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 $\ZZ$-rational non commutating 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 regular finite automata converting between various numeration systems. A practical implementation for Walnut is provided.

翻译：抽象计数系统利用无限正则语言的基数有序词对自然数进行编码，线性递推序列在其赋值中起着关键作用。本文引入序列自动机——一种在每条转移上附加线性递推序列的确定性有限自动机——用于计算抽象计数系统中各类$\ZZ$-有理非交换形式级数。当递推序列满足特定Pisot条件时，这些级数的支撑集是正则的。该性质可用于推导多种同步关系，包括计算各类Dumont-Thomas计数系统加法关系的确定性有限自动机，以及在不同计数系统间进行转换的正则有限自动机。文中提供了适用于Walnut系统的实际实现方案。