Purpose-based access control models are widely applied in both governments and industry. According to users' attributes, users are required to dynamically act as various roles. Access permissions with roles are assigned to users that shows what can do for the purposes with certain conditions. Conditional role involved purpose-based access control models (CPAC) are designed for users to dynamically act with conditional roles. For efficiency of an access method, it is a challenging problem to find an optimal set of roles from an existing access control model. This paper devises a set of roles in implementing a CPAC model that can deal with the problem. We define a conditional role mining problem (CRMP) model that includes conditional roles and purposes. Algorithms are developed to discover an optimal set of roles from existing permissions in the access control model. Furthermore, based on practical situations, two different variations from CRMP are introduced. One is named δ-approx CRMP while the other is Minimal Noise CRMP. The two variations are not analysed in this paper but in future work.