Weak Cost Register Automata are Still Powerful

TL;DR

This paper demonstrates that even very weak variants of cost register automata over the tropical semiring can simulate counter machines with zero-tests, leading to undecidability results and revealing their surprising computational power.

Contribution

It proves that copyless cost register automata over natural numbers with min and increment updates can simulate counter machines with zero-tests, contradicting prior conjectures.

Findings

Undecidability of equivalence for the model

Simulation of counter machines with zero-tests

Representation as restricted linearly-ambiguous weighted automata

Abstract

We consider one of the weakest variants of cost register automata over a tropical semiring, namely copyless cost register automata over $\mathbb{N}$ with updates using $\min$ and increments. We show that this model can simulate, in some sense, the runs of counter machines with zero-tests. We deduce that a number of problems pertaining to that model are undecidable, in particular equivalence, disproving a conjecture of Alur et al. from 2012. To emphasize how weak these machines are, we also show that they can be expressed as a restricted form of linearly-ambiguous weighted automata.