Series which are both max-plus and min-plus rational are unambiguous

TL;DR

This paper proves that unambiguous rational series are equivalent to series that are both max-plus and min-plus rational, providing a unified understanding and an effective method to construct automata recognizing these series.

Contribution

It establishes the equivalence between unambiguous rational series and the intersection of max-plus and min-plus rational series, along with an effective automaton construction method.

Findings

Unambiguous rational series coincide with max-plus and min-plus rational series.

Decidability of equality is unified for these series classes.

An effective procedure for automaton construction is provided.

Abstract

Consider partial maps from the free monoid into the field of real numbers with a rational domain. We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand. The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture. We give an effective procedure to build an unambiguous automaton from a max-plus automaton and a min-plus one that recognize the same series.