# Apply all the rules at the same time? How is it possible?

Posted 4 months ago
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Hi everyone, Since I first read about rulial space and the concept that every possible rule may be applied at the same time, I was dubious about the idea.

You can find these concepts explained in the Project Announcement, or also in Exploring Rulial Space, to get some context.

I had always thought that I had misinterpreted what Wolfram was saying. Maybe he didn't mean that every step, every possible rule is applied. Maybe he just meant that every Turing complete rule is able to emulate the behavior of every other rule, and therefore every rule is equivalent. Then, to us (computationally bounded observers) it is like every rule could be applied every step.

But I recently came across the bulletin Why Does the Universe Exist? where it seems that Wolfram is indeed considering to apply every rule simultaneously.

The reason of my concern about the simultaneous application of all the possible rules is that I feel that this would lead to a trivial universe, that doesn't evolve. Imagine for example to start from the simplest initial condition, an empty hypergraph $\{\}$. You then apply all the possible rules that have $\{\}$ as LHS. For every hypergraph $a$, there will be a rule $\{\} \rightarrow a$, and therefore the second step of the multiway graph will contain every possible hypergraph. For the third step you are now free to apply really every possible rule, since you have every possible LHS available. As a consequence the third step will consist, again, of every possible hypergraph. And each subsequent step will be exactly the same. Hence the system is not evolving anymore. Time is frozen.

This argument may not convince you because treating infinities may be slippery and the state of the a multiway system isn't described just by the collection of states in a particular slice, but also the events that lead there (the causal relations) carry information. I will now address these issues by formalizing the argument.

I will start by defining what I mean by "step" and what I mean when I say that a system is frozen in time, or that it is not evolving anymore.

### Definition: multiway step

Given a set of rules $\mathcal{R}$ and a set of spatial hypergraphs $C$, a step is a triplet $(C,E, C')$, where $C'$ is the set of all the spatial hypergraphs that can be obtained by applying a rule $r\in \mathcal{R}$ to an element of $C$, and $E$ is the set of causal events generated by the application of such rules.

In other words, to complete a step you have to apply the rules everywhere they are applicable. This definition of step induces a natural way to foliate a multiway graph, and defines a global time coordinate.

### Definition: causally inverted event

Given a rule $r$, one can define the inverted rule $\bar{r}$ that is obtained from $r$ by swapping left hand side and right and side. A similar operation can be done to events. Given two hypergraphs $a$ and $b$, and an event $e: a \rightarrow^r b$, the causally inverted event $\bar{e}$ will be $\bar{e}: b \rightarrow^\bar{r} a$

### Definition: frozen system

Consider a multiway system that has evolved for n steps $(C_i,E_i, C_{i+1})$ for $i=1,...,n$. The system at the step n is considered frozen if:

1. $C_{n-1} = C_n = C_{n+1}$
2. $E_{n}$ can be obtained by $E_{n-1}$ by causally inverting every event of $E_n$.

In a certain sense this definition says that if you invert the direction of time (go from n to n-1 instead that to n+1) and don't notice any difference, then it means that time has stopped, because you cannot distinguish the future from the past.

By this definition, the system on the left is not frozen, because it is asymmetric under temporal inversion. The system on the right is instead frozen.

### Definition: complete set of rules

Given a set of hypergraphs $C$, the complete set of rules associated with $C$ is the set $\mathcal{R}_C$ defined such that for every couple of hypergraphs $a,b \in C$ there exist a rule $r \in \mathcal{R}_C$, that brings the state $a$ to the state $b$ (in symbols $a\rightarrow^r b$).

It is obvious that if $r \in \mathcal{R}_C$, then also $\bar{r} \in \mathcal{R}_C$.

### Theorem: A multiway system with a complete set of rules freezes after the second step

Consider a set of hypergraphs $C$ and its complete set of rules $\mathcal{R}_C$. Choose an hypergraph $c \in C$ as initial condition and evolve the multiway system. Since $\mathcal{R}_C$ is complete, $c$ can be linked to any other hypergraph in $C$, and the first step will then be $( \{ c \} , E_1, C)$, where $E_1$ is the set of events that link $c$ to any hypergraph in $C$.

The second step will be $(C , E_2, C)$ and the third will be $(C , E_3, C)$.

For every $a,b \in C$ there exist a rule $r \in \mathcal{R}_C$ such that the event $e: a \rightarrow^r b$ belongs to $E_2$ and to $E_3$. This means that $E_2=E_3$. Moreover, for every event $e: a \rightarrow^r b$ that belongs to $E_2$, there will also be in $E_3$ the inverted event $\bar{e}: b \rightarrow^\bar{r} a$. In other words, $E_2=E_3$ and they coincide with the set of their inverted events. As a consequence, by the third step the system has frozen.

It is important to notice that I have nowhere assumed that the set of hypergraph or the set of rules must be finite. The whole argument fully applies to infinite set, and in particular it applies to the set of all possible hypergraphs, completed by the set of all possible rules.

Therefore, if we model a universe where, every step, every possible rule is applied, then we can only obtain a trivial, non evolving universe, deprived of any of the complex structures that we observe today in our universe.

In the figure it is shown an example of complete set of rules, which are: $A \rightarrow A$, $A \rightarrow B$, $A \rightarrow C$, $A \rightarrow D$, $B \rightarrow B$, $B \rightarrow C$, $B \rightarrow D$, $C \rightarrow C$, $C \rightarrow D$, $D \rightarrow D$, plus their inverse. As you can see, by the second step the system started repeating itself, and every step is the same.

In Why Does the Universe Exist?, Wolfram wants to address this particular critic that someone may have raised:

What if each piece of our hypergraph is updated according to all possible rules—generating many different possible histories? Or, in other words, what if the universe in some sense “simultaneously runs” all possible rules—generating all possible resulting histories?

Our first instinct might be that if all these possibilities are allowed then there could never be anything definite said about what would happen in the universe. But it turns out that this is far from correct. And it all has to do with the entangling of different possibilities associated with the repeated application of rules.

Given a particular state of the universe, applying different rules can lead to different states. But applying rules to different states can also potentially lead to the same state. Or, in other words, the “rulial multiway graph” that represents how one state leads to another can exhibit both branching and merging.

I am not able to see how the merging of the multiway branches could provide structure to a multiway system with a complete set of rules, which, by the theorem just proven, is required to freeze by the third step. Additionally, Wolfram makes two examples, but the first does not have a complete set of rules, so I don't think it is suited to prove that complete sets of rules can generate nontrivial structure. The second example is instead complete, and therefore freezes after the second step. What Wolfram refers to as a "more symmetrical structure", is just a complete graph, the signature that the system is not evolving anymore.

Any thoughts on that? Have I misinterpreted Wolfram? Does the concept of simultaneous application of every possible rule need to be abandoned?

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Posted 4 months ago
 I'm no expert on this, but maybe sharing my own thoughts gives you some new ideas at least.The missing part here is, I think, the arrow of time. Your description is just so coarse-grained that it practically hides it. The simplest way to point out the "devil in the details" in this case is perhaps to give you an example of a reversible setup with an arrow of time.One way to set this up is to pick a rule in which the RHS has one or more parts that will match the LHS pattern. For example, consider a rule (1,2)(1,3)->(1,2)(1,4)(2,4)(3,4) and its inverse (1,2)(1,4)(2,4)(3,4)->(1,2)(1,3). This setup is reversible. However, as every possible sequence of events is realized, and as we keep sampling from them, there will eventually be increasingly more locations on which to apply the rule in one direction as compared to the other direction. This imbalance will emerge whether you start from (1,1)(1,1), a random initial graph, or a complete graph.Now, you might argue that this statistical arrow of time is not enough, because every state is still in fact realized in one step. One answer might be that the part of the system that has the arrow of time is the only part in which evolution, life etc. is possible. Therefore, what happens outside of that part is just irrelevant to us who must exist (be spread out) inside the arrow.This last point might also be the reason why it is possible - and often useful - to introduce some fundamental principles in our models. If they are picked right, they will effectively limit the number of possible rules without affecting the part in which we live - the observed laws of physics. As an example, excluding the inverse part of my first example would give a good approximation of the system with less computational effort.
Posted 3 months ago

Thank you for your input. I realize that I may have explained myself poorly. I am not arguing that no multiway system has an arrow of time, of course. Your example is clear in this regard, it features an obvious arrow of time, I am not arguing against that.

I am saying that there is a class of systems, that I called "complete multiway systems" or "multiway systems with a complete set of rules", for which after a few steps the arrow of time necessarily ceases to exist.

I am also claiming that a universe where every possible rule can be applied (such as the one referred to by Wolfram) is a complete multiway system, and therefore it may not have an arrow of time and may not describe our own universe.

While writing this, I realized that in my previous post I defined a "complete set of rules", but I didn't give a definition of "complete multiway system". I thought that it could be inferred from the context, but this may instead be a source of misunderstanding. I will define the concept here:

### definition: complete multiway system

Consider a multiway system that evolves from some initial conditions through the application of a set of rules $\mathcal{R}$. Let $C$ be the set of all possible hypergraphs that are reached by the multiway system during its evolution. The multiway system is complete iff $\mathcal{R} = \mathcal{R}_C$, i.e. if the set of rules used is a complete set of rules of $C$.

In other words, the rules must be able to go from every element in $C$ to every other element, and at the same time, the application of a rule to an element of $C$ always returns another element of $C$.

A couple of examples made with string substitution systems:

• The set of rules $\mathcal{R} = \{a \rightarrow b, b \rightarrow a\}$ is a complete set of rules of $C = \{a,b\}$. Evolving from the initial condition $\{a\}$ generates a complete multiway system.

• The set of rules $\mathcal{R} = \{a \rightarrow b, b \rightarrow a, a \rightarrow c\}$ is still a complete set of rules of $C = \{a,b\}$. But this time, evolving from the initial condition $a$ leads to the state $c \notin C$ and therefore the multiway system is not complete.

Some less trivial examples:

• Consider set of integer numbers $\mathbb{Z}$ coupled with the set of rules $\{\text{add}_i \forall i \in \mathbb{Z}\}$ that add an integer number, i.e. $\text{add}_i: n \rightarrow n+i$. Starting from some integer number as initial condition you obtain a complete multiway system.

• The set of all possible planar graphs, coupled with all the possible rules that preserve planarity. This too generates complete multiway systems.

I hope it is now clear why your example is not a complete multiway system and therefore is not subject to the theorem I proved in the previous post.

I hope it is also clear why the system with all possible rules of which Wolfram speaks is indeed a complete multiway system and therefore the theorem applies.

Posted 3 months ago
 Thanks for your reply. I think this whole issue of "all possible rules" is an interesting one.And yes, my first example was just to show that even a reversible setup can have an arrow of time. It was not, as you rightly pointed out, a complete multiway system the way you define it.Having now read your reply carefully, I can try to give you a counterexample that is, by your definition, a complete multiway system and still has an arrow of time:Let the set of rules be { (1,1)->(1,2), (1,2)->(1,1) }. Start from (1,1) and apply the rules simultaneously. All the possible two states are visited. However, since the LHS pattern (1,2) matches both (1,1) and (1,2), the system ends up spending more time at state (1,1). Therefore, the system has a statistical arrow of time and it never ceases to exist.
Posted 3 months ago
Posted 3 months ago
 Actually, if you add the rules (1,1)->(1,1) and (1,2)->(1,2) into my example, the imbalance increases. You might have forgotten that the RHS is a pattern too. So, when the rule (1,2)->(1,2) matches the (1,1), the outcome is (1,1) not (1,2). I made you a little sketch from which you can see all the possible outcomes.As for your own example, I think it doesn't have an arrow time, because not all four rules match your initial condition as you have indicated in your diagram. The relative directions of the two edges matter here. That said, the more complicated these setups get, the more my intuition fails me, so I urge you to simulate the system and actually count the paths.And yes, in Wolfram models the magnitude of the quantum amplitude is associated with path weights for geodesics in the multiway graph. So basically it is path counting according to your second interpretation.I encourage you to read these technical papers if you haven't already. Attachments:
Posted 3 months ago
 Actually I now get your point of there being two ways to apply one rules. That's right.