Energy Complexity of Regular Language Recognition
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The erasure of each bit of information 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 use a general quantum finite automaton model to study the thermodynamic cost per step associated with the recognition of different regular languages. We show that zero-error quantum finite automata have no energy cost advantage over their classical deterministic counterparts, and prove an upper bound for the cost that holds for all regular languages. We also demonstrate languages for which "error can be traded for energy", i.e. whose zero-error recognition is associated with provably bigger energy cost per step when compared to their bounded-error recognition by real-time finite-memory quantum devices.
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