The ruliad may be defined as the entangled limit of everything that is computationally possible, i.e., the result of following all possible computational rules in all possible ways (Wolfram 2020, 2021).

The ruliad can be considered as the ultimate abstraction and generalization involving any aspects of the physical universe. In particular, while a computational system or mathematical theory requires certain choices to be made, there are no choices or outside inputs in a ruliad because it already contains everything.


In the context of a rulial multiway system, the ruliad traces out the entangled consequences of progressively applying all possible computational rules. At each step, the rules are applied in all possible ways to each state. This process often generates multiple new states, leading to branching in the graph, but there can also be merging from multiple states being transformed to the same state. The illustration at left above shows an ordinary multiway system based on the string replacement rules A->AB, BB->A. In contrast, the illustration at right above shows the rulial multiway system given by "1->2, 2->1 A, B string rules" (see Wolfram 2021 for details).

While a ruliad can be coordinatized and sampled in different ways, it has the important underlying property of uniqueness. This follows from the principle of computational equivalence, which says that almost all rules lead to computations that are equivalent (or in other words, thereÕs only one ultimate equivalence class for computations) (Wolfram 2020).

The ruliad builds on a tower of ideas that includes the computational paradigm, the exploration of the computational universe of simple programs, the principle of computational equivalence, the Wolfram Physics Project, and the multicomputational paradigm.

The multiplicad provides a simple example of an idea like the ruliad.

See also

Multiplicad, Multiway System, Principle of Computational Equivalence

Explore with Wolfram|Alpha


Wolfram, S. "Exploring Rulial Space: The Case of Turing Machines." Jun. 9, 2020., S. "The Concept of the Ruliad." Nov. 10, 2021. S. "Expression Evaluation and Fundamental Physics: The Rulial Case." Sep. 29, 2023.

Cite this as:

Weisstein, Eric W. "Ruliad." From MathWorld--A Wolfram Web Resource.

Subject classifications