Pumping-Like Results for Copyless Cost Register Automata and Polynomially Ambiguous Weighted Automata

TL;DR

This paper introduces Pumping Sequence Families to analyze the expressiveness of polynomially ambiguous weighted automata and copyless cost register automata over various fields, revealing their differences and computational complexities.

Contribution

It develops a novel pumping technique applicable to these automata classes over arbitrary fields, and demonstrates their expressiveness divergence over certain fields and alphabet sizes.

Findings

The two classes coincide over rationals with 1-letter alphabets.

Pumping Sequence Families enable analysis over unrestricted alphabets.

Zeroness and equivalence are polynomial-time for weighted automata but PSPACE-complete for copyless automata.

Abstract

In this work we consider two rich subclasses of weighted automata over fields: polynomially ambiguous weighted automata and copyless cost register automata. Primarily we are interested in understanding their expressiveness power. Over the field of rationals and $1$-letter alphabets, it is known that the two classes coincide; they are equivalent to linear recurrence sequences (LRS) whose exponential bases are roots of rationals. We develop a tool we call Pumping Sequence Families, which, by exploiting the simple single-letter behaviour of the models, yields two pumping-like results over arbitrary fields with unrestricted alphabets, one for each class. As a corollary of these results, we present examples proving that the two classes become incomparable over the field of rationals with unrestricted alphabets. We complement the results by analysing the zeroness and equivalence problems. For weighted automata (even unrestricted) these problems are well understood: there are polynomial time, and even NC$^2$ algorithms. For copyless cost register automata we show that the two problems are \textsc{PSpace}-complete, where the difficulty is to show the lower bound.