Cellular automata are used as simple mathematical models to investigate self-organization in statistical mechanics. A detailed analysis is given of "elementary" cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to definite rules involving the values of its nearest neighbors. With simple initial configurations, the cellular automata either tend to homogeneous states, or generate self-similar patterns with fractal dimensions \ensuremath{\simeq} 1.59 or \ensuremath{\simeq} 1.69. With "random" initial configurations, the irreversible character of the cellular automaton evolution leads to several self-organization phenomena. Statistical properties of the structures generated are found to lie in two universality classes, independent of the details of the initial state or the cellular automaton rules. More complicated cellular automata are briefly considered, and connections with dynamical systems theory and the formal theory of computation are discussed.

Statistical mechanics of cellular automata

https://www.ibisc.univ-evry.fr/~hutzler/Cours/IMBI_MPS/Kari05.pdf

Theory of cellular automata: a survey

Self-organizing behaviour in cellular automata is discussed as a computational process. Formal language theory is used to extend dynamical systems theory descriptions of cellular automata. The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages. Many examples are given. The sizes of the minimal grammars for these languages provide measures of the complexities of the sets. This complexity is usually found to be non-decreasing with time. The limit sets generated by some classes of cellular automata correspond to regular languages. For other classes of cellular automata they appear to correspond to more complicated languages. Many properties of these sets are then formally non-computable. It is suggested that such undecidability is common in these and other dynamical systems.

/pdf/computation-theory-of-cellular-automata-4snltftqhc.pdf

Computation theory of cellular automata

1: Introduction.- 1.1. Systems Science.- 1.2. Systems Problem Solving.- 1.3. Hierarchy of Epistemological Levels of Systems.- 1.4. The Role of Mathematics.- 1.5. The Role of Computer Technology.- 1.6. Architecture of Systems Problem Solving.- Notes.- 2: Source and Data Systems.- 2.1. Objects and Object Systems.- 2.2. Variables and Supports.- 2.3. Methodological Distinctions.- 2.4. Discrete Versus Continuous.- 2.5. Image Systems and Source Systems.- 2.6. Data Systems.- Notes.- Exercises.- 3: Generative Systems.- 3.1. Empirical Investigation.- 3.2. Behavior Systems.- 3.3. Methodological Distinctions.- 3.4. From Data Systems to Behavior Systems.- 3.5. Measures of Uncertainty.- 3.6. Search for Admissible Behavior Systems.- 3.7. State-Transition Systems.- 3.8. Generative Systems.- 3.9. Simplification of Generative Systems.- 3.10. Systems Inquiry and Systems Design.- Notes.- Exercises.- 4: Structure Systems.- 4.1. Wholes and Parts.- 4.2. Systems, Subsystems, Supersystems.- 4.3. Structure Source Systems and Structure Data Systems.- 4.4. Structure Behavior Systems.- 4.5. Problems of Systems Design.- 4.6. Identification Problem.- 4.7. Reconstruction Problem.- 4.8. Reconstructability Analysis.- 4.9. Simulation Experiments.- 4.10. Inductive Reasoning.- 4.11. Inconsistent Structure Systems.- Notes.- Exercises.- 5: Metasystems.- 5.1. Change versus Invariance.- 5.2. Primary and Secondary Systems Traits.- 5.3. Metasystems.- 5.4. Metasystems versus Structure Systems.- 5.5. Multilevel Metasystems.- 5.6. Identification of Change.- Notes.- Exercises.- 6: Complexity.- 6.1. Complexity in Systems Problem Solving.- 6.2. Three Ranges of Complexity.- 6.3. Measures of Systems Complexity.- 6.4. Bremermann's Limit.- 6.5. Computational Complexity.- 6.6. Complexity Within GSPS.- Notes.- Exercises.- 7: Goal-Oriented Systems.- 7.1. Primitive, Basic, and Supplementary Concepts.- 7.2. Goal and Performance.- 7.3. Goal-Oriented Systems.- 7.4. Structure Systems as Paradigms of Goal-Oriented Behavior Systems.- 7.5. Design of Goal-Oriented Systems.- 7.6. Adaptive Systems.- 7.7. Autopoietic Systems.- Notes.- Exercises.- 8: Systems Similarity.- 8.1. Similarity.- 8.2. Similarity and Models of Systems.- 8.3. Models of Source Systems.- 8.4. Models of Data Systems.- 8.5. Models of Generative Systems.- 8.6. Models of Structure Systems.- 8.7. Models of Metasystems.- Notes.- Exercises.- 9: GSPS: Architecture, USE, Evolution.- 9.1. Epistemological Hierarchy of Systems: Formal Definition.- 9.2. Methodological Distinctions: A Summary.- 9.3. Problem Requirements.- 9.4. Systems Problems.- 9.5. GSPS Conceptual Framework: Formal Definition.- 9.6. Overview of GSPS Architecture.- 9.7. GSPS Use: Some Case Studies.- 9.8. GSPS Evolution.- Notes.- Exercises.- Appendices.- A: List of Symbols.- B: Glossary of Relevant Mathematical Terms.- C: Some Relevant Theorems.- D: Refinement Lattices.- E: Classes of Structures Relevant to Reconstructability Analysis.- References.- Author Index.

Architecture of Systems Problem Solving

Invertible cellular automata: a review

Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures

The concepts of structural and behavioral isomorphism on tessellation automata are investigated. Certain equivalence relations preserving one or both forms of isomorphism lead to standardizations of neighborhood structure. The concepts of blocking and the blocked structure play a central role. A weaker form of behavioral isomorphism is also introduced leading to further simplifications of standard neighborhood structure. Finally, a concept of simulation is investigated.

Serafino Amoroso

Papers

Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures

Structural and behavioral equivalences of tessellation automata

Tessellation structures for reproduction of arbitrary patterns

A completeness problem for pattern generation in tessellation automata

Some clarifications of the concept of a Garden-of-Eden configuration