Relative Universality

1. Relative Universality On classes of Real Computation Hector ZenilUniversity of Lille 1 Université de Paris 1 (Panthéon-Sorbonne)

2. 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

3. 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

4. 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.

5. 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

6. 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.

7. 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

8. 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.

9. 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

10. 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.

11. 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…

12. 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

13. 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

14. 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

15. 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.

16. 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.

17. 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…

18. 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.

19. 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.

20. 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.

21. 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.