A bitstring  exists if some physical configuration encodes it. In a computable universe where  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),  then occurs twice, because it is specified in some way on the hardware of the computer on which that simulation is run. If  refers to the simulated bitstring:

  • What should distinguish , as a bitstring existing in a real external world, and ?
  • If our simulation was run on a hypercomputer such that it could infinitely recur in perfect fidelity, is  'more real' than ?

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  where  is the sample space,  is the event space, and  is the probability measure, mapping events in  to the interval , let  refer to individual outcomes in , each of which describe a discrete simulacrum, and  as individual discrete events in . Classifying simulacra as programs, we denote the maximum Kolmogorov complexity  with respect to a universal Turing machine  for any given space  as .

Proceeding, let  be the complete set of simulacra for some Cartesian object , where individual simulacra are addressed by  as sets of actions indexed by . Let  be a function that maps from choices for each  to a world , where $= refers to the  world in the set of possible worlds for the  object, and  indexes the object  describes possible worlds for, culminating in .

By modeling the coupling of the probability space  and its contained simulacra as a dynamical system, the following are considered to describe sampling tokens from a simulation state  at time-step  given the complete simulation history prior  as a trajectory through states, where states are given by the set of worlds for all objects  realized in the set of actualized objects  at time-step :

  • The token selection function , where  is a distribution over all tokens in an alphabet .
  • The evolution operator  which evolves a trajectory  to  by appending the token sampled with .

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

 is applied for one time-step:

  •  selects  from  under , aggregating the set of realized worlds in  as 
  • The token selection function is applied to the current state as 
     

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