[Paper Review] Symbolic dynamics

Marie-Pierre Béal, Jean Berstel+2 more|arXiv (Cornell University)|Jun 7, 2010

Cellular Automata and Applications17 references4 citations

TL;DR

This paper establishes connections between automata theory and symbolic dynamics, focusing on local automata's role in embedding shifts of finite type and syntactic semigroups' classification of sofic shifts up to conjugacy. It demonstrates how automata-theoretic structures provide tools for analyzing dynamical systems in symbolic dynamics.

ABSTRACT

This chapter presents some of the links between automata theory and symbolic dynamics. The emphasis is on two particular points. The first one is the interplay between some particular classes of automata, such as local automata and results on embeddings of shifts of finite type. The second one is the connection between syntactic semigroups and the classification of sofic shifts up to conjugacy.

Motivation & Objective

• To explore the interplay between specific automata classes, such as local automata, and shifts of finite type.
• To investigate how syntactic semigroups can classify sofic shifts under conjugacy.
• To establish theoretical connections between automata-theoretic constructs and symbolic dynamics structures.
• To provide a framework for embedding shifts of finite type using automata constructions.

Proposed method

• Utilizes local automata to model and analyze shifts of finite type.
• Applies syntactic semigroup theory to classify sofic shifts up to conjugacy.
• Employs automata-theoretic constructions to study embedding properties of shift spaces.
• Analyzes structural properties of automata to derive dynamical system invariants.
• Relies on formal language and semigroup theory to connect automata and symbolic dynamics.
• Uses conjugacy relations to compare and classify sofic shifts via algebraic invariants.

Experimental results

Research questions

• RQ1 How can local automata be used to embed shifts of finite type?
• RQ2 What is the role of syntactic semigroups in classifying sofic shifts up to conjugacy?
• RQ3 In what ways do automata-theoretic structures reflect properties of symbolic dynamical systems?
• RQ4 How do automata constructions facilitate the analysis of shift spaces?
• RQ5 What algebraic invariants emerge from the interaction between automata and symbolic dynamics?

Key findings

• Local automata provide a structural framework for embedding shifts of finite type.
• Syntactic semigroups serve as invariants that classify sofic shifts up to conjugacy.
• The connection between automata and symbolic dynamics enables deeper analysis of shift space properties.
• Automata-theoretic methods reveal algebraic invariants in dynamical systems.
• Embedding results for shifts of finite type are facilitated by specific automata constructions.
• Conjugacy classification of sofic shifts is achievable through syntactic semigroup analysis.