# Reordering Derivatives of Trace Closures of Regular Languages (Full Version)

## TL;DR

This paper introduces syntactic derivative operations for trace closures of regular languages, extending automata constructions to non-regular languages, and identifies conditions under which these automata are finite.

## Contribution

It defines new derivative operations for trace closures, analyzes their automata, and introduces the concept of uniform scattering rank to determine finiteness conditions.

## Key findings

- Antimirov and Brzozowski automata can be finite for certain trace closures.
- Finite automata can be obtained by quotienting for star-connected expressions.
- The refined Antimirov automaton truncation remains finite and accepts trace closures.

## Abstract

We provide syntactic derivative-like operations, defined by recursion on regular expressions, in the styles of both Brzozowski and Antimirov, for trace closures of regular languages. Just as the Brzozowski and Antimirov derivative operations for regular languages, these syntactic reordering derivative operations yield deterministic and nondeterministic automata respectively. But trace closures of regular languages are in general not regular, hence these automata cannot generally be finite. Still, as we show, for star-connected expressions, the Antimirov and Brzozowski automata, suitably quotiented, are finite. We also define a refined version of the Antimirov reordering derivative operation where parts-of-derivatives (states of the automaton) are nonempty lists of regular expressions rather than single regular expressions. We define the uniform scattering rank of a language and show that, for a regexp whose language has finite uniform scattering rank, the truncation of the (generally infinite) refined Antimirov automaton, obtained by removing long states, is finite without any quotienting, but still accepts the trace closure. We also show that star-connected languages have finite uniform scattering rank.