Energy Complexity of Regular Languages

TL;DR

This paper investigates the intrinsic energy costs of recognizing regular languages using finite automata, comparing classical and quantum models, and classifying languages based on their energy requirements.

Contribution

It provides a comprehensive analysis of the energy complexity of regular languages and establishes bounds for classical and quantum finite automata.

Findings

Zero-error quantum automata have no energy advantage over classical automata.

Some languages require more energy for zero-error recognition than for bounded-error recognition.

Regular languages can be classified by their intrinsic energy requirements as input length varies.

Abstract

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 prove that zero-error quantum finite automata have no advantage over their classical deterministic counterparts in terms of the maximum obligatory thermodynamic cost associated by any step during the recognition of different regular languages. We also 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. 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.