Abstract In a seminal paper, Lin and Reiter introduced the notion of progression of basic action theories. Unfortunately, progression is second-order in general. Recently, Liu and Lakemeyer improve on earlier results and show that for the local-effect and normal actions case, progression is computable but may lead to an exponential blow-up. Nevertheless, they show that for certain kinds of expressive ?rst-order knowledge bases with disjunctive information, called proper+ , it is ef?cient. However, answering queries about the resulting state is still undecidab In this paper, we continue this line of research and ext proper+ KBs to include functions. We prove that their progression wrt local-effect, normal actions, and rangerestricted theories, is ?rst-order de?nable and ef?cientl computable. We then provide a new logically sound and complete decision procedure for certain kinds of queries.