We prove the existence of universal convergent and tuatara machines.
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced.
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
Every self-delimiting Turing machine is tuatara, but the converse is not true.
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
Also, there exist universal convergent and tuatara machines; there is a tuatara machine universal for the class of convergent machines.
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
Every self-delimiting Turing machine is a tuatara machine.
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
Tuatara has not changed its form much in over 225 million years! Its relatives died out about 60 million years ago.
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
We have seen that every self-delimiting Turing machine is a tuatara machine (Proposition 9), but the converse is not true (Fact 15, b)).
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
For each self-delimiting Turing machine C there effectively exists a tuatara machine V such that ζV = ΩC .
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
Actually, we can describe a more precise simulation of a self-delimiting Turing machine with a tuatara machine.
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
Given a self-delimiting Turing machine C we can effectively construct a tuatara machine V such that ζV = ΩC .
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
We deﬁne the domain of the tuatara machine V to be dom(V ) = [p∈dom(C ) where X (p) is the set {p} ∪ {p0i |pi = 1} and pi is the ith bit of p, numbering from the left and starting with i = 1.
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
Given a universal self-delimiting Turing machine U we can effectively construct a tuatara machine W universal for al l self-delimiting Turing machines such that ζW = ΩU .
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
Next we turn our attention to universal convergent/tuatara machines.
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
The sets of convergent machines and tuatara machines are c.e.
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
Let (Ci )i≥1 be an enumeration of tuatara machines.
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
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