Nondegenerate 2-State 3-Symbol Reversible Logic Elements Are All Universal

In the investigation of minimal machinery in reversible computing, we proved that each nondegenerate 2-state 3-symbol reversible logic element is logically universal. So far, a 2-state 4-symbol element called “rotary element” was shown to be logically universal in the framework of reversible logic element with memory. The main result in this paper not only improves the previous result with respect to the number of symbols, but also shows all the 2-state 3-symbol reversible logic elements except degenerate ones are logically universal. It is known that there are 24 essentially different 2-state 3-symbol reversible logic elements. Among them, 10 are degenerate ones, which are equivalent to 2-state 2-symbol ones or simple connecting wires. The other 14 are nondegenerate ones, and thus they are “proper” 2-state 3-symbol reversible logic elements. For each of the 14 elements we construct a circuit composed only of it that simulates a Fredkin gate, a logically universal gate.