# On the complexity of the Leibniz hierarchy

## TL;DR

This paper establishes the computational complexity of determining whether a finite logical matrix corresponds to various classes of algebraizable logics, proving EXPTIME-completeness for most cases.

## Contribution

It provides the first complexity classification for the problem of identifying algebraizable and related logics from finite matrices.

## Key findings

- Determined EXPTIME-completeness for algebraizable logic recognition
- Extended results to order algebraizable, weakly algebraizable, equivalential, and protoalgebraic logics
- Showed truth-equational logic recognition is EXPTIME-hard

## Abstract

We prove that the problem of determining whether a finite logical matrix determines an algebraizable logic is complete for EXPTIME. The same result holds for the classes of order algebraizable, weakly algebraizable, equivalential and protoalgebraic logics. Finally, the same problem for the class of truth-equational logic is shown to be hard for EXPTIME.