Open-universe probability models, representable by a variety of probabilistic programming languages (PPLs), handle uncertainty over the existence and identity of objects-forms of uncertainty occurring in many real-world situations. We examine the problem of extending a declarative PPL to define decision problems (specifically, POMDPs) and identify non-trivial representational issues in describing an agent's capability for observation and action-issues that were avoided in previous work only by making strong and restrictive assumptions. We present semantic definitions that lead to POMDP specifications provably consistent with the sensor and actuator capabilities of the agent. We also describe a variant of point-based value iteration for solving open-universe POMDPs. Thus, we handle cases-such as seeing a new object and picking it up-that could not previously be represented or solved.