The Ruliad

A schematic figure of the Ruliad

The Ruliad is a concept introduced by Stephen Wolfram. It is defined as the entangled limit of everything that is computationally achievable: the result of following all possible computational rules in all possible ways. It can be seen as an attempt to abstract and generalize every aspect of the physical universe through the use of structures such as graphs.

The term is derived from the concept of the Rulial Space, a space defined by allowing all practicable rules of a given class to be followed between states of a system.

Basic structure

At the lowest level, the structures on which the models operate consist of collections of relations between identical discrete elements. These structures can be represented as graphs or hypergraphs, with such collections being represented as the edges, and those elements are nodes. An example of a simple rule: $$\begin{array}{rl}\{\{x,y\}\}& ⟶\{\{x,y\}\},\\ \{\{x,y\}\}& ⟶\{\{x,y\},\{y,z\}\},\\ \{\{0,1\}\}& ⟶\{\{0,1\},\{1,z\}\}.\end{array}$$ with $x$, $y$, $z$ as three arbitrary elements; the arrow signifies a replacement or transformation between relations. Note that the element $z$ can be anything, as long as it is distinct from 0 and 1, which have already been used. Continuing to apply this rule, obtains $\{\{0,1\},\{1,a\},\{1,z\},\{z,b\}\}$ The values $a$ and $b$ are necessarily introduced because when operating on $\{0,1\}$ and $\{1,z\}$, which may be obtained in the following: $$\begin{array}{rl}\{0,1\}& ⟶\{\{0,1\},\{1,z\}\},\\ \{1,z\}& ⟶\{\{1,z\},\{z,a\}\}.\end{array}$$ Henceforth, the second replacement requires a different element than the previous $z$, and the same reasoning holds for $b$.

When imagining to continue this process (iteratively or recursively), the branching in the graph's space may be noticed, expanding both in terms of the number of edges and nodes. In the example, with a preference for an organized top-to-bottom layout, it becomes apparent that the rule generates a binary tree.

In more complex cases, the Ruliad presents correlations with the world of fractal geometry, cellular automata (like The Game Of Life by John Conway), Turing machines or growth processes of biological organisms like the Physarum Polycephalum. These are Rulial multiway graphs, after 3-steps for Turing machines (with various numbers of states):.

The full Ruliad involves taking the infinite limits of all possible rules, initial conditions and steps.

References

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