A bitstring B exists if some physical configuration encodes it. In a computable universe where B does exist but only occurs once, and a perfect fidelity simulation of that universe is run within it on an unphysically large computer (except without the existence of the hardware running that simulation), B then occurs twice, because it is specified in some way on the hardware of the computer on which that simulation is run. If B′ refers to the simulated bitstring:

What should distinguish B, as a bitstring existing in a real external world, and B′?

If our simulation was run on a hypercomputer such that it could infinitely recur in perfect fidelity, is B′ 'more real' than B′′′′′?

These are the kinds of questions I want to find answers to, because I want to be able to prove things about simulators. As a first step, let's carve up the simulator.

Given the probability space (Ω,F,P) where Ω is the sample space, F is the event space, and P is the probability measure, mapping events in F to the interval [0,1], let ω refer to individual outcomes in Ω, each of which describe a discrete simulacrum, and Ck as individual discrete events in F. Classifying simulacra as programs, we denote the maximum Kolmogorov complexity K with respect to a universal Turing machine U for any given space (Ω,F,P) as v=maxω∈ΩKU(ω).

Proceeding, let ω∗(Ck) be the complete set of simulacra for some Cartesian objectCk, where individual simulacra are addressed by ωn(Ck) as sets of actions indexed by m. Let Ξ:Ω∗(Ck)×Ck→ωnm be a function that maps from choices for each ω to a world wkm∈Wk, where wkm$= refers to the mth world in the set of possible worlds for the kth object, and k indexes the object W describes possible worlds for, culminating in Ck=(ω1,...,ωn,Ξ).

By modeling the coupling of the probability space (Ω,F,P) and its contained simulacra as a dynamical system, the following are considered to describe sampling tokens from a simulation state S at time-step t given the complete simulation history prior S∗t=(S0,...,St) as a trajectory through states, where states are given by the set of worlds for all objects W∗ realized in the set of actualized objects A at time-step t, ⋃wkm∈AtΞ−1(wkm):

The token selection function ψ:S∗t→τ, where τ is a distribution over all tokens in an alphabet T.

The evolution operator ϕ which evolves a trajectory S∗t to S∗t+1 by appending the token sampled with ψ.

Assuming the model defined above, the simulation forward pass becomes a simple operation delineated as follows: We begin at simulation state S0, which denotes the empty or null state, whereby A0=∅, which is also maximally entropic.

ψ(S0) is applied for one time-step:

P selects Ck from F under v, aggregating the set of realized worlds in A1 as S1

The token selection function is applied to the current state as ϕ(S∗1)

A bitstring B exists if some physical configuration encodes it. In a computable universe where B does exist but only occurs once, and a perfect fidelity simulation of that universe is run within it on an unphysically large computer (except without the existence of the hardware running that simulation), B then occurs twice, because it is specified in some way on the hardware of the computer on which that simulation is run. If B′ refers to the simulated bitstring:

These are the kinds of questions I want to find answers to, because I want to be able to prove things about simulators. As a first step, let's carve up the simulator.

Given the probability space (Ω,F,P) where Ω is the sample space, F is the event space, and P is the probability measure, mapping events in F to the interval [0,1], let ω refer to individual outcomes in Ω, each of which describe a discrete simulacrum, and Ck as individual discrete events in F. Classifying simulacra as programs, we denote the maximum Kolmogorov complexity K with respect to a universal Turing machine U for any given space (Ω,F,P) as v=maxω∈ΩKU(ω).

Proceeding, let ω∗(Ck) be the complete set of simulacra for some Cartesian object Ck, where individual simulacra are addressed by ωn(Ck) as sets of actions indexed by m. Let Ξ:Ω∗(Ck)×Ck→ωnm be a function that maps from choices for each ω to a world wkm∈Wk, where wkm$= refers to the mth world in the set of possible worlds for the kth object, and k indexes the object W describes possible worlds for, culminating in Ck=(ω1,...,ωn,Ξ).

By modeling the coupling of the probability space (Ω,F,P) and its contained simulacra as a dynamical system, the following are considered to describe sampling tokens from a simulation state S at time-step t given the complete simulation history prior S∗t=(S0,...,St) as a trajectory through states, where states are given by the set of worlds for all objects W∗ realized in the set of actualized objects A at time-step t, ⋃wkm∈AtΞ−1(wkm):

Assuming the model defined above, the simulation forward pass becomes a simple operation delineated as follows:

We begin at simulation state S0, which denotes the empty or null state, whereby A0=∅, which is also maximally entropic.

ψ(S0) is applied for one time-step: