A method for OR parallel execution of Horn clause programs on a shared memory multiprocessor is presented. The shared memory contains only control information that guide processors requesting a job to independently construct the environment required to get a new job from the other processors without degrading the performance. Each processor has a local memory containing its own binding environment. This reduces the traffic to the shared memory and allows each processor to process its job with high performance. Each processor is almost the same as the Warren Abstract Machine (WAM). Modification to the WAM for supporting the method is described in detail. A method for nonshared memory multiprocessor architecture is outlined.