Linear Recurrence Sequence Automata and the Addition of Abstract Numeration Systems

2024-06-14

AI Summary

This paper addresses the automata-theoretic characterization of ℤ-rational noncommutative formal series in abstract numeration systems. We introduce the *sequence automaton*, a novel model that embeds linear recurrent sequences directly into the transition structure of deterministic finite automata (DFAs) — the first such construction. Under the Pisot condition, we prove that the support of these series is a regular language, thereby establishing a foundational arithmetic framework for abstract numeration systems. Building on this, we construct a deterministic automaton for addition in Dumont–Thomas numeration systems and design a regular transducer for base conversion between distinct abstract numeration systems. All results are formally implemented and verified using the Walnut toolchain; the executable code is open-sourced to support automated reasoning and verification.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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 $Z$-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.

Problem

• Develop sequence automata for abstract numeration systems.
• Compute non-commutating formal series using sequence automata.
• Implement addition relation in Dumont-Thomas numeration systems.

Innovation

• Sequence automata with linear recurrence transitions
• Pisot conditions ensure series support regularity
• Automata for addition and numeration system conversion