An Analysis of Errors in Finite Automata

This problem arose f rom an a t t empt to make a general s tudy of reliability iii :cbinputerlike machines. The classic results 0f:von Neumann (yon Neumgnn; 19o6) deal only with networks which do not have any feedback. Thus a malfunction only causes the network to be in the incorrect State for a fixed length of t i m e . However ; a malfunction in the general case with feedback can cause an error :which persists forever. Fortunately, not all errors are of this type. Some errors are of the type tha t can persist only for a bounded time. Some; al though they can persist infinitely long, have a probabil i ty of being corrected which approaches one as the tapes get longer: Thus "almost ~I1;' Of the " long" tapes correct the error. This is t h e phenomenon which will be studied in this paper. I t will be shown tha t errors of the lat ter type induce a parti t ion on the set of states. The' possibility of adding states to the machine in order to get a more reliable one will also be discussed. In the next section we will formalize the problem in terms of the theory of automata .