Myhill-Nerode Relation for Sequentiable Structures

TL;DR

This paper extends the Myhill-Nerode relation to functions over sequentiable structures, showing that finite index functions can be represented with subsequential transducers with a number of states equal to their index.

Contribution

It introduces a novel generalization of the Myhill-Nerode relation for sequentiable structures and proves a state-representation theorem for finite index functions.

Findings

Finite index functions can be represented with subsequential transducers.

The Myhill-Nerode relation is generalized to sequentiable structures.

Representation size equals the function's Myhill-Nerode index.

Abstract

Sequentiable structures are a subclass of monoids that generalise the free monoids and the monoid of non-negative real numbers with addition. In this paper we consider functions $f:\Sigma^*\rightarrow {\cal M}$ and define the Myhill-Nerode relation for these functions. We prove that a function of finite index, $n$, can be represented with a subsequential transducer with $n$ states.