On Shuffling and Splitting Automata

TL;DR

This paper introduces automata models over the Shuffling Monoid to represent shuffling and splitting words, proving decidability of functionality and equivalence, with efficient algorithms in the deterministic case.

Contribution

It formalizes automata for shuffling and splitting words over the Shuffling Monoid and establishes decidability results for their functionality and equivalence.

Findings

Functionality is decidable for splitters.

Equivalence between functional splitters is decidable.

In the deterministic case, the equivalence algorithm runs in polynomial time.

Abstract

We consider a class of finite state three-tape transducers which models the operation of shuffling and splitting words. We present them as automata over the so-called Shuffling Monoid. These automata can be seen as either shufflers or splitters interchangeably. We prove that functionality is decidable for splitters, and we also show that the equivalence between functional splitters is decidable. Moreover, in the deterministic case, the algorithm for equivalence is polynomial on the number of states of the splitter.