Classification of Reversible Logic Elements with Memory and Their Universality

Abstract

A reversible logic element with memory (RLEM) is a logical primitive useful for composing reversible machines. It is a kind of reversible sequential machine (RSM), and thus has finite numbers of states and input/output symbols. First, we prove that every non-degenerate 2-state k-symbol RLEM is universal if k > 2. Hence, any RSM can be constructed by such a universal RLEM. On the other hand, it is shown that RLEMs Nos. 2-2, 2-3 and 2-4 among the four non-degenerate 2-state 2-symbol RLEMs are non-universal, and thus we obtain a simple hierarchy among 2-state RLEMs. We then show that there is a compact realization method of RSMs by RLEM 4-31 or RLEM 3-7. Hence, these RLEMs are useful for constructing reversible machines, as well as rotary element (RE), a typical 2-state 4-symbol RLEM. We also give a simple and systematic method of realizing a reversible 4-symbol RLEM with arbitrary number of states in the billiard ball model (BBM), a reversible physical model of computing.