Models of Computation (Advanced Course 2024 – 25)

Lecturer

Clemens Grabmayer

Room

Library, Ex-INPS building, GSSI, L'Aquila.

Lecture times

Monday, July 7, 10.30—12.30
Tuesday, July 8, 10.30—12.30
Wednesday, July 9, 10.30—12.30
Thursday, July 10, 10.30—12.30
Friday, July 11, 14.30—16.30

Overview at a glance

Course Aims

The course aims at familiarizing attendees with basic concepts of computability theory and with several and diverse models of computation. Following the historical development, three classical models will be presented first:

Turing machines,
recursive functions,
Lambda Calculus.

Subsequently, more recent models in computer science and other fields like science and biology will be mentioned. The focus will be on understanding underlying intuitions rather than on exhaustively formal presentations. Specific attention will be directed to:

the comparison of the computational power of different models by finding, whenever possible, simulations between them, modulo reasonable encodings.

By recognizing that some quite disparate models are computationally equally powerful, we will survey some of the ample empirical evidence that computability is a fundamental concept, the Church-Turing Thesis: every informally computable function is Turing-computable (and equivalently, is definable in Lambda Calculus), modulo reasonable encodings.

Book

Maribel Fernández: Models of Computation (An Introduction to Computability Theory), Springer-Verlag London, 2009.

Links to Resources

Böhm's full precision calculator and Recursive Real Analysis (RRA):
- Popular science article by Chad Nauseam (link)
Post machines, and Post's computability `working hypothesis':
- Emil Leon Post: "Finite Combinatory Processes — Formulation 1" (pdf via wolframscience), The Journal of Symbolic Logic, Vol. 1, No. 3. (Sep., 1936), pp. 103-105.
Turing machines:
- Alan M. Turing: "On Computable Numbers, with an Application to the Entscheidungsproblem" (pdf via wolframscience), Proceedings of the London Mathematical Society, 1936.
  - General interest (recent report): Wartime codebreaker Alan Turing’s scientific papers sell for £465,000 at auction (Guardian article, Tue 17 Jun 2025)
Church's thesis and effective calculability:
- A. Church: "An Unsolvable Problem of Elementary Number Theory" (read pdf at JSTOR)
From partial recursive functions to λ-definable functions:
- Handout (pdf).
Comparing Computational Power of Models of Computation
- Handout (pdf).
Post's Correspondence Problem: "A Variant of a Recursively Unsolvable Problem", Bulletin of the American Mathematical Society, 1946
- Emil Leon Post: pdf via projectEuclid).
Interaction Nets:
- Yves Lafont: "Interaction Nets", Proceedings of POPL'90, 1990 (pdf via ACM).

Lectures

Introduction and Overview (slides)
Post and Turing machines, Turings analysis of computability (slides)
Partial-Recursive Functions (slides)
Lambda Calculus (slides, handout from-recursive-to-λ-definable)
Three More Models (slides, handout comparing computational power)

Clemens Grabmayer / www: https://clegra.github.io / mailto: c one dot a one dot grabmayer one at gmail one dot com / Last modified: Tue 1 Jul 2025 10:23 CEST /