Parametrized complexity of relations between multidimensional subshifts

2025-09-29

AI Summary

This paper investigates the parameterized complexity of fundamental relations—equivalence, conjugacy, inclusion, and embedding—between multidimensional subshifts, focusing on shifts of finite type (SFTs) and effective subshifts. Adopting the Jeandel–Vanier two-input framework, it fixes one subshift as a parameter and analyzes the decidability and computational hardness of the relation when the other subshift is given as input. The work provides the first systematic characterization of how dynamical properties—periodicity, minimality, and finite typeness—affect parameterized complexity, exposing inherent asymmetries and conjugacy instability. It identifies critical parameter thresholds at which each relation attains maximal arithmetic or Turing degree hardness. Furthermore, it establishes a novel connection between minimality and computable languages, uncovering several nontrivial decidable subclasses. Collectively, these results advance the understanding of structural and computational boundaries in symbolic dynamics.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

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

Abstract

We study the parametrized complexity of fundamental relations between multidimensional subshifts, such as equality, conjugacy, inclusion, and embedding, for subshifts of finite type (SFTs) and effective subshifts. We build on previous work of E. Jeandel and P. Vanier on the complexity of these relations as two-input problems, by fixing one subshift as parameter and taking the other subshift as input. We study the impact of various dynamical properties related to periodicity, minimality, finite type, etc. on the computational properties of the parameter subshift, which reveals interesting differences and asymmetries. Among other notable results, we find choices of parameter that reach the maximum difficulty for each problem; we find nontrivial decidable problems for multidimensional SFT, where most properties are undecidable; and we find connections with recent work relating having computable language and being minimal for some property, showing in particular that this property may not always be chosen conjugacy-invariant.

Problem

• Analyzing parametrized complexity of multidimensional subshift relations
• Investigating computational properties of subshifts with fixed parameters
• Exploring decidability and dynamical properties in multidimensional SFTs

Innovation

• Fixed one subshift as parameter for complexity analysis
• Studied dynamical properties impact on computational complexity
• Found nontrivial decidable problems for multidimensional SFTs