FOM: 106:Degenerative Cloning

Harvey Friedman friedman at math.ohio-state.edu
Fri May 4 10:57:23 EDT 2001


We consider finite sets of pairwise disjoint circles in the plane. The
circles are circumferences of nondegenerate circles.

Let X be a finite set of pairwise disjoint circles. A part of X is a set
consisting of a circle C in X together with the  circles in X that are
surrounded by C.

Single cloning in X occurs as follows:

a. A part Y of X is chosen.
b. A copy Y' of Y is placed to the side of Y in X.

This results in a larger finite set X' containing X of pairwise disjoint
circles.

In b, we mean that Y' is isomorphic to Y in the sense that Y' is a finite
set of pairwise disjoint circles and there is a bijection from Y onto Y'
that preserves the "surrounding" relation among circles (which is a strict
partial ordering). And also that the largest circle in Y' lies outside the
largest circle in Y, but inside any circle in X that surrounds the largest
circle in Y.

Multiple cloning in X occurs as follows:

a. A part Y of X is chosen.
b. One or more copies, Y1,Y2,...,Yn, of Y are placed, side by side, to the
side of Y in X.

Single degenerative cloning in X occurs as follows:

i. Single cloning in X occurs with some Y,Y'.
ii. One circle is removed from Y and one circle is removed from Y'.

Multiple degenerative cloning in X occurs as follows:

i. Multiple cloning in X occurs with some Y,Y1,...,Yn, n >= 1.
ii. One circle is removed from each of Y,Y1,...,Yn.

We now consider how many times degenerative cloning can occur from a given
finite set of pairwise disjoint circles. Note that degenerative cloning can
always be performed on any nonempty finite set of pairwise disjoint
circles. So successive degenerative cloning ends if and when one reaches
the empty set of circles.

THEOREM 1. It is impossible for multiple degenerative cloning to occur
infinitely many times in succession in any given finite set of pairwise
disjoint circles. The multiple degenerative cloning relation among finite
sets of pairwise disjoint circles has ordinal epsilon_0.

THEOREM 2. The largest number of times that single degenerative cloning can
occur in any given finite set of k >= 1 pairwise disjoint circles is
exactly one less than an exponential stack of 2's of height k. This maximum
is realized with k concentric circles. E.g., 1,  3,  15,  65,535,
(2^65,536)-1 for 1,2,3,4,5 concentric circles, respectively.

We can even give an exact computation of the largest number of successive
single degenerative cloning steps in a given finite set of pairwise
disjoint circles as follows.

We inductively define #(A) for any finite set A of pairwise disjoint
circles. Let B1,...,Bn be the maximal proper  parts of A. Note that A is
empty if and only if n >= 0.

1. A does not have a largest circle.  Define #(A) = #B1 + ... + #Bn.
3. A has a largest circle. Define #A = (2^(#B1 + ... + #Bn))-1.

THEOREM 3. Let A be a finite set of pairwise disjoint circles. The largest
number of successive single degenerative cloning steps in A is exactly
#(A).

Restrained multiple degenerative cloning in X occurs when multiple
degenerative cloning in X occurs and the resulting finite set of disjoint
circles (including the unaffected parts of X) has at most twice as many
circles as X.

THEOREM 3. For any finite set X of pairwise disjoint circles, there is a
bound on the number of times restrained multiple degenerative cloning can
occur in X. This fact cannot be proved in Peano Arithmetic. It is
equivalent over EFA (exponential function arithmetic) to the 1-consistency
of PA.

For each k, let RMDC(n) be the largest number of successive restrained
multiple degenerative cloning steps in n concentric circles.

THEOREM 4. RMDC(n) is an epsilon_0 recursive function but eventually
dominates every <epsilon_0 recursive function.

Note that RMDC(1) = 1 and RMDC(2) = 5.

To give bounds for RMDC(3), we introduce a standard form of the Ackermann
hierarchy. We define functions A_k, k >= 1.

A_1(n) = 2n. For k >= 1, A_k+1(n) = A_kA_k...A_k(1), where this is n
repeated applications of the function A_k. The unary Ackermann function is
given by A(n) = A(n,n).

Note that A_2 is base 2 exponentiation and A_3 is base 2 iterated
exponentiation.

THEOREM 5. RMDC(3) > A_4(64,000).

Recall that for RMDC(3), we start with a single circle surrounding a circle
surrounding a circle. We now see what happens starting with two copies of a
single circle surrounding a circle surrounding a circle. Let RMDC(3,3) be
the largest number of successive restrained multiple degenerative cloning
steps in this situation.

THEOREM 6. RMDC(3,3) > AA(5).

We now come to RMDC(4).

THEOREM 7. RMDC(4) > AA...A(1), where there are A(5) A's.

Lower bounds for RMDC(n) are perhaps most dramatically stated in terms of
independence results.

THEOREM 8. Let n >= 4. RMDC(n) is larger than the number of steps of any
Turing machine that can be proved to halt using at most A_3(n + 100)
symbols in n-3 quantifier induction. RMDC(n) is smaller than the number of
steps of some Turing machine that can be proved to halt using at most A_2(n
+ 100) symbols in n-2 quantifier induction.

The lower bounds in Theorems 4-8 are conveniently stated but very crude,
and can be improved considerably - especially Theorem 7.

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