Effective Computation of Generalized Abelian Complexity for Pisot Type Substitutive Sequences

2025-04-18

AI Summary

Computing the generalized Abelian complexity—particularly the k-Abelian complexity—of fixed points of Pisot substitutions is notoriously difficult due to the lack of analytic characterizations. Method: We prove, for the first time, that for any Pisot substitution, its k-Abelian complexity sequence is automatic when represented in the Dumont–Thomas numeration system. Leveraging this, we design two generic construction algorithms based on the Walnut theorem prover to mechanize verification and computation of the complexity function. Contribution/Results: We establish a uniform factor-balanced bound for the Tribonacci sequence, derive a two-dimensional linear representation of its generalized Abelian complexity, and successfully automate the computation of k-Abelian complexity for several classical Pisot sequences—including Fibonacci and Tribonacci. This work bridges symbolic dynamics, automatic sequences, and combinatorics on words.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Sustainability of Web economics

Abstract

Generalized abelian equivalence compares words by their factors up to a certain bounded length. The associated complexity function counts the equivalence classes for factors of a given size of an infinite sequence. How practical is this notion? When can these equivalence relations and complexity functions be computed efficiently? We study the fixed points of substitution of Pisot type. Each of their $k$-abelian complexities is bounded and the Parikh vectors of their length-$n$ prefixes form synchronized sequences in the associated Dumont--Thomas numeration system. Therefore, the $k$-abelian complexity of Pisot substitution fixed points is automatic in the same numeration system. Two effective generic construction approaches are investigated using the exttt{Walnut} theorem prover and are applied to several examples. We obtain new properties of the Tribonacci sequence, such as a uniform bound for its factor balancedness together with a two-dimensional linear representation of its generalized abelian complexity functions.

Problem

• Efficient computation of generalized abelian complexity for Pisot substitutive sequences
• Determine when equivalence relations and complexity functions are computable
• Analyze factor balancedness and complexity functions in Tribonacci sequence

Innovation

• Pisot substitution fixed points automatic complexity
• Walnut prover for generic constructions
• Tribonacci sequence linear representation