Article Outline
Glossary
Definition of the Subject
Introduction
The Basic Model
Emerging Patterns and Behaviors
SDCA as Models of Computation
Generalized SDCA Models
Related Graph Dynamical Systems
SDCA as Models of Fundamental Physics
Future Directions and Speculations
Bibliography
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Abbreviations
- Adjacency matrix:
-
The adjacency matrix of a graph with N sites is an \( { N\times N } \) matrix \( { [a_{i\!j}] } \) with entries \( { a_{i\!j}=1 } \) if i and j are linked, and \( { a_{i\!j}=0 } \) otherwise. The adjacency matrix is symmetric (\( { a_{i\!j}=a_{ji} } \)) if the links in the graph are undirected.
- Coupler link rules:
-
Coupler rules are local rules that act on pairs of next‐nearest sites of a graph at time t to decide whether they should be linked at \( { t+1 } \). The decision rules fall into one of three basic classes – totalistic (T), outer‐totalistic (OT) or restricted‐totalistic (RT) – but can be as varied as those for conventional cellular automata.
- Decoupler link rules:
-
Decoupler rules are local rules that act on pairs of linked sites of a graph at time t to decide whether they should be unlinked at \( { t+1 } \). As for coupler rules, the decision rules fall into one of three basic classes – totalistic (T), outer‐totalistic (OT) or restricted‐totalistic (RT) – but can be as varied as those for conventional cellular automata.
- Degree:
-
The degree of a node (or site, i) of a graph is equal to the number of distinct nodes to which i is linked, and where the links are assumed to possess no directional information. In general graphs, the in-degree (= number of incoming links towards i) is distinguished from the out-degree (= number of outgoing links originating at i).
- Effective dimension:
-
A quantity used to approximate the dimensionality of a graph. It is defined as the ratio between the average number of next‐nearest neighbors to the average degree, both averaged over all nodes of the graph. The effective dimension equals the Euclidean dimension d, in cases where the graph is the familiar d‑dimensional hypercubic lattice.
- Graph:
-
A graph is a finite, nonempty set of nodes (referred to as “sites” throughout this article), together with (a possibly empty) set of edges (or links). The links may be either directed (in which case the edge from a site i, say, is directed away from i toward another site j, and is considered distinct from another directed edge originating at j and pointed toward i) or undirected (in which case if a link exists between sites i and j it carries no directional information).
- Graph grammar:
-
Graph grammars (sometimes also referred to as graph rewriting systems) apply formal language theory to networks. Each language specifies the space of “valid structures”, and the production (or “rewrite”) rules by which given graphs may be transformed into other valid graphs.
- Graph metric function:
-
The graph metric function defines the distance between any two nodes, i and j. It is equal to the length of the shortest path between i and j. If no path exists (such as when i and j are on two disconnected components of the same graph), the distance is assumed to be equal to ∞.
- Graph‐rewriting automata:
-
Graph‐rewriting automata are generalized CA-like systems in which both (the number of) nodes and links are allowed to change.
- Next‐nearest neighbor:
-
Two sites i and j are next‐nearest neighbors in a graph if (1) they are not directly linked (so that \( { a_{i\!j}=0 } \); see adjacency matrix), and (2) there exists at least one other site k such that \( { k\notin \left\{{i,j}\right\} } \), and i and j are both lined to k.
- Random dynamics approximation:
-
The long-term behavior of structurally dynamic cellular automata may be approximated in certain cases (in which the structure and value configurations are both sufficiently random and uncorrelated) by a random dynamics approximation: values of sites are replaced by the probability p σ of a site having value σ (and is assumed to be equal for all sites), and links between sites are replaced by the probability \( { p_{\ell} } \) of being linked (and also assumed to be the same for all pairs of sites).The approximation often yields qualitatively correct predictions about how the real system evolves under a specific set of rules; for example, to predict whether one expects unbounded growth or that the lattice will eventually settle onto a low periodic state or simply decay.
- Restricted totalistic rules:
-
Restricted totalistic rules are a generalized class of link rules (operating on pairs of sites, i and j), analogous to “outer totalistic” rules (that operate on site values) used in conventional CA. The local neighborhood around i and j is first partitioned into three sets: (1) the two sites, i and j; (2) sites connected to either i or j, but not both; and (3) sites connected to both i and j. The restricted totalistic rule is then completely defined by associating a specific action with each possible 3-tuple of site-value sums (where the individual components represent a unique sum in each of the three neighborhoods).
- Structurally dynamic cellular automata:
-
Structurally dynamic cellular automata are generalizations of conventional cellular automata models in which the underlying lattice structure is dynamically coupled to the local site-value configurations.
- SDCA model hierarchy:
-
The SDCA model hierarchy is a set of eight related structurally dynamic cellular automata models, defined explicitly for studying their formal computational capabilities. The hierarchy is ordered (from lowest to highest level) according to their relative computational strength. For example, the SDCA model at the top of the hierarchy is capable of simulating a conventional CA with a speedup factor of two.
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Ilachinski, A. (2012). Structurally Dynamic Cellular Automata. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_194
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