On Shuffling and Splitting Automata

TL;DR

The paper introduces the Shuffling Monoid $U$ and automata over it (spliff ers) to model shuffling and splitting of words, unifying two operational perspectives. It develops a decision framework for the equivalence of functional splitters by adapting squaring techniques and employing valuations via the Lead or Delay action and its bidimensional variant $\Delta$. The main contributions are the decidability of equivalence for functional splitters and a polynomial-time algorithm for equivalence of deterministic splitters, along with a detailed analysis of the closure properties of $\text{Rat}(U)$ and $\text{DRat}(U)$. This work extends prior results on transducers and provides a rigorous method to compare splitting behaviors in a monoid-theoretic setting, with potential implications for independence notions and related automata-theoretic models.

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.

Figures (5)

• Figure 1: A simple spliffer accepting the set $\{(aa,b,aba),(a,ab,aba)\}$
• Figure 2: Two different deterministic Spliffers accepting the same set
• Figure 3:
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Theorems & Definitions (32)

• Definition : The Shuffling Monoid
• Definition : Spliffer
• Definition : Deterministic splitter
• Definition
• Lemma 1.1
• Corollary 1.1
• Corollary 1.2
• Definition
• Proposition 1.1
• proof