Fixed points are a recurring theme in computer science and related fields: shortest paths, game equilibria, semantics and verification, social choice theory, or dynamical systems are only a few of many instances. Various fixed point theorems - e.g. the famous Kleene fixed point theorem - state that fixed points emerge as limits of suitably seeded fixed point iterations. 

I will showcase in this talk a purely algebraic way of reasoning about fixed points which we call the algebra of iterative constructions (AIC). It lead us to discover novel fixed point theorems as well as prove fully algebraically well-established ones (e.g. Kleene, Tarksi-Kantorovic, and k-induction). As for the novel fixed point theorems, we will (given a suitable setting) obtain a method that maps any point to two canonical corresponding fixed points of a function by way of a limit of some fixed point iteration.

Our algebra is mechanized in Isabelle/HOL. Isabelle's sledgehammer tool is able to find proofs of fixed point theorems fully automatically whereas sledgehammer is not able to find such proofs relying only on Isabelle’s standard libraries.

From the audience I would like to learn: Do you encounter fixed points or fixed point problems in your work? Do you perhaps work on algorithms for solving fixed point problems?

I look forward to talking to you!

It has been described as the most important problem whose decidability is still open: the Skolem Problem asks how to determine algorithmically whether a given integer linear recurrence sequence (such as the Fibonacci numbers) has a zero term. This deceptively simple question arises across a wide range of topics in computer science and mathematics, from program verification and automata theory to number theory and logic. This talk traces the history of the Skolem Problem: from the early 1930s to the current frontier of one of the most enduring open questions in computer science.