# On the proof theoretic strength of cyclic reasoning

## Anupam Das, University of Birmingham

Cyclic (or circular) proofs are now a common technique for demonstrating metalogical properties of systems incorporating (co)induction, including modal logics, predicate logics, type systems and algebras. Inspired by automaton theory, cyclic proofs encode a form of self-dependency of which induction/recursion comprise special cases. An overarching question of the area, the so-called ‘Brotherston-Simpson conjecture’, asks to what extent the converse holds.

In this talk I will discuss a line of work that attempts to understand the expressivity of circular reasoning via forms of proof theoretic strength. Namely, I address predicate logic in the guise of first-order arithmetic, and type systems in the guise of higher-order primitive recursion, and establish a recurring theme: circular reasoning buys precisely one level of ‘abstraction’ over inductive reasoning.

This talk will be based on the following works: