Concurrent compositions of recursive programs with finite data are a natural abstraction model for concurrent programs. Since reachability is undecidable for this class, a restricted form of reachability has become popular in the formal verification literature, where the set of states reached within k context-switches, for a fixed small constant k, is explored. In this paper, we consider the language theory of these models: concurrent recursive programs with finite data domains that communicate using shared memory and work within k round-robin rounds of context-switches, and where further the stack operations are made visible (as in visibly pushdown automata). We show that the corresponding class of languages, for any fixed k, forms a robust subclass of context-sensitive languages, closed under all the Boolean operations. Our main technical contribution is to show that these automata are determinizable as well. This is the first class we are aware of that includes non-context-free languages, and yet has the above properties.