Finite Sequentiality of Finitely Ambiguous Max-Plus Tree Automata

Erik Paul

doi:10.1007/s00224-021-10049-6

Abstract

AbstractWe show that the finite sequentiality problem is decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton.

Highlights

A max-plus automaton is a finite automaton whose transitions are weighted by real numbers
In the form of min-plus automata, they were originally introduced by Imre Simon as a means to show the decidability of the finite power property [6, 7] and they enjoy a continuing interest [8,9,10,11,12,13,14]
Theory of Computing Systems to determine the star height of a language [15], to prove the termination of certain string rewriting systems [16], and to model discrete event systems [17]. They appear in the context of natural language processing [18], where probabilities are often computed in the min-plus semiring as negative log-likelihoods for reasons of numerical stability

Summary

Introduction

A max-plus automaton is a finite automaton whose transitions are weighted by real numbers. We provide an honest attempt to compare our approach to his but note that our interpretation might not be accurate In his proof for max-plus word automata, Bala first introduces the A-Fork property and proceeds to show that this property is a decidable criterion characterizing the finite sequentiality of a finitely ambiguous max-plus automaton. To show that an automaton is not finitely sequential if the AFork property is satisfied, he uses pumping techniques similar to those employed in [25] for the finite sequentiality problem of unambiguous max-plus word automata This part of his proof most likely employs Ramsey’s Theorem [33] as it involves “colorings of finite hypercubes’. We provide full proofs together with detailed illustrations for these

Preliminaries

The Criterion for Finite Sequentiality

Decidability

H the set of all subsets of

Sufficiency