Nanomagnetic logic, which makes use of arrays of dipolar-coupled single-domain nanomagnets for computation, holds promise as a low-power alternative to traditional computation with CMOS. Beyond the use of nanomagnets for Boolean logic, nanomagnets can also be exploited for nondeterministic computational schemes such as edge detection in images and for solving the traveling salesman problem. Here, we demonstrate the potential of arrangements of thermally active nanomagnets based on artificial spin ice for both deterministic and probabilistic computation. This is achieved by engineering structures that follow particular thermal relaxation pathways consisting of a sequence of reorientations of magnet moments from an initial field-set state to a final low-energy output state. Additionally, we demonstrate that it is possible to tune the probability of attaining a particular final low-energy state, and therefore the likelihood of a given output, by modifying the intermagnet distance. Finally, we experimentally demonstrate a scheme to connect several computational building blocks for complex computation.