September 9, 2013

Thursday, 8:30–10:00, T 642, beginning October 3 2013

In contrast to courses like Design and Analysis of Algorithms, where specific algorithms are sought for specific problems, this course deals with the very notion of (algorithmic) computability. What precisely does it mean to say that certain mathematical problems cannot be solved algorithmically even in principle? Negative results of this kind are typical for computability theory. As it turns out, various attempts to formalize the notion of computability all give rise to the same notion of partial recursive function, and there are good arguments to regard it as a precise mathematical equivalent for the informal notion of algorithmically computable partial function. Certain considerations in computability theory also cocern the old question whether or not computers may completely replace humans (as mathematical problem solvers).

One natural sequel is the subject of abstract complexity, one of the topics in the follow-up course Decidability- and Complexity Classes.

Chapters 1–6 deal with the following:

- Definition of the basic concepts of computability theory. For instance, the notion of recursive function is regarded as being equivalent to the intuitive notion of algorithmically computable function, but it is defined in a mathematically precise way.
- In order to support the claim that the class of recursive functions coincides with that of algorithmically computable functions, it is shown that it is closed under numerous operations, and that many computable functions which occur in real life are indeed recursive.
- The first group of results are established which are really typical for computability theory; they include the basic enumeration- and S-m-n-theorems, various characterizations of recursively enumerable sets, and the first undecidability result. The diagonal method is introduced as a basic proof method.
- Some alternative ways are discussed how the class of recursive functions could have been defined in the first place; two of the most well-known ones use μ-recursion and Turing machines.
- Further undecidability results are established by the method of reduction. This chapter contains other typical results from recursive function theory, for instance, the famous fixed-point theorem.
- Elementary theory of reducibility. Intuitively, a problem P is reducible to a problem Q if any (hypothetical) algorithm for solving Q yields an algorithm for solving P. This intuitive concept can be made mathematically precise in various (nonequivalent) ways.

Lecture notes will be handed out. Among textbooks we mention

- G. Tourlakis: Computability. Reston Publishing Company 1984.
- W. Brainerd, L. Landweber: Theory of Computation. Wiley 1974.
- M. Sipser: Introduction to the Theory of Computation. PWS Publishing Company 2005.

Oral exam by agreement. Just consult me when you are ready.