The Assumption-Based Truth Maintenance System (ATMS) is an important tool in A I . So far its wider use has been limited due to the enormous computational resources which it requires. We investigate the possibility of speeding it up by using a modest number of processors in parallel. We begin with a highly efficient sequential version written in C and then extend this version to allow parallel execution on the Encore Multimax, a 16 node shared-memory multiprocessor. We describe our experiences in implementing this shared-memory parallel version of the ATMS, present detailed results of its execution, and discuss the factors which limit the available speedup.