Linear Recurrence Sequence Automata and the Addition of Abstract Numeration Systems

TL;DR

This paper introduces sequence automata for abstract numeration systems, enabling the computation of addition and conversion relations, with theoretical regularity results and practical implementation for Walnut.

Contribution

It defines sequence automata for abstract numeration systems and demonstrates their ability to compute addition and conversion relations under Pisot conditions.

Findings

Support of series is regular under Pisot conditions

Automata for addition relations are constructed

Implementation provided for Walnut

Abstract

Abstract numeration systems encode natural numbers using radix ordered words of an infinite regular language and linear recurrence sequences play a key role in their valuation. Sequence automata, which are deterministic finite automata with an additional linear recurrence sequence on each transition, are introduced to compute various $\mathbb{Z}$-rational non commutative formal series in abstract numeration systems. Under certain Pisot conditions on the recurrence sequences, the support of these series is regular. This property can be leveraged to derive various synchronized relations including a deterministic finite automaton that computes the addition relation of various Dumont-Thomas numeration systems and deterministic finite automata converting between various numeration systems. A practical implementation for Walnut is provided.