Samuel Mimram

CSC_51051_EP – Computational logic: from Artificial intelligence to Zero bugs

Computational logic has been used in a wide range of application in computer science, ranging from the deductive approach to Artificial Intelligence advocated by AI’s founder John McCarthy, to proving the absence of bugs in large industrial software such as the 14th metro line in Paris, or checking difficult theorems the as the one of Feit-Thompson in the classification of finite simple groups.

The goal of this course is to explain how logic can be used in order for modeling problems of computational or mathematical nature, and how computers can be used to achieve this. In particular, we will present proof assistants, which allow to formalize human reasoning by interactively constructing proofs, and explain their use to certify the absence of bugs in programs.

This is based on the so-called Curry-Howard correspondence: a program is the same as a proof (or, more precisely, a typed functional program corresponds to a derivation of its type). We will gradually introduce the required notions:

Then in order to reach realistic applications, we will present the proof assistant Agda and dependent logic which underlies it. Time permitting, we will also present various important related notions and techniques such as set theory or homotopy type theory.

This course is also about logic in practice: all the TDs will be on computer, using OCaml and then Agda.

Pre-requisites: some basic knowledge of logic, the CSC_41012_EP course being perfect for this, otherwise reading some introductory book on logic over the summer is recommended (see the references below).

Course notes

The course notes are available here.

You can order a printed copy on Amazon (bibtex).

Slides

Here are the slides for the course:

  1. Functional programming
  2. Natural deduction
  3. Pure λ-calculus
  4. Simply typed λ-calculus
  5. Agda
  6. First-order logic
  7. Dependent types
  8. Homotopy types

TD

Before beginning with the first TD, you are advised to read getting started with OCaml if you are not so familiar with this language.

  1. Typing a simple programming language
  2. Satisfiability of boolean formulas
  3. The λ-calculus
  4. Implementing a proof assistant
  5. Propositional logic in Agda
  6. Inductive types (over two sessions)
  7. Sorting (over two sessions)
  8. Typing a simple programming language (again)

References

Apart from the course notes, good references on the subject include:

Planning

The course takes places on Tuesday afternoon, in room 102-20-04 of Bâtiment d’Enseignement Mutualisé located here:

The first course is on 24 September 2024 at 14:00, followed by a lab at 16:15.

Videos

Machine Assisted Proofs by Terence Tao:

Conferences

Thierry Coquand will be lecturing at College de France on type theory and proof assistants starting from 13 March 2025 (conferences are open and free)!

OCaml

Agda

Editors

Other proof assistants

Additional resources

Where to find internships

Research labs:

Companies:

Topics:

People (by no means exhaustive, ask me!):

Other resources:

Where to find master 2 courses

If you liked this course I would strongly recommend that you follow the MPRI next year. In particular, the following courses are about closely related topics:

(this is especially true for 2.4, 2.7.1 and 2.7.2).

Other interesting and related masters abroad include:

Where to find a PhD thesis

A good way to find offers for internships, PhD thesis and post-docs is through mailing-lists. In particular: