March 18, 2024

Thomas Seiller (CNRS, LIPN)

What is a model of computation? What is a program, an algorithm? While theoretical computer science has been founded on computability theory, the latter does not answer these questions. Indeed, it is a mathematical theory of computable functions, and does not account for computation itself. A symptomatic consequence is the notion of Turing-completeness. This standard (sole?) equivalence between models of computation is purely extensional: it does only care about what is computed and not how, blind to complexity aspects and the question of algorithmic completeness. More importantly, the theory of computation is continuously growing further from how actual machines compute.

I will present a proposal for alternative foundations more faithful to computer science practice and interests. This mathematisation of computer science is grounded within the theory of dynamical systems, focussing on *how* computation is performed rather than only on *what* is computed. I will argue that it generalises computability theory while still allowing to recover standard results.

If time permits, I can then explain how this theory can be used to (I will let he audience decide which items will be discussed):
• provide a uniform account of several lower bounds from algebraic complexity and strengthen them
• define static analysis methods which can be implemented in usable tools
• define families of linear realisability models (realisability models for linear logic)
• lead to a semantic approach of implicit computational complexity
• propose a formal definition of the notion of algorithm