On the Structure of the Capacity Formula for General Finite State Channels with Applications

Finite state channels (FSCs) model discrete channels with memory where the channel output depends on the channel input and the actual channel state. The capacity of general FSCs has been established as the limit of a sequence of multi-letter expressions; a corresponding finite-letter characterization is not known to date. In this paper, it is shown that it is indeed not possible to find such a finite-letter entropic characterization for FSCs whose input, output, and state alphabets satisfy |X| $\geq2$, |Y| $\geq2$, and |${S}$| $\geq2$. Further, the algorithmic computability of the capacity of FSCs is studied. To account for this, the concept of a Turing machine is adopted as it provides fundamental performance limits for today’s digital computers. It is shown that the capacity of a FSC is not Banach-Mazur computable and therewith not Turing computable for $|\mathcal {X}| \geq 2, |\mathcal {Y}| \geq 2, |S| \geq 2$.