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

TL;DR

This work investigates the expressive relationship between two sophisticated weight-based automata models on fields: polynomially ambiguous weighted automata and copyless cost register automata. By introducing Pumping Sequence Families (PSF), it derives pumping-like constraints that apply across unrestricted alphabets and uses them to prove incomparability of the two classes over $\mathbb{Q}$, complementing known 1-letter alphabet results tied to linear recurrence sequences. The paper further shows that, for PA WA, PSF imposes a finitely generated structure on exponential bases, while for CCRA the zeroness and equivalence problems are $\mathrm{PSPACE}$-complete over $\mathbb{Q}$, in contrast to the polynomial-time/NC$^2$ results for WA. Together, these findings clarify the boundary between these two rich formalisms and illuminate the computational complexity of fundamental decision problems, using EPS-based tools to connect algebraic and automata-theoretic properties.

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.

Figures (7)

• Figure 1: Weighted automata over the field of rationals $\mathbb{Q}(+,\cdot)$. For clarity, we omit zero labels.
• Figure 2: Two simple single-state CCRAs on a single-letter alphabet (\ref{['example:basic_ccra']}).
• Figure 3: A polynomially ambiguous weighted automaton with no equivalent CCRA (\ref{['exm:wfa-not-ccra']}).
• Figure 4: A simple CCRA and its variable flow graph. Red nodes are constant registers; blue nodes are updating ones (\ref{['definition:simple']}).
• Figure 8: A CCRA recognising a function that, at first glance, may seem unrecognisable by a CCRA (\ref{['exm:looks-unrecognisable-but-is-recognisable']}).

Theorems & Definitions (58)

• Definition 1
• Example 2
• Example 3
• Theorem 4
• Example 5
• Example 6
• Example 7
• Example 8
• Example 9
• Example 10