Relative Universality On classes of Real Computation Hector Zenil University 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

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. Relative universality

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: 1.The scope of operation of M is arbitrary bound up to a certain computability degree in the AH 2.M computes at most the class of the recursive functions (M is Turing equivalent) (a case of 1) 3.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 1.Suppose U is such universal machine. Then U is capable to behave as any other machine M in R 2.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: 1.The class of recursive functions 2.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) 3.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.