DECIDABILITY OF BEING A UNION-SPLITTING

Abstract Many logical properties are known to be undecidable for normal modal logics, with few exceptions such as the consistency and the coincidence with $\mathsf {K}$ . This article shows that the property of being a union-splitting in $\mathop {\mathsf {NExt}}{\mathsf {K}}$ , the lattice of normal modal logics, is decidable, thus answering Problem 2 in [F. Wolter and M. Zakharyaschev, 2007]. This is done by providing a semantic characterization of union-splittings in terms of finite modal algebras. Moreover, by clarifying the connection to union-splittings, we show that in $\mathop {\mathsf {NExt}}{\mathsf {K}}$ , having a decidable axiomatization problem and being a (un)decidable formula are also decidable. The latter answers Problem 17.3 in [A. Chagrov and M. Zakharyaschev, 1997] for $\mathop {\mathsf {NExt}}{\mathsf {K}}$ . 1