We introduce the concept of Logically Reconfigurable Physical Unclonable Functions (LR-PUFs). In contrast to classical Physically Unclonable Functions (PUFs) LR-PUFs can be dynamically 'reconfigured' after deployment such that their challenge/response behavior changes in a random manner. To this end, we amend a conventional PUF with a stateful control logic that transforms challenges and responses of the PUF. We present and evaluate two different constructions for LR-PUFs that are simple, efficient and can easily be implemented. Moreover, we introduce a formal security model for LR-PUFs and prove that both constructions are secure under reasonable assumptions. Finally, we demonstrate that LR-PUFs enable the construction of securely recyclable access tokens, such as electronic tickets: LR-PUFs enhance security against manipulation and forgery, while reconfiguration allows secure re-use of tokens for subsequent transactions.