On tree sources, finite state machines, and time reversal

We investigate the effect of time reversal on tree models of finite-memory processes. This is motivated in part by the following simple question that arises in some data compression applications: when trying to compress a data string using a universal source modeler, can it make a difference whether we read the string from left to right or from right to left? We characterize the class of finite-memory two-sided tree processes, whose time-reversed versions also admit tree models. Given a tree model, we present a construction of the tree model corresponding to the reverse process, and we show that the number of states in the reverse tree might be, in the extreme case, quadratic in the number of states of the original tree. This answers the above motivating question in the affirmative.