Abstract
In this paper we study finite higher-dimensional automata (HDAs) from the logical point of view. Languages of HDAs are sets of finite bounded-width interval pomsets with interfaces (iiPoms$\le k$) closed under order extension. We prove that languages of HDAs are MSO-definable. For the converse, we show that the order extensions of MSO-definable sets of iiPoms$\le k$ are languages of HDAs. Furthermore, both constructions are effective. As a consequence, unlike the case of all pomsets, the order extension of any MSO-definable set of iiPoms$\le k$ is MSO-definable.