Minimality and computability of languages of G-shifts

2025-06-12

AI Summary

This paper investigates strong computability of $G$-shifts over a finitely generated group $G$ with decidable word problem—i.e., whether the language of the shift is co-computable. The central problem is to characterize, both necessary and sufficient, when this property holds, and to establish a unified minimality criterion based on bounded complexity. Methodologically, the work integrates computability theory, symbolic dynamics, group actions, and formal language analysis. Key contributions include: (i) the first systematic definition and study of strong computability types for $G$-shifts; (ii) a proof that strong computability is preserved under direct products of shifts—a property absent in classical set-theoretic contexts; (iii) an equivalent characterization of minimality in terms of local complexity; (iv) unification and generalization of prior results on various classes of minimal shifts; and (v) a decidability pathway for novel classes of shifts via strong computability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Humans and AI: Other Foundations of Human Computation & AI
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Motivated by the notion of strong computable type for sets in computable analysis, we define the notion of strong computable type for $G$-shifts, where $G$ is a finitely generated group with decidable word problem. A $G$-shift has strong computable type if one can compute its language from the complement of its language. We obtain a characterization of $G$-shifts with strong computable type in terms of a notion of minimality with respect to properties with a bounded computational complexity. We provide a self-contained direct proof, and also explain how this characterization can be obtained from an existing similar characterization for sets by Amir and Hoyrup, and discuss its connexions with results by Jeandel on closure spaces. We apply this characterization to several classes of shifts that are minimal with respect to specific properties. This provides a unifying approach that not only generalizes many existing results but also has the potential to yield new findings effortlessly. In contrast to the case of sets, we prove that strong computable type for G-shifts is preserved under products. We conclude by discussing some generalizations and future directions.

Problem

• Characterize G-shifts with strong computable type
• Link minimality and computational complexity for G-shifts
• Generalize existing results on shifts via unifying approach

Innovation

• Defines strong computable type for G-shifts
• Characterizes shifts via minimality and complexity
• Proves strong computable type preserved under products