Maximum entropy rate of Markov sources for systems with non-regular constraints

Using the concept of discrete noiseless channels, it was shown by Shannon in A Mathematical Theory of Communication that the ultimate performance of an encoder for a constrained system is limited by the combinatorial capacity of the system if the constraints define a regular language. In the present work, it is shown that this is not an inherent property of regularity but holds in general. To show this, constrained systems are described by generating functions and random walks on trees.