State Complexity of Reversals of Deterministic Finite Automata with Output

TL;DR

This paper explores the maximum state complexity of reversals in deterministic finite automata with output, revealing the influence of transition monoids and output mappings, and establishing bounds for different alphabet sizes.

Contribution

It generalizes the known bounds for ordinary DFA reversals to DFAOs, providing new bounds and conjectures based on alphabet size and output values.

Findings

Maximum reversal complexity can reach $k^n$ with a ternary alphabet.

Bound cannot be reached with binary alphabet unless $k=2$.

Reversal complexity depends on transition monoid and output mapping.

Abstract

We investigate the worst-case state complexity of reversals of deterministic finite automata with output (DFAOs). In these automata, each state is assigned some output value, rather than simply being labelled final or non-final. This directly generalizes the well-studied problem of determining the worst-case state complexity of reversals of ordinary deterministic finite automata. If a DFAO has $n$ states and $k$ possible output values, there is a known upper bound of $k^n$ for the state complexity of reversal. We show this bound can be reached with a ternary input alphabet. We conjecture it cannot be reached with a binary input alphabet except when $k = 2$, and give a lower bound for the case $3 \le k < n$. We prove that the state complexity of reversal depends solely on the transition monoid of the DFAO and the mapping that assigns output values to states.