion theory as it is traditionally known to mathematicians). Dr Cutland begins with a mathematical characterisation of computable functions using a simple idealised computer (a register machine); after some comparison with other characterisations, he develops the mathematical theory, including a full discussion of non-computability and undecidability, and the theory of recursive and recursively enumerable sets. The later chapters provide an introduction to more advanced topics such as Godel's incompleteness theorem, degrees of unsolvability, the Recursion theorems and the theory of complexity of computation. Computability is thus a branch of mathematics which is of relevance also to computer scientists and philosophers. Mathematics students with no prior knowledge of the subject and computer science students who wish to supplement their practical expertise with some theoretical background will find this book of use and interest.

Computability: An Introduction to Recursive Function Theory

The study of artificial self-replicating structures or machines has been taking place now for almost half a century. My goal in this article is to present an overview of research carried out in the domain of self-replication over the past 50 years, starting from von Neumann's work in the late 1940s and continuing to the most recent research efforts. I shall concentrate on computational models, that is, ones that have been studied from a computer science point of view, be it theoretical or experimental. The systems are divided into four major classes, according to the model on which they are based: cellular automata, computer programs, strings (or strands), or an altogether different approach. With the advent of new materials, such as synthetic molecules and nanomachines, it is quite possible that we shall see this somewhat theoretical domain of study producing practical, real-world applications.

Ira Pohl

Papers

Theories of Abstract Automata