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The capacity of finite state channels (FSCs) with feedback has been shown tobe a limit of a sequence of multi-letter expressions. Despite many efforts, aclosed-form single-letter capacity characterization is unknown to date. In thispaper, the feedback capacity is studied from a fundamental algorithmic point ofview by addressing the question of whether or not the capacity can bealgorithmically computed. To this aim, the concept of Turing machines is used,which provides fundamental performance limits of digital computers. It is shownthat the feedback capacity of FSCs is not Banach-Mazur computable and thereforenot Borel-Turing computable. As a consequence, it is shown that eitherachievability or converse is not Banach-Mazur computable, which means thatthere are computable FSCs for which it is impossible to find computable tightupper and lower bounds. Furthermore, it is shown that the feedback capacitycannot be characterized as the maximization of a finite-letter formula ofentropic quantities.

Capacity of Finite State Channels with Feedback: Algorithmic and Optimization Theoretic Properties