# Orbit-Finite-Dimensional Vector Spaces and Weighted Register Automata

6 Apr 2021

Mikołaj Bojańczyk, Bartek Klin and Joshua Moerman

LICS 2021

Distinguished paper at LICS 2021.

## Abstract

We develop a theory of vector spaces spanned by orbit-finite sets. Using this theory, we give a decision procedure for equivalence of weighted register automata, which are the common generalization of weighted automata and register automata for infinite alphabets. The algorithm runs in exponential time, and in polynomial time for a fixed number of registers. As a special case, we can decide, with the same complexity, language equivalence for unambiguous register automata, which improves previous results in three ways: (a) we allow for order comparisons on atoms, and not just equality; (b) the complexity is exponentially better; and (c) we allow automata with guessing.