# $\Sigma^{\mu}_2$ is decidable for $\Pi^{\mu}_2$

## TL;DR

This paper proves that it is decidable whether a given $\Pi^{\mu}_2$ formula in the modal $\mu$ calculus is equivalent to a $\Sigma^{\mu}_2$ formula, advancing understanding of formula equivalence in modal logic.

## Contribution

It establishes the decidability of equivalence between $\Pi^{\mu}_2$ and $\Sigma^{\mu}_2$ formulas in the modal $\mu$ calculus, a previously unresolved problem.

## Key findings

- Decidability of $\Pi^{\mu}_2$ and $\Sigma^{\mu}_2$ formula equivalence.
- Provides an algorithm for checking equivalence.
- Advances theoretical understanding of modal $\mu$ calculus.

## Abstract

Given a $\Pi^{\mu}_2$ formula of the modal $\mu$ calculus, it is decidable whether it is equivalent to a $\Sigma^{\mu}_2$ formula.