Sofic random processes

For a finite alphabet Σ, a subshift Y ⊂ Σ ^ℤ and an ergodic shift invariant probability measure with support Y, the future measures of the process (Y;n) are the conditional measures of v on the future, given the past. We introduce sofic processes as the processes that have finitely many future measures. We characterize the weighted Shannon graphs that canonically present sofic processes that are finitarily Markovian. With an example of Furstenberg as a starting point, we characterize the weighted Shannon graphs that canonically present sofic processes that are not finitarily Markovian. The semigroup measures of Kitchens and Tuncel yield sofic processes. We show that the class of sofic processes with semigroup measures is closed under measure preserving topological conjugacy.