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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
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
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.
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
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-