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### Relative Universality

On classes of Real Computation

Hector ZenilUniversity of Lille 1

Université 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.

- Lets say that a subset S in R is complete if S contains at least one representative member of each complexity degree of R. by instance any non-empty open or closed ball [x,y) in R with x!=y

Arithmetical Hierarchy

- A formula is in Sigma_1 if it is of the form:
- En_1,En_2,…,En_mQ(n_1,n_2,…n_i) with Q a free quantifier recursive formula.
- A formula is in Pi_1 if it is of the form:
- An_1,An_2,…,An_mQ(n_1,n_2,…n_i) with Q a free quantifier recursive formula.
- Inductively:
- A formula is in Sigma_n if it is of the form:
- E^m Q_n-1 where Q has n-1 quantifiers alternations.
- And in Pi_n if
- A^m Q_n-1 where Q has n-1 quantifiers alternations.
- Delta_n=Sigma_n Intersection Pi_n

Oracle Turing machine

- A Turing machine with an aditional tape called oracle tape and three new states: q_?, q_y and q_n. With a Turing machine gets into the state q_? The oracle answers using the oracle tape and entering into the “yes” -q_y- or “no” states -q_n-.
- The oracle tape is made by 0’s and 1’s depending if the answer to q_? is yes or no.

Post Theorem

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

Real numbers and oracles

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.

Degree of computability of a real number

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

Model of Computation

- A model of computation M={D,F}. A domain of operation D and a set of operators F.
- An automata T in M is an abstract machine taking inputs from D and applying a set of Functions F’ in F. So if M is closed under F, T is closed under F’ in M.

Intrinsic Universality

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

Relative universality

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.

Universality Hierarchy

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…

Universality Jump Operator

- Let be F the set of functions allowed in a model of computation M with inputs at most in Delta_n^m for some n and m.
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

Collapsing the hierarchies

- If jump functions are allowed, the hierarchies or at least some parts of them collapse under those models of computation.
- Lets call a universality jump a complete universal jump if at least one function f is allowed such that for an input i and deg_T(i)<Delta_n^m :
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

Automata Complexity

- The complexity of an automata denoted by C(A) is the measure defined by the set of maximal* Turing degrees of the set of all possible outputs O in D -either M is closed under F or not-. If M is not closed under F deg_T(O)>deg_T(D). If M is closed the complexity of A is at most the maximal Turing degree of D.

* Because the Turing degrees are a partial order

Universality Loss

Finally, if a model of computation M is intrinsically universal then:

- The scope of operation of M is arbitrary bound up to a certain computability degree in the AH
- M computes at most the class of the recursive functions (M is Turing equivalent) (a case of 1)
- M is non-closed under its functions F=f_1,f_2,…,f_n because a subset G in F performs a complete universal jump*.

* 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.

Relative Universality (sketch proof)

- Take any level of the AH e.g. Delta_n. If M is a closed model of computation:
(building the Delta_n-universal machine)

- Take the usual universal Turing machine and an oracle O^n. Because the Turing degree of an automata M with domain Delta_n is at most in Delta_n F:Delta_n->Delta_n and applying Post’s theorem O^n computes x for all x real number in Delta_n, then M is universal.

Universality Loss (sketch proof)(lack of absolute universality in R)

- Assuming the complete set of real numbers denoted by R (or any complete subset as defined before) we claim that:
There is no absolute universal machine in R

- Suppose U is such universal machine. Then U is capable to behave as any other machine M in R
- Lets take MAX the set of maximal Turing degrees of U which defines C(U). Then take r a real number such that deg_T(r)>C(U), then build M an abstract automata able to compute r, then U is not able to compute “Mx”, where x is the input for M. Thus U is not a universal machine for R. Therefore there is no possible U in R.
Unless…

Universality conditions in real computation

- The model of real computation can be intrinsically universal only if it allows:
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.

Conclusions

- Three possible scenarios for intrinsic universality are possible:
- The class of recursive functions
- The class of any level of the AH (an infinite number of models of different power) with non-recursive functions up to that level in the AH (case 1 is naturally in the first level of this case)
- The class of real computation with access to all real numbers and functions able to reach any level of the AH and the hyper AH.

Conclusions 2

- There is a universal model of computation for each level of the AH and the hyper AH picking the correct set of functions F for each level.
- In a Discrete/Continuum dichotomy the election of an special set F and an arbitrary level in the AH does not seem natural. If the Continuum is taken as R -as usual- only arbitrary powerful functions makes the model intrinsically universal otherwise the model does not allow intrinsic universality.

Conclusions 3

- In other words:
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.

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