# Solving of Regular Equations Revisited (extended version)

## TL;DR

This paper revisits the solving of regular equations using Arden's Lemma, providing a detailed algorithm, a computational interpretation, and applications to convert automata to unambiguous expressions and express operations algebraically.

## Contribution

It offers a concise algorithmic specification, a novel interpretation of solving via coercions, and new algebraic methods for automata operations without conversion.

## Key findings

- Identifies conditions for unambiguous solutions of regular equations.
- Provides a method to convert DFA to unambiguous regular expressions.
- Expresses subtraction and shuffling operations algebraically via regular equations.

## Abstract

Solving of regular equations via Arden's Lemma is folklore knowledge. We first give a concise algorithmic specification of all elementary solving steps. We then discuss a computational interpretation of solving in terms of coercions that transform parse trees of regular equations into parse trees of solutions. Thus, we can identify some conditions on the shape of regular equations under which resulting solutions are unambiguous. We apply our result to convert a DFA to an unambiguous regular expression. In addition, we show that operations such as subtraction and shuffling can be expressed via some appropriate set of regular equations. Thus, we obtain direct (algebraic) methods without having to convert to and from finite automaton.