# [FOM] 435: Kernel Structure Theory 1

Harvey Friedman friedman at math.ohio-state.edu
Mon Jul 5 09:57:30 EDT 2010

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION.

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This is essentially a reworking of Digraph Kernel Structure Theory 1 http://www.cs.nyu.edu/pipermail/fom/2010-July/014853.html

We are dropping "Digraph" from the name, only because KST is a nice
acronym - like BRT.

This reworking, we believe, is particularly important. It lays out a
very attractive and deep program with initial results. The setup is
very well motivated, borrowing from the fundamental definition of
kernel in digraph theory.

In the interest of brevity, we have removed some background material,
which will be mentioned in more elaborate presentations.

Furthermore, there is a plausible path to incrementally carrying out
the program in full.

Kernels are very fundamental, and we believe that the program will
jump naturally into diverse areas of mathematics.

THIS POSTING IS ENTIRELY SELF CONTAINED.

KERNEL STRUCTURE THEORY 1
by
Harvey M. Friedman
July 5, 2010

1. DIGRAPHS AND KERNELS.
2. ORDER INVARIANCE AND PIECEWISE LINEARITY.
3. f CLOSED SETS AND TOWERS.
4. INITIAL RESULTS.
5. THE FUTURE.

1. DIGRAPHS AND KERNELS.

Kernels in digraphs are intensively studied with a large literature,
including applications to game theory, computer science, and elsewhere.

A digraph is a pair G = (V,R), where R is contained in V x V. The
vertices of G are the elements of V, and the edges of G are the
elements of R.

A dag is a digraph in which there are no cycles.

We say that E is independent in G if and only if E is a subset of V,
where there is no edge from any element of E to any element of E.

We say that E is a kernel in G if and only if E is independent in G,
and for all x in V\E, there is an edge from x to some element of E.

The following is due to J. Von Neumann 1944, in his book with
Morgenstern on Game Theory.

THEOREM 1.1. There is a unique kernel in every finite dag.

A kernel of A in G is a kernel of the induced subdigraph G|A.

2. ORDER INVARIANCE AND RATIONAL SEMILINEARITY.

Let Q be the set of all rational numbers.

We say that x,y in Q^k are order equivalent if and only if for all
1 <= i,j <= k, x_i < x_j iff y_i < y_j.

We say that A contained in Q^k is order invariant if and only if for
all order equivalent x,y in Q^k, x in A iff y in A.

An order invariant digraph on Q^k is a digraph (Q^k,R), where R is an
order invariant subset of Q^2k.

We say that (Q^k,R) is downward if and only if x R y implies max(x) >
max(y).

We say that P contained in Q^r is rational semilinear linear if and
only if P is a Boolean combination of linear inequalities with
rational coefficients.

3. P CLOSED SETS AND TOWERS.

Let P be contained in Q^3k. We say that A contained in Q^k is P closed
if and only if A is nonempty and P[A^2] is contained in A.

TEMPLATE I. Let P,J contained in Q^3k be rational semilinear. In every
downward order invariant graph on Q^k, some P closed set has a J
closed kernel.

A P closed tower is a nonempty finite sequence of nonempty subsets
A_1,...,A_n of Q^k, where for all 1 <= i < n, A_i union P[A_i^2] is
contained in A_i+1.

TEMPLATE II. Let P,J contained in Q^3k be piecewise linear. In every
downward order invariant graph on Q^k, some P closed tower of finite
sets of length r has a J closed tower of kernels.

TEMPLATE III. Let P,J contained in Q^3k be rational semilinear with
coefficients from {0,1}. In every downward order invariant graph on
Q^k, some P closed tower of sets of length r, involving only
numerators and denominators of magnitude at most (8kr)!!, has a J
closed tower of kernels.

Note that each instance of Template II is explicitly Pi02, and each
instance of Template III is explicitly Pi01.

4. INITIAL RESULTS.

We have specific choices of P,J such that all three Templates result
in Pi01 independent statements.

For each k, define P_k(x,y,z) if and only if

i. x,y,z are in Q^k.
ii. Every coordinate of z is a coordinate of x or a coordinate of y or
0.

For each k, define J_k(x,y,z) if and only if

i. x,y,z are in Q^k.
ii. z is the result of starting with x or y, and adding 1 to all of
its nonnegative coordinates.

RENARK. P_k and J_k are particularly simple rational semilinear sets.
Both can be presented in terms of order, 0, and the +1 function.

PROPOSITION 4.1. In every downward order invariant graph on Q^k, some
P_k closed set has a J_k closed kernel.

PROPOSITION 4.2. In every downward order invariant graph on Q^k, some
P_k closed tower of finite sets of length r has a J_k closed tower of
kernels.

PROPOSITION 4.3. In every downward order invariant graph on Q^k, some
P_k closed tower of sets of length r, involving only numerators and
denominators of magnitude at most (8kr)!!, has a J_k closed tower of
kernels.

Note that Proposition 4.2 is explicitly Pi02, and Proposition 4.3 is
explicitly Pi01.

THEOREM 4.4. Propositions 4.1 - 4.3 are provable in SRP+ but not from
any consequence of SRP that is consistent with RCA_0 (EFA can be used
here for 4.2, 4.3). Propositions 4.1 - 4.3 are provably equivalent,
over WKL_0, to Con(SRP) (EFA can be used here for 4.2, 4.3). These
results hold if we fix k to be any particular sufficiently large
integer in these Propositions.

Here SRP+ = ZFC + "for all k there exists a limit ordinal with the k-
SRP. SRP = ZFC + {there exists a limit ordinal with the k-SRP}_k. The
k-SRP asserts that every partition of the unordered k-tuples from
lambda into two pieces has a homogeneous set that is a stationary
subset of lambda.

EFA is exponential function arithmetic.

5. THE FUTURE.

The Holy Grail is to present an algorithm for determining the truth
value of all instances of Templates A,B,C, and prove in SRP+ that it
is correct. Closely related, is to show in EFA that every instance of
these Templates is provable or refutable in SRP. We know that the
former is impossible with SRP (unless SRP is inconsistent) - because
of Theorem 4.4. We know that the latter is impossible in a finite
fragment of SRP (unless SRP is inconsistent).

The plan is to accomplish this for incrementally increasing fragments
of the rational semilinear subsets of the Q^3k.

Note that remark in section 4. More specifically, suppose we are given
a set W of rational linear terms. W may be finite or finite.

The W sets are given by a Boolean combination of inequalities s < t
between terms s,t from V.

For Propositions 4.1 - 4.3, we have used only the set W_0 of terms

x_i, i >= 1.
x_i + 1, i >= 1.
0.

This is a very impoverished set of terms, yet we obtain the SRP level
from Propositions 4.1 - 4.3.

The first order of business should be a complete analysis of Templates
I - III with W_0 sets only.

I have some emergencies to deal with, but I expect that this is going
to be pretty easy to do.

Next, is a very long stretch of incrementally expanding W_0.

This setup has the feel of permanence. But I have said that before.
This time it may well be true.

A parallel enterprise is to return to much larger cardinals. This is
also a high priority. The more it becomes clear that this is the setup
of permanent value, the earlier I will return to the very large
cardinals.

**********************

manuscripts. This is the 435th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-349 can be found athttp://www.cs.nyu.edu/pipermail/fom/2009-August/014004.html
in the FOM archives.

350: one dimensional set series  7/23/09  12:11AM
351: Mapping Theorems/Mahlo/Subtle  8/6/09  10:59PM
352: Mapping Theorems/simpler  8/7/09  10:06PM
353: Function Generation 1  8/9/09  12:09PM
354: Mahlo Cardinals in HIGH SCHOOL 1  8/9/09  6:37PM
355: Mahlo Cardinals in HIGH SCHOOL 2  8/10/09  6:18PM
356: Simplified HIGH SCHOOL and Mapping Theorem  8/14/09  9:31AM
357: HIGH SCHOOL Games/Update  8/20/09  10:42AM
358: clearer statements of HIGH SCHOOL Games  8/23/09  2:42AM
359: finite two person HIGH SCHOOL games  8/24/09  1:28PM
360: Finite Linear/Limited Memory Games  8/31/09  5:43PM
361: Finite Promise Games  9/2/09  7:04AM
362: Simplest Order Invariant Game  9/7/09  11:08AM
363: Greedy Function Games/Largest Cardinals 1
364: Anticipation Function Games/Largest Cardinals/Simplified 9/7/09
11:18AM
365: Free Reductions and Large Cardinals 1  9/24/09  1:06PM
366: Free Reductions and Large Cardinals/polished  9/28/09 2:19PM
367: Upper Shift Fixed Points and Large Cardinals  10/4/09 2:44PM
368: Upper Shift Fixed Point and Large Cardinals/correction 10/6/09
8:15PM
369. Fixed Points and Large Cardinals/restatement  10/29/09 2:23PM
370: Upper Shift Fixed Points, Sequences, Games, and Large Cardinals
11/19/09  12:14PM
371: Vector Reduction and Large Cardinals  11/21/09  1:34AM
372: Maximal Lower Chains, Vector Reduction, and Large Cardinals
11/26/09  5:05AM
373: Upper Shifts, Greedy Chains, Vector Reduction, and Large
Cardinals  12/7/09  9:17AM
374: Upper Shift Greedy Chain Games  12/12/09  5:56AM
375: Upper Shift Clique Games and Large Cardinals 1graham
376: The Upper Shift Greedy Clique Theorem, and Large Cardinals
12/24/09  2:23PM
377: The Polynomial Shift Theorem  12/25/09  2:39PM
378: Upper Shift Clique Sequences and Large Cardinals  12/25/09 2:41PM
379: Greedy Sets and Huge Cardinals 1
380: More Polynomial Shift Theorems  12/28/09  7:06AM
381: Trigonometric Shift Theorem  12/29/09  11:25AM
382: Upper Shift Greedy Cliques and Large Cardinals  12/30/09 2:51AM
383: Upper Shift Greedy Clique Sequences and Large Cardinals 1
12/30/09  3:25PM
384: THe Polynomial Shift Translation Theorem/CORRECTION 12/31/09
7:51PM
385: Shifts and Extreme Greedy Clique Sequences  1/1/10  7:35PM
386: Terrifically and Extremely Long Finite Sequences  1/1/10 7:35PM
387: Better Polynomial Shift Translation/typos  1/6/10  10:41PM
388: Goedel's Second Again/definitive?  1/7/10  11:06AM
389: Finite Games, Vector Reduction, and Large Cardinals 1 2/9/10
3:32PM
390: Finite Games, Vector Reduction, and Large Cardinals 2 2/14/09
10:27PM
391: Finite Games, Vector Reduction, and Large Cardinals 3 2/21/10
5:54AM
392: Finite Games, Vector Reduction, and Large Cardinals 4 2/22/10
9:15AM
393: Finite Games, Vector Reduction, and Large Cardinals 5 2/22/10
3:50AM
394: Free Reduction Theory 1  3/2/10  7:30PM
395: Free Reduction Theory 2  3/7/10  5:41PM
396: Free Reduction Theory 3  3/7/10  11:30PM
397: Free Reduction Theory 4  3/8/10  9:05AM
398: New Free Reduction Theory 1  3/10/10  5:26AM
399: New Free Reduction Theory 2  3/12/10  9:36AM
400: New Free Reduction Theory 3  3/14/10  11:55AM
401: New Free Reduction Theory 4  3/15/10  4:12PM
402: New Free Reduction Theory 5  3/19/10  12:59PM
403: Set Equation Tower Theory 1  3/22/10  2:45PM
404: Set Equation Tower Theory 2  3/24/10  11:18PM
405: Some Countable Model Theory 1  3/24/10  11:20PM
406: Set Equation Tower Theory 3  3/25/10  6:24PM
407: Kernel Tower Theory 1  3/31/10  12:02PM
408: Kernel tower Theory 2  4/1/10  6:46PM
409: Kernel Tower Theory 3  4/5/10  4:04PM
410: Kernel Function Theory 1  4/8/10  7:39PM
411: Free Generation Theory 1  4/13/10  2:55PM
412: Local Basis Construction Theory 1  4/17/10  11:23PM
413: Local Basis Construction Theory 2  4/20/10  1:51PM
414: Integer Decomposition Theory  4/23/10  12:45PM
415: Integer Decomposition Theory 2  4/24/10  3:49PM
416: Integer Decomposition Theory 3  4/26/10  7:04PM
417: Integer Decomposition Theory 4  4/28/10  6:25PM
418: Integer Decomposition Theory 5  4/29/10  4:08PM
419: Integer Decomposition Theory 6  5/4/10   10:39PM
420: Reduction Function Theory 1  5/17/10   2:53AM
421: Reduction Function Theory 2  5/19/10   12:00PM
422: Well Behaved Reduction Functions 1  5/23/10  4:12PM
423: Well Behaved Reduction Functions 2  5/27/10  3:01PM
424: Well Behaved Reduction Functions 3  5/29/10  8:06PM
425: Well Behaved Reduction Functions 4  5/31/10  5:05PM
426: Well Behaved Reduction Functions 5  6/2/10  12:43PM
427: Finite Games and Incompleteness 1  6/10/10  4:08PM
428: Typo Correction in #427  6/11/10  12:11AM
429: Finite Games and Incompleteness 2  6/16/10  7:26PM
430: Finite Games and Incompleteness 3  6/18/10  6:14PM
431: Finite Incompleteness/Combinatorially Simplest  6/20/10  11:22PM
432: Finite Games and Incompleteness 4  6/26/10  8:39PM
433: Finite Games and Incompleteness 5  6/27/10  3:33PM
434: Digraph Kernel Structure Theory 1  7/4/10  3:17PM

Harvey Friedman