Memristor offers a promising logic-in-memory (LIM) functionality to achieve the futuristic in-memory computing machine, which may solve the problem of the "von Neumann bottleneck" in the conventional computer architecture. A sequential logic concept is capable of achieving LIM based on the finite-state machine (FSM) using a single bipolar (BRS) or complementary resistive switching (CRS) memristor, where 14 of the 16 two-input Boolean logic functions (XOR and XNOR are missing) are mapped between the resistance state and the terminal voltages. In this paper, a new FSM-LIM concept based on a reconfigurable finite-state machine (RFSM logic) is proposed, which is experimentally confirmed by a diode-integrated single unipolar memristor. The scope of application of the proposed concept is further extended to three other asymmetric resistive switching devices, demonstrating the universality of the RFSM logic. With the use of different voltage conditions to denote logic input "1," the function of FSM is reconfigured between the two state-transition equations. Through the triggering of the two state-transition equations alternatively, all the 16 Boolean logic functions can be achieved in three steps at most, without the assistance of reading operation. These correspond to the logical completeness and nonvolatility.