Exploring Rulial Space: The Case of Turing Machines

01/22/2021

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by   Stephen Wolfram, et al.

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As an example of the concept of rulial space, we explore the case of simple Turing machines. We construct the rulial multiway graph which represents the behavior of all possible Turing machines with a certain class of rules. This graph (which is a Cayley graph of a "Turing machine group") gives a map of the space of non-deterministic Turing machines. We investigate the subgraph formed by deterministic machines, and explore the relationship to the P vs. NP problem. We also consider the implications of features of rulial space for physics, including estimating the maximum speed h̊o̊ in rulial space, relations between rulial black holes and computational reducibility, and interpretations of hypercomputation.