A hierarchy of reversible finite automata

TL;DR

This paper establishes a hierarchy of reversible finite automata based on their expressive power, comparing various models and showing how their capabilities differ in recognizing formal languages.

Contribution

It introduces a detailed hierarchy of reversible automata variants, clarifying their relative expressive powers and collapsing properties, with new results on their relationships.

Findings

One-way reversible automata with multiple initial states are more powerful than sweeping reversible automata.

Three passes over input suffice for the hierarchy of sweeping reversible automata to collapse.

In the unary case, several automata classes become equivalent in expressive power.

Abstract

In this paper, different variants of reversible finite automata are compared, and their hierarchy by the expressive power is established. It is shown that one-way reversible automata with multiple initial states (MRFA) recognize strictly more languages than sweeping reversible automata (sRFA), which are in turn stronger than one-way reversible automata with a single initial state (1RFA). The latter recognize strictly more languages than one-way permutation automata (1PerFA). It is also shown that the hierarchy of sRFA by the number of passes over the input string collapses: it turns out that three passes are always enough. On the other hand, MRFA form a hierarchy by the number of initial states: their subclass with at most $k$ initial states (MRFA$^k$) recognize strictly fewer languages than MRFA$^{k + 1}$, and also MRFA$^k$ are incomparable with sRFA. In the unary case, sRFA, MRFA$^k$ and MRFA become equal in their expressive power, and the inclusion of 1RFA into sRFA remains proper.