[Paper Review] On injective cellular automata over schemes

Tullio Ceccherini‐Silberstein, Michel Coornaert+1 more|arXiv (Cornell University)|Dec 15, 2017

Geometric and Algebraic Topology8 references8 citations

TL;DR

This paper introduces cellular automata over schemes, generalizing algebraic cellular automata over algebraic varieties defined over algebraically closed fields. It establishes key results on surjunctivity, reversibility, and invertibility, proving that injective cellular automata over such schemes are necessarily bijective, extending classical results to a broader geometric framework.

ABSTRACT

We introduce a notion of cellular automata over schemes which generalize algebraic cellular automata. We establish several results related to surjunctivity, reversibility, and invertibility for cellular automata over algebraic varieties with coefficients in an algebraically closed field.

Motivation & Objective

• To extend the theory of cellular automata from finite configurations to algebraic geometry by defining cellular automata over schemes.
• To generalize existing results on surjunctivity and invertibility from algebraic cellular automata to a broader class of schemes.
• To investigate the structural properties of injective cellular automata in the context of algebraic varieties over algebraically closed fields.
• To establish conditions under which injectivity implies reversibility and invertibility in this geometric setting.

Proposed method

• The authors define cellular automata over schemes by generalizing the local rule concept to morphisms of schemes, using the structure sheaf and étale neighborhoods.
• They employ tools from algebraic geometry, particularly the theory of algebraic varieties and morphisms over algebraically closed fields.
• The framework incorporates the notion of finite type morphisms and local finitely generated algebras to ensure computability and finiteness conditions.
• They apply the Garden of Eden theorem and its generalizations, adapting them to the scheme-theoretic setting.
• The proof strategy relies on the use of the Ax-Grothendieck theorem and properties of endomorphisms on algebraic varieties.
• Injectivity is analyzed via the behavior of the global transition map on the space of configurations, interpreted as sections of sheaves over schemes.

Experimental results

Research questions

• RQ1 Under what conditions is an injective cellular automaton over a scheme necessarily surjective or invertible?
• RQ2 How do classical results on cellular automata over finite configurations extend to algebraic varieties over algebraically closed fields?
• RQ3 What role does the scheme-theoretic structure play in determining the reversibility of cellular automata?
• RQ4 Can the surjunctivity property be generalized from finite-state systems to algebraic geometric settings?
• RQ5 How do morphisms of schemes influence the dynamics of cellular automata defined on them?

Key findings

• Injective cellular automata over algebraic varieties defined over algebraically closed fields are necessarily bijective, extending the classical surjunctivity result to this geometric setting.
• The reversibility of a cellular automaton over such schemes is equivalent to its injectivity, generalizing the finite-state case.
• The global transition map of an injective cellular automaton is an isomorphism of schemes, indicating a strong structural constraint.
• The results hold under the assumption that the underlying scheme is of finite type over an algebraically closed field, ensuring finiteness and geometric control.
• The framework allows for a uniform treatment of cellular automata dynamics across different algebraic and geometric contexts.