Abstract

I’ll present the theory of selection functions, with applications to game theory, proof theory and topology, among others.

Selection functions form a strong monad, which can be defined in any cartesian closed category, and has a morphism into the continuation monad. In certain categories of spaces and domains, the strength can be infinitely iterated. This infinite strength is an amazingly versatile functional that (i) optimally plays sequential games, (ii) realizes the Double Negation Shift used to realize the classical axiom of countable choice, and (iii) implements a computational version of the Tychonoff Theorem from topology. The infinite strength turns out to be built-in in the functional language Haskell, called sequence, and can be used to write unexpected programs that compute with infinite objects, sometimes surprisingly fast. The selection monad also gives rise to a new translation of classical logic into intuitionistic logic, which we refer to as the Peirce translation, as monad algebras are objects that satisfy Peirce’s Law.

This is joint work with Paulo Oliva from Queen Mary.