FAPKC [17, 18, 19, 20, 22] is a public key cryptosystem based weakly invertible nite automata. Weak invertibility of FAs is the key to understand and analyze this scheme. In this paper a set of algebraic terminologies describing FAs is developed, and the theory of weak invertibility of FAs is studied. Based on this, a cryptanalysis on FAPKC is made. It is shown that the keys proposed in [17, 18, 19, 20, 21] for FAPKCs are insecure both in encrypting and in signing.