arXiv:0810.0033v1
[cs.CC]
30 Sep 2008
COMPLEXITY CLASSES AS MATHEMATICAL AXIOMS
MICHAEL H. FREEDMAN
Abstract.
Treating a conjecture,
P
#
P
negationslash
=
NP
, on the separation of complex
ity classes as an axiom, an implication is found in three manifold topology with
little obvious connection to complexity theory. This is reminiscent of Harvey
Friedman’s work on finitistic interpretations of large cardinal axioms.
1.
Introduction
This short paper introduces a new subject with a simple example. The theory
of computation defines a plethora of complexity classes.
While the techniques
of diagonalization and oracle relativization have produced important separation
results, for nearly forty years the most interesting (absolute) separation conjectures,
such as
P
negationslash
=
NP
remain unproven, and with the invention of ever more complexity
classes, analogous separation conjectures have multiplied in number.
With no prospect in sight for proving these conjectures (within
ZFC
), we pro
pose considering them instead as potential axioms and looking for what implications
they might have in mathematics as a whole. This program is analogous to the search
for interesting “finitistic” consequences of large cardinal axioms, an area explored
by Harvey Friedman and collaborators ([4]). (Although, in the latter case, the large
cardinal axioms are actually known to be independent of
ZFC
.)
What would be the best possible theorem in this subject? It would be to postu
late a very weak separation “axiom,” say
P
negationslash
=
PSPACE
, and prove the Riemann
hypothesis, i.e.
an important mathematical result apparently far removed from
complexity theory. Of course, we should be more modest. We will assume a more
technical but well accepted separation “axiom”
P
#
P
negationslash
=
NP
, which we call Axiom
A, and prove a theorem, Theorem A, in knot theory.
The theorem is easily and
briefly expressed in terms of classical notions such as “girth” and “Dehn surgery”
and appears to be as close to current research topics in knot theory as the known
finitistic implications of the large cardinal axioms are to research in Ramsey theory,
to continue that analogy. Theorem A is extremely believable but seems to exist in
a “technique vacuum.” What makes the theorem interesting is that it sounds both
“very plausible” and “impossible to prove.”
2.
Theorem A
We consider smooth links
L
of finitely many components in
R
3
. The girth of
L
,
girth
(
L
) =
g
(
L
), is defined as the minimum over all knot diagrams
D
for
L
(say
in the
xz
plane) of the maximum over all lines
z
= constant of the cardinality of
the horizontal intersection

D
∩
(
z
= constant)

. We also consider link diagrams
D
′
for
L
which may have larger girth
g
(
D
′
)
≥
g
(
L
) defined by the same formula. The
Date
: 9/22/2008, Version 0.02.