Unified inverse correspondence for DLE-Logics

TL;DR

This paper introduces an algorithm that inverts the ALBA correspondence procedure for distributive lattice expansions, enabling the derivation of inductive formulas from their first order correspondents within DLE logics.

Contribution

It presents a novel algorithm that reverses ALBA's steps, facilitating the computation of inductive formulas from first order correspondents in DLE logics.

Findings

Successfully inverts ALBA steps within DLE setting

Enables derivation of inductive formulas from first order correspondents

Enhances the algebraic and order theoretic tools for DLE logics

Abstract

By exploiting the algebraic and order theoretic mechanisms behind Sahlqvist correspondence, the theory of unified correspondence provides powerful tools for correspondence and canonicity across different semantics and signatures, covering all the logics whose algebraic semantics are given by normal (distributive) lattice expansions (referred to as (D)LEs). In particular, the algorithm ALBA, parametric in each (D)LE, effectively computes the first order correspondents of (D)LE-inductive formulas. We present an algorithm that makes use of ALBA's rules and algebraic language to invert its steps in the DLE setting; therefore effectively computing an inductive formula starting from its first order correspondent.