  A new approach to the design of massively parallel and interactive
programming languages has been recently proposed using rv-systems (interactive
systems with registers and voices) and Agapia programming. In this paper we
present a few theoretical results on FISs (finite interactive systems), the
underlying mechanism used for specifying control and interaction in these
systems. First, we give a proof for the undecidability of the emptiness problem
for FISs, by reduction to the Post Correspondence Problem. Next, we use the
construction in this proof to get other undecidability results, e.g., for the
accessibility of a transition in a FIS, or for the finiteness of the language
recognized by a FIS. Finally, we present a simple proof of the equivalence
between FISs and tile systems, making explicit that they precisely capture
recognizable two-dimensional languages.
