Computability theory

Abstract

Why do we need a formalization of the notion of algorithm or effective computation? In order to show that a specific problem is algorithmically solvable, it is sufficient to provide an algorithm that solves it in a sufficiently precise manner. However, in order to prove that a problem is in principle not solvable by an algorithm, a rigorous formalism is necessary that allows mathematical proofs. The need for such a formalism became apparent in the studies of David Hilbert (1900) on the foundations of mathematics and Kurt Gödel (1931) on the incompleteness of elementary arithmetic.

The first investigations in the field were conducted by the logicians Alonzo Church, Stephen Kleene, Emil Post, and Alan Turing in the early 1930s. They have provided the foundation of computability theory as a branch of theoretical computer science. The fundamental results established Turing computability as the correct formalization of the informal idea of effective calculation. The results have led to Church’s thesis stating that ”everything computable is computable by a Turing machine”. The theory of computability has grown rapidly from its beginning. Its questions and methods are penetrating many other mathematical disciplines. Today, computability theory provides an important theoretical background for logicians, pure mathematicians, and computer scientists. Many mathematical problems are known to be undecidable such as the word problem for groups, the halting problem, and Hilbert’s tenth problem.

This book is a development of class notes for a two-hour lecture including a one-hour lab held for second-year Bachelor students of Computer Science at the Hamburg University of Technology during the last two years. The course aims to present the basic results of computability theory, including mathematical models of computability, primitive recursive and partial recursive functions, Ackermann’s function, Gödel numbering, universal functions, smn theorem, Kleene’s normal form, undecidable sets, theorems of Rice, and word problems. The manuscript has partly grown out of notes taken by the author during his studies at the University of Erlangen-Nuremberg. I would like to thank again my teachers Martin Becker† and Volker Strehl for giving inspiring lectures in this field.

The second edition contains minor changes. In particular, the section on Gödel numbering has been rewritten and a glossary of terms has been added.

Das Buch gibt eine kurze Einführung in die Theorie der Berechenbarkeit.