  The realization and study of arrays of interacting magnetic nanoislands, such
as artificial spin ices, have reached mature levels of control that allow
design and demonstration of exotic, collective behaviors not seen in natural
materials. Advances in the direct manipulation of their local, binary moments
also suggest a use as nanopatterned, interacting memory media, for computation
{\it within} a magnetic memory. Recent experimental work has demonstrated the
possibility of building logic gates from clusters of interacting magnetic
domains, and yet the possibility of large scale integration of such gates can
prove problematic even at the theoretical level. Here we introduce
theoretically complete sets of logical gates, in principle realizable in an
experiment, and we study the feasibility of their integration into tree-like
circuits. By evaluating the fidelity control parameter between their collective
behavior and their expected logic functionality we determine conditions for
integration. Also, we test our numerical results against the presence of
disorder in the couplings, showing that the design gate structure is robust to
small coupling perturbations, and thus possibly to small imperfections in the
fabrication of the islands.
