Myhill-Nerode Relation for Sequentiable Structures

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

Space-Efficient Bimachine Construction Based on the Equalizer Accumulation Principle

Algorithms for building bimachines from functional transducers found in the literature in a run of the bimachine imitate one successful path of the input transducer. Each single bimachine output exactly corresponds to the output of a single transducer transition. Here we introduce an alternative construction principle where bimachine steps take alternative parallel transducer paths into account, maximizing the possible output at each step using a joint view. The size of both the deterministic left and right automaton of the bimachine is restricted by $2^{\vert Q\vert}$ where $\vert Q\vert$ is the number of transducer states. Other bimachine constructions lead to larger subautomata. As a concrete example we present a class of real-time functional transducers with $n+2$ states for which the standard bimachine construction generates a bimachine with at least $\Theta(n!)$ states whereas the construction based on the equalizer accumulation principle leads to $2^n + n +3$ states. Our construction can be applied to rational functions from free monoids to "mge monoids", a large class of monoids including free monoids, groups, and others that is closed under Cartesian products

Characterisation of (Sub)sequential Rational Functions over a General Class Monoids

In this technical report we describe a general class of monoids for which (sub)sequential rational can be characterised in terms of a congruence relation in the flavour of Myhill-Nerode relation. The class of monoids that we consider can be described in terms of natural algebraic axioms, contains the free monoids, groups, the tropical monoid, and is closed under Cartesian