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Problem of large rules in rulial space

Posted 1 year ago
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In Section 8.22 of Technical introduction concerning rulial space we read:

In principle there are an infinite number of such rules, but any rule that involves rewriting a hypergraph that is larger than the hypergraph that represents the whole universe can never apply, so at least at any given point in the evolution of the system, the number of rules to consider is finite.

Is this in fact true? Given any graph rewriting rule of the form $n_k\rightarrow m_k$, value of $n$ is bounded by the current size of the hypergraph, but there seem to be no bounds on $m$. If so, there is actually infinite number of rules that can be applied at any given state of the hypergraph representing the whole universe. The consequence of this is that the evolution graph with use of possible rules is actually fully connected infinite graph, since there is always a rule transforming any possible state of universe (for given $k$) into any other state, both "future" (with hypergraph larger than one in the current state) and "past" (smaller hypergraph), for example by taking the entire hypergraph and rewriting it entirely into another one.

Possible arising problem is whether such evolution of the universe agrees with what we experience as observers? For simplicity, let us assume $m$ fixed to be comparable to the size of the hypergraph at the given stage of evolution. Comparatively small rules, allowing for elegant evolution and producing (hopefully) emergent properties of our universe constitute smaller and smaller fraction of all possible rules as evolution progresses, while large rules almost always transform current hypergraph into universe with no special structure, since among all possible hypergraphs, the structured ones are only tiny fraction. One can answer that conscious observer cannot survive rewriting structured hypergraph into unstructured one, since it is the structure of hypergraph that allows for existence of conscious observer, and that is why we observe only well-structured universe. But there is still highly probable that we find ourselves in one of this almost-normal universes, where maybe simple rules dominate evolution but occasionaly there happens something caused by large-rule rewrite, which does not stop our existence (perhaps some item on my desk suddenly disappears or creates itself out of nothing :)). The problem seems to follow in some sense the Bolzmann brain argument.

So can there by any natural constraint on $m$ in "all possible" rules $n_k\rightarrow m_k$?

2 Replies
Posted 1 year ago

Perhaps an "out of control m" is a trigger to big/bang and/or inflation. Eg, small rules dominate until the some LARGE hypergraph arises somewhere in the universe triggering tons of "new" rules, or just the specific initial conditions leading to same, maybe a new "bubble universe"?

Posted 1 year ago

It seems like we need a better understanding of what nodes are, if they can be freely created and destroyed with no consequences, and/or if they get reused. It seems like either nodes must be freely creatable to a more or less nearly infinite amount, or there must be some method of perfect data storage that has an effectively unlimited capacity (a Turing tape?). Either way, things get kind of weird.

If nodes can be created and destroyed as needed, then there is no reason to apply different rules to an existing graph and mess up all its nodes, and it seems counter to the basic assumption of this entire system, which is causality (who knew the Merovingian was right?).

So the observer wouldn't constantly be in this pingponging state between existing and not and existing as a shoe or a goldfish or a bald guy with a chromebook and some spare time at work.

Now, if you assume that your node supply is limited, but you have infinite storage, then you can just kind of apply some kind of ur-verse principle of attention deficit disorder, where you progress a branchial construct through rulial (I want to call this Rulian space but for now I'll skip that bit) space, save it to Turing tape, and then load a different branchial construct and iterate it for a while, then go back to the other one (side note: is every character in Skyrim ending up as a stealthing archer an example of causal invariance??) and progress a bit further, etc. But this feels really inelegant and requires a Demon of some sort to hang around and constantly swap tapes for no good reason.

Since we have to assume at least one infinite resource for all for this to work we might as well assume that nodes are infinite and branchial spaces run in paralell in rulial (Rulian!) space, as that fits in with our models much better and also keeps with the central theme of all of it, that internally consistent causality is the underlying organizational principle of basically everything that we could recognize as any layer of the universe (it also hints that there might be a hypercomputer at the top, and that someone suggested to it the idea of causality and then asked it to compute every possible and impossible expression of that idea, which would look like it took time if you were inside the computation but which would be instantaneous from outside of it).

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