Relative Universality. On classes of Real Computation Hector Zenil University of Lille 1 Universit é de Paris 1 (Panthéon-Sorbonne). Real numbers. Definitions: A real number is of the form: X.{0,1}* {0,1}* can be always seen as a subset of natural numbers.
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Relative Universality
On classes of Real Computation
Hector ZenilUniversity of Lille 1
Université de Paris 1 (Panthéon-Sorbonne)
Definitions:
{0,1}* can be always seen as a subset of natural numbers.
Establishes a close connection between the arithmetical hierarchy of sets of natural numbers and the Turing degrees.
A formula F is in Sigma_n iff F is a recursively enumerable set with an oracle O^n
If O is an oracle machine O=r=X.r_1,r_2,… for some r real number since f(r_n)=1 if O answers yes for a question concerning the n-th string of a language, and f(r_n)=1 otherwise. And where f is the characteristic function for O and a language L.
The Turing degree of a real number r_L is defined as the Turing degree of its binary expansion since it can be seen as a subset of natural numbers
We are going to say that a class L is intrinsically universal if there exists an abstract machine U in L capable to behave as any other abstract machine M in L given the transition table and input for M.
Intrinsic universality = universality in the Turing sense
The class of recursive functions is
evidently intrinsically universal
Assuming a Delta_n^m set of real numbers D:
D allows Delta_n^m-universality
In other words, each level of the arithmetical and the hyper arithmetical hierarchies admit a universal abstract machine for that level.
Because the arithmetical and hyper arithmetical hierarchies do not collapse, there is no Delta_n universal machine able to behave as any other Delta_n+1 machine
Unless…
Lets call f a universal jump operator if a model of computation M allows at least one function f such that:
f(i)=j
with deg_T(i) <deg_T(j)
In other words f converts inputs of certain degree into outputs of higher Turing degree.
Then M is non-closed under its operators
f(i)=j
with deg_T(j)= Delta_n’^m’ for n’ and m’ arbitrary non-negative numbers.
In other words M reaches any AH and hyper AH level
through f. Both AH and hyper AH collapse under M
* Because the Turing degrees are a partial order
Finally, if a model of computation M is intrinsically universal then:
* Another possible case would be F as infinite, however a finite set F’ could be built from F composing all the functions from F into a new function f’ in F’. The result would be a non-recursive function f’ and a finite set F which would falls into the case number 3.
(building the Delta_n-universal machine)
There is no absolute universal machine in R
Unless…
A set of functions F of arbitrary power going through all the AH and hyper AH reaching any possible level.
Only in such case M can be closed under its operators and intrinsically universal. In other words, the model is not a field since it is not closed under the set of functions F unless it allows the whole complexity class of R.
The appropriate hierarchy for computations with
real numbers are beyond the scope of the AH and the HAH.
There is no universal machine for R at any level of the AH or the HAH, therefore the notion of real computation is not well-founded in the sense of lack of an abstract universality machine.
Hyper-models do not converge like those Turing-equivalent which supports the Church-Turing thesis. Therefore there is no a single Church-Turing thesis for real computation but an infinite hierarchy of Church-Turing type thesis based on each of the universal devices at each level of the AH and HAH.