This paper proposes a synthesis method to obtain speed-independent asynchronous circuits directly from signal transition graph (STG) specifications with single cycle signals which can be non-persistent and have free-choice operations. The resulting circuits are implemented with basic gates and asynchronous latches, and operate correctly under finite but unbounded gate delays and the zero wire delay assumptions. The proposed method introduces 5 types of lock relations to implement a non-persistent STG. A non-persistent STG can be implemented if every non-persistent signal to a signal t is super-locked with t. The resulting circuits are optimized by extracting of literals, mapping onto asymmetric C-elements, etc. Experimental results show that the proposed synthesis method outperforms the existing synthesis systems such as SYN and SIS.