The Entangled Limit of Everything I call it the ruliad. Thi…

The Entangled Limit of Everything
I call it the ruliad. Think of it as the entangled limit of everything that is computationally possible: the result of following all possible computational rules in all possible ways. It’s yet another surprising construct that’s arisen from our Physics Project. And it’s one that I think has extremely deep implications—both in science and beyond.
In many ways, the ruliad is a strange and profoundly abstract thing. But it’s something very universal—a kind of ultimate limit of all abstraction and generalization. And it encapsulates not only all formal possibilities but also everything about our physical universe—and everything we experience can be thought of as sampling that part of the ruliad that corresponds to our particular way of perceiving and interpreting the universe.
We’re going to be able to say many things about the ruliad without engaging in all its technical details. (And—it should be said at the outset—we’re still only at the very beginning of nailing down those technical details and setting up the difficult mathematics and formalism they involve.) But to ground things here, let’s start with a slightly technical discussion of what the ruliad is.
In the language of our Physics Project, it’s the ultimate limit of all rulial multiway systems. And as such, it traces out the entangled consequences of progressively applying all possible computational rules.
Here is an example of an ordinary multiway system based on the string replacement rules {A → AB, BB → A} (indicated respectively by blueish and reddish edges):
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At each step, the rules are applied in all possible ways to each state. Often this generates multiple new states, leading to branching in the graph. But, importantly, there can also be merging—from multiple states being transformed to the same state.
The idea of a rulial multiway system is not just to apply particular rules in all possible ways, but to apply all possible rules of a given form. For example, if we consider “1 → 2, 2 → 1 A, B string rules”, the possible rules are
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and the resulting multiway graph is (where now we’re using purple to indicate that there are edges for every possible rule):
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Continuing a little longer, and with a different layout, we get:
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This may already look a little complicated. But the ruliad is something in a sense infinitely more complicated. Its concept is to use not just all rules of a given form, but all possible rules. And to apply these rules to all possible initial conditions. And to run the rules for an infinite number of steps.
The pictures above can be thought of as coarse finite approximations to the ruliad. The full ruliad involves taking the infinite limits of all possible rules, all possible initial conditions and all possible steps. Needless to say, this is a complicated thing to do, and there are many subtleties yet to work out about how to do it.
Perhaps the most obviously difficult issue is how conceivably to enumerate “all possible rules”. But here we can use the Principle of Computational Equivalence to tell us that whatever “basis” we use, what comes out will eventually be effectively equivalent. Above we used string substitution systems. But here, for example, is a rulial multiway system made with 2-state 2-color Turing machines:
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And here is a rulial multiway system made from hypergraph rewriting of the kind used in our Physics Project, using all rules with signature :
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As another example, consider a multiway system based on numbers, in which the rules multiply by each possible integer:
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Here’s what happens starting with 1 (and truncating the graph whenever the value exceeds 100):
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Even with this simple setup, the results are surprisingly complicated (though it’s possible to give quite a bit of analysis in this particular case, as described in the Appendix at the end of this piece).
The beginning of the multiway graph is nevertheless simple: from 1 we connect to each successive integer. But then things get more complicated. To see what’s going on, let’s look at a fragment of the graph:
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In a sense, everything would be simple if every path in the graph were separate:
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But the basic concept of multiway systems is that equivalent states should be merged—so here the “two ways to get 6” (i.e. 1 × 2 × 3 and 1 × 3 × 2) are combined, and what appears in the multiway graph is:
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For integers, the obvious notion of equivalence is numerical equality. For hypergraphs, it’s isomorphism. But the important point is that equivalence is what makes the multiway graph nontrivial. We can think about what it does as being to entangle paths. Without equivalence, different paths in the multiway system—corresponding to different possible histories—would all be separate. But equivalence entangles them.
The full ruliad is in effect a representation of all possible computations. And what gives it structure is the equivalences that exist between states generated by diffe…