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RSS Feed for Wolfram Community showing any discussions in tag Wolfram Science sorted by activeProblem of large rules in rulial space
https://community.wolfram.com/groups/-/m/t/1965281
In [Section 8.22 of Technical introduction][1] 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][2].
So can there by any natural constraint on $m$ in "all possible" rules $n_k\rightarrow m_k$?
[1]: https://www.wolframphysics.org/technical-introduction/potential-relation-to-physics/multiway-systems-in-the-space-of-all-possible-rules/
[2]: https://en.wikipedia.org/wiki/Boltzmann_brain#Identifying_whether_oneself_is_a_Boltzmann_observerDominik Rzepka2020-05-05T09:22:08Z