On shuffle products, acyclic automata and piecewise-testable languages

TL;DR

This paper proves that shuffling a piecewise-testable language with a finite language results in a language that remains piecewise-testable, using automata theory, and discusses related generalizations and complexity bounds.

Contribution

It introduces a novel automata-theoretic proof showing the closure of piecewise-testable languages under shuffle with finite languages.

Findings

Shuffle of piecewise-testable and finite languages is piecewise-testable

Provides bounds on piecewise complexity after shuffle

Discusses generalizations of the main result

Abstract

We show that the shuffle $L \unicode{x29E2} F$ of a piecewise-testable language $L$ and a finite language $F$ is piecewise-testable. The proof relies on a classic but little-used automata-theoretic characterization of piecewise-testable languages. We also discuss some mild generalizations of the main result, and provide bounds on the piecewise complexity of $L \unicode{x29E2} F$.