MAGIC107: Computability Theory and Applications

Course details

A specialist MAGIC course


Spring 2022
Monday, January 31st to Friday, March 25th; Monday, April 25th to Friday, May 6th


Live lecture hours
Recorded lecture hours
Total advised study hours


12:05 - 12:55 (UK)


This is an introductory course on computability theory, the branch of mathematical logic concerned with classifying sets of natural numbers by how much extra information is needed to describe them by computer programs.  The computational strength of a set is intimately tied to the complexity of its definition, thus, more generally, computability theory is the study of definable sets of natural numbers.  Any countable mathematical object, such as a countable ring or a countable graph, can be coded by a set of natural numbers.  A continuous function on the reals, for example, can be coded by a set of natural numbers too.  Thus the techniques and results of computability theory can be applied to describe the complexities of all manner of mathematical objects.

The computational strength of a set is measured by its Turing degree.  The collection of all Turing degrees has a rich structure and is the central object of study in computability theory.  This course introduces the basic theory of the Turing degrees.  Time permitting, the course also surveys contemporary research directions that connect to other areas of mathematics, such as reverse mathematics and algorithmic randomness. 


The following prerequisites are preferable but not essential:
  • A basic knowledge of first-order logic. 
  • Familiarity with a formal model of computation, such as Turing machines, register machines, while programs, or similar. 
  • A basic knowledge of undecidability, such as many-one reducibility and the undecidability of the halting problem. 


Core topics:
  • Review of computable functions, computably enumerable sets, and undecidability.
  • Turing reducibility.
  • The arithmetical hierarchy.
  • Basic structure of the Turing degrees.
  • Basic structure of the computably enumerable Turing degrees.
  • Trees and diagonally non-computable functions.
Additional topics and applications:
  • Reverse mathematics.
  • Algorithmic randomness.


  • PS

    Dr Paul Shafer

    University of Leeds


Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

  • Computability Theory (S. Barry Cooper, book)
  • Computability Theory (Rebecca Weber, book)
  • Degrees of Unsolvability (Manuel Lerman, book)
  • Recursively Enumerable Sets and Degrees (Robert I. Soare, book)
  • Subsystems of Second Order Arithmetic (Stephen George Simpson, book)
  • Theory of Recursive Functions and Effective Computability (Hartley Rogers, book)
  • Turing Computability (Robert I. Soare, book)


The assessment for this course will be released on Monday 9th May 2022 at 00:00 and is due in before Monday 23rd May 2022 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.


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