Memristive switches are able to act as both storage and computing elements, which make them an excellent candidate for beyond-CMOS computing. In this paper, multi-input memristive switch logic is proposed, which enables the function X OR (Y NOR Z) to be performed in a single-step with three memristive switches. This ORNOR logic gate increases the capabilities of memristive switches, improving the overall system efficiency of a memristive switch-based computing architecture. Additionally, a computing system architecture and clocking scheme are proposed to further utilize memristive switching for computation. The system architecture is based on a design where multiple computational function blocks are interconnected and controlled by a master clock that synchronizes system data processing and transfer. The clocking steps to perform a full adder with the ORNOR gate are presented along with simulation results using a physics-based model. The full adder function block is integrated into the system architecture to realize a 64-bit full adder, which is also demonstrated through simulation.