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A hierarchy of reversible finite automata

Maria Radionova, Alexander Okhotin

TL;DR

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).

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.

A hierarchy of reversible finite automata

TL;DR

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).

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 initial states (MRFA) recognize strictly fewer languages than MRFA, and also MRFA are incomparable with sRFA. In the unary case, sRFA, MRFA and MRFA become equal in their expressive power, and the inclusion of 1RFA into sRFA remains proper.

Paper Structure

This paper contains 5 sections, 9 theorems, 19 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Main hierarchy
  4. Is there a hierarchy of sRFA by the number of passes over an input string?
  5. Hierarchy of MRFA with a bounded number of initial states

Key Result

Lemma 1

For every sRFA with acceptance at both sides, with a set of states $P_+ \cup P_-$, and with accepting states $E \subseteq P_+ \cup P_-$, there exists an ordinary sRFA (with acceptance at one side), which has the set of states $Q_+ \cup Q_-$, where $Q_+ = P_+ \cup \{ \, p' \mid p \in E \cap P_- \, \}

Figures (5)

  • Figure 1: The hierarchy of reversible and permutation automata.
  • Figure 2: Hierarchy in the unary case.
  • Figure 3: Computations of an sRFA recognizing the language $(aa)^* \cup \{ a \}$.
  • Figure 4: Constructing the function $f$ from the function $g$.
  • Figure 5: Cycles in sRFA

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 1
  • proof : A sketch of a proof.
  • Definition 6
  • Definition 7: Pin Pin1987
  • Lemma 2: Pin Pin1987
  • ...and 25 more