  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<=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<=k are languages of HDAs.
As a consequence, unlike the case of all pomsets, order extension of
MSO-definable sets of iiPoms<=k is also MSO-definable.
