Section: New Results

Invertible-rule-free sequent calculi and an intuitionistic arithmetical hierarchy

Participants : Taus Brock-Nannestad, Danko Ilik.

In sequent calculi, proof rules can be divided into two groups: invertible (asynchronous) proof rules and non-invertible (synchronous) proof rules. Even in focusing sequent calculi the two groups of rules are present, albeit grouped together in synthetic rules (we speak of the synchronous and asynchronous phase).

In this work, we used the exp-log decomposition (described above) in the context of logic in order to obtain a version of sequent caclulus which contains synchronous rules only, a first such formalism for intuitionistic logic.

We extended the picture from the setting of propositional to the one of first-order intuitionistic logic, where the exp-log decomposition provided us with an intuitionistic hierarchy of formulas analogous to the classical arithmetical hierachy; although the classical arithmetical hierarchy exists since the 1920s, a correspondingly versatile notion for intuitionistic logic has been elusive up to this day.

This work is described in the manuscript [37], submitted to an academic journal.