Energy complexity of regular languages

Each step that results in a bit of information being "forgotten" by a computing device has an intrinsic energy cost. Although any Turing machine can be rewritten to be thermodynamically reversible without changing the recognized language, finite automata that are restricted to scan their input once in "real-time" fashion can only recognize the members of a proper subset of the class of regular languages in this reversible manner. We study the energy expenditure associated with the computations of deterministic and quantum finite automata. We show that regular languages can be classified according to the intrinsic energy requirements on the recognizing automaton as a function of input length, and prove upper and lower bounds. We also provide a classification in terms of the maximum obligatory thermodynamic cost associated by any step during the recognition of different regular languages. We prove that zero-error quantum finite automata have no advantage over their classical deterministic counterparts in terms of this "peak load" criterion. We demonstrate languages for which "error can be traded for energy", i.e. whose zero-error recognition is associated with computation steps having provably bigger obligatory energy cost when compared to their bounded-error recognition by real-time finite memory quantum devices.