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A Course in Universal Algebra

Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis. It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics. Many examples illustrate the new concepts while numerous exercises of varying difficulty extend the material and stimulate readers to work actively on the text.

Computable Analysis : An Introduction

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Decidability of second-order theories and automata on infinite trees.

Algorithmic model building in artificial intelligence provides a contemporary method for explaining certain kinds of human behavior. The paranoid mode of thought and action represents a kind of pathological behavior that has been well recognized for over twenty centuries. This article describes a computer simulation model embodying a theory that attempts to explain the paranoid mode of behavior in terms of strategies for minimizing and forestalling shame-induced distress. The model consists of two parts, a parsing module and an interpretation-action module. To bring the model into contact with the conditions of a psychiatric diagnostic interview, the parsing module attempts to understand the interview input of clinicians communicating in unrestricted natural language. The meaning of the input is passed to an interpretation-action module made up of data structures and production rules that size up the current state of the interview and decide which (linguistic) actions to perform in order to fulfill the model's intentions. This module consists of an object system which deals with interview situations and a metasystem which evaluates how well the object system is performing to attain its ends. The fidelity of the simulation has been tested using Turing-like indistinguishability tests in which clinical judges attempt to distinguish the paranoid model-patient from an actual paranoid patient. Since clinicians are unable to make the relevant distinctions, the simulation is considered successful at the input-output level of functional equivalence. Issues of underlying structural equivalence and the nature of generative pattern explanations are discussed in the light of the model's potential value in guiding clinicial decisions and intervention strategies in paranoid disorders.

Modeling a paranoid mind

This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.

Andrzej Grzegorczyk

Papers

Some classes of recursive functions

On the definitions of computable real continuous functions

Undecidability of Some Topological Theories

On the definition of computable functionals

A Philosophically Plausible Formal Interpretation of Intuitionistic Logic