Prediction with Restricted Resources and Finite Automata

We obtain an index of the complexity of a random sequence by allowing the role of the measure in classical probability theory to be played by a function we call the generating mechanism. Typically, this generating mechanism will be a finite automata. We generate a set of biased sequences by applying a finite state automata with a specified number, m, of states to the set of all binary sequences. We detail optimal algorithms to predict sequences generated in this way. We explore a finite setting for the problem of prediction.