From free algebras to proof bounds

Abstract

(This is joint work with Nick Bezhanishvili).In the first part of our contribution, we review and compare existing constructions of finitely generated free algebras in modal logic focusing on step-by-step methods. We discuss the notions of step algebras and step frames arising from recent investigations, as well as the role played by finite duality.In the second part of the contribution, we exploit the potential of step frames for investigating proof-theoretic aspects. This includes developing a method which detects when a specific rule-based calculus Ax axiomatizing a given logic L has the so-called bounded proof property. This property is a kind of an analytic subformula property limiting the proof search space. We prove that every finite conservative step frame for Ax is a p-morphic image of a finite Kripke frame for L iff Ax has the bounded proof property and L has the finite model property. This result, combined with a `step version' of the classical correspondence theory turns out to be quite powerful in applications.