Wolfram Model Training Course

NOTE : WORK IN PROGRESS

Part 1 : Prerequisites

Before diving into the work of Stephen Wolfram, the Wolfram Physics Project, or most projects from the Wolfram Institute, it's helpful to have some background knowledge in certain fields and concepts. If you're unfamiliar with these terms, below are some helpful video links and concise definitions.

Complex Systems

A basic introduction to complexity theory, covering network graphs and how those are used to model systems.

Isomorphism & Graph Isomorphism

An isomorphism is an equivalence relation between different objects, structures, or systems, proven through a structure-preserving mapping. It ensures that operations, relationships, and transformations in one structure correspond directly to those in another.

Transformations

A transformation is an operation that changes an object, structure, or system from one form or state to another. Such operations are called mappings or functions and can include translation, rotation, scale, reflection. In physics, the preservation of an object's structure under transformation proves invariance, meaning physical laws remain unchanged under different transformations.

Invariance

Invariance is the property that an object, structure, or system remains unchanged under transformations. In physics, invariance means that despite changes in form, position, or other aspects, specific properties or laws stay the same.

Turing Machines

The Turing machine is a construct invented by Alan Turing, capable of computing all computable functions. It is the basis for today's modern computer.

Turing Universality

The idea that something (typically a scripting language) is capable of computing any computable function. To prove that a system is Turing Universal, one must show that the system can emulate a Turing Machine.

Halting Problem

A famous contradiction proof by Alan Turing : that a Universal Truth Machine, a machine that can tell when a program will continue forever or halt, is impossible to construct and can not exist.

Godel Incompleteness

Explains the importance of Consistency, Decidability, and completeness in formal systems, and the history of Hilberts Program vs Godel’s Program and how Hilberts program [A complete, decidable and consistent theory of mathematic] was shown to be false.

Part 2 : New Kind of Science

A New Kind of Science, a book written by Stephan Wolfram in the 1980’s and published in 2002 set the foundations of the Wolfram Model. The book is split into three distinct arcs:

The first arc is a set of experiments where Wolfram exhaustively runs classes of rules, making general observations about their behaviour. He observers three key properties about these systems:

1. That many simple rules can perform arbitrarily complicated behaviour, specifically behaviour that can not be described by mathematical equations.
2. That rules typically fall into 4 classes of behaviour : Homogenous, Patterned, Random and Complex.
3. That rules can emulate the behaviour of other rules.

The second arc of the book is spent making connections between computational rules and how they show up in nature, in system analysis and in fundamental physics.

The third and final arc of the book is spent formalizing the observations made in these experiments, by first showing the Elementary Cellular Automata Rule Class emulating each other under different initial conditions. Wolfram and Cook then prove the universality of Rule 110, strongly implying that any rule within this rule class, when given the right initial conditions can lead to an emulation of Rule 110, therefore proving the entire rule class as Turing Universal.

Computational Equivalence

In the Final Chapter, Wolfram poses this as strong evidence for The Principle of Computational Equivalence : That practically any system that follows rules, are computationally equivalent to a Turing universal machine, capable of computing any computable function.

“The answer is the Principle of Computational Equivalence, which states that, in effect, most rules will be equivalent in their computational capabilities—and in particular they will be capable of universal computation, so that any given rule can always “run a program” that will make it emulate any other rule.”

Excerpt from : The Concept of the Ruliad - Stephen Wolfram

Computational Irreducibility

The Principle of Computational Equivalence then gives explanatory power to The Phenomenon of Computational Irreducibility, explaining why many rules fail to be described by mathematical equations : Trying to predict or condense what a system following rules will do, is as hard as trying to solve the halting problem, and therefore one can not construct a universal truth machine that always gives us the answer. Operationally, a system can under different initial conditions emulate arbitrarily complicated rules and therefore to describe what they do, is nothing short of “they do everything.”

Computational Reducibility

In the same vein, Computational Reducibility also takes form: Since all systems are capable of computing any computable function under Computational Equivalence, then, there is also an infinite number of different patterns and regularities that these systems have access to. This feature forms the basis for doing science and mathematics.

These three ideas are core to understanding the Wolfram Model and will play major roles in practically every aspect of it, from the Ruliad, to Observer Theory, Multi-Computation and Ruliology.

In-Depth

I personally recommend watching the 16 part lecture series Wolfram made on the book, because he explains things with his point of view 20 years after its release, in the context of the 2020 Physics Model.

Part 3 : The Ruliad

The work of A New Kind of Science and the Principle of Computational Equivalence tells us that systems can emulate each other's behavior and, therefore, all perform computations as sophisticated as any other arbitrarily complicated system, which also makes them as sophisticated as a Turing universal machine. In some sense, we can state colloquially that all systems are just this single Turing machine.

To dig deeper into this, we can ask a logical question: If a single rule can compute the universe, and all rules are computationally equivalent to each other, then does it actually matter what rule the universe is running on? Why would it be running on this rule over here rather than that rule over there? The logical conclusion to this is that the universe isn’t just running any single rule—it is running all possible rules. What exists is not a collection of separate systems or particular rules; rather, what actually exists is this abstract Turing Machine space of all possible rules.

Imagine we successfully identify a rule that describes everything about our universe. Then the obvious next question will be: “Why this rule, and not another?” Well, how about if actually the universe in effect just runs every possible rule? What would this mean? It means that in a sense the “full story” of the universe is just the ruliad.

Excerpt from : The Concept of the Ruliad - Stephen Wolfram

So when we look at any system, we aren’t really looking at just the system itself—we are looking at something deeper. Something we can’t fully see. We are looking at this Ruliad object: a very thin slice, a sample, a limited perception of it.

Properties

The Ruliad is “the entangled limit of all possible rules” and is the most important "true ontology" of the wolfram model. So to conduct faithful science, we study our interface with this object.

Some properties of the Ruliad are that it is maximally symmetric and countably infinite. From the perspective of the entire object you can think of all mappings as bijective (one-to-one), but for all systems within the Ruliad, all mappings are surjective. In other words, subsystems embedded in this object cannot perceive it without consolidating information about its structure. This will play an important role in observer theory and, subsequently, the Wolfram Model of Physics.

Rulial Space

Another way to think about the Ruliad is like a state-space “made” of rules, such that every coordinate in that space is occupied by a unique rule or function. One can then imagine traversing that space, moving from one coordinate to another. Although this is an oversimplification, it can be useful to think about it this way when trying to parametrize and visualize the structure and connection to other rules in that space.

Once we start imagining the Ruliad as a space, having a sort of geometry, we can start attributing to it physical properties and formal analysis such as geodesics (shortest paths), maxima and minima, propagation speeds, and emergent features.

In-Depth

To take a much deeper dive, i recommend reading these papers written by Wolfram on the topic The Concept of the Ruliad—Stephen Wolfram Writings Why Does the Universe Exist? Some Perspectives from Our Physics Project—Stephen Wolfram Writings