FOM: Clarification to my Pi^0_2 comment:

Soren Riis sriis at fields.fields.utoronto.ca
Mon Apr 20 11:41:44 EDT 1998

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Clarification to my Pi^0_2 comment:
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On reflection I realize that my comment on
Pi^0_2-conjectures [Riis; 19/4/98] was confusing
and could be misunderstood. For a given Pi^0_2
conjecture B one can of course "conjecture" it is
provable in some specific subsystem of ZFC. This
"conjecture" could then be expressed as a purely
existential conjecture. I did not intend to claim
that any conjecture follows from a "plausible"
stronger, but purely existential conjecture.

The (very elementary) point I was trying to make
concerns "real life" Pi^0_2-conjectures. Consider
for example the conjecture P \neq NP. This sentence is
although certainly Pi^0_2 in some very strong sense
close in being Pi^0_1.

The reason is that we actually expect P to be quite
different from NP and it seems reasonable that any
algorithm running in time n^k, and having program
size s, are doomed to fail deciding the satisfiability
problem already for satisfiability problems of size
\leq Q(k)+ log(s) where Q is some low degree polynomial.

All algorithms running in time n^1000 will probably
already fail on input of size (1,000)^2= 1,000,000.
Provided of course we do not build in some huge
database (of size of order 2^1,000,000) into the
program. Well of course we do not know whether they will
all fail. Perhaps n^k programs first must fail for
inputs of size F_{\epsilon_0}(k)+s in which case the
P \neq NP of course would be independent of PA.

Now P \neq NP might or might not be probable in PA
(or ZFC). This I have virtually no intuition about.
However I will certainly strongly support that:

(*) "all algorithms with program size s and running in
time n^k will fail deciding the satisfiability problem
for some inputs of size \leq 2^k+s+1,000".

The elementary point I was trying to make is that though
the exponential function does not belong to the language
L=L(0,1,+,\cdot) of arithmetic, we can rephrase (*)
[which are of the form \forall x \exists y<F(x) A(x,y)]
and write it as the Pi^0_1 sentence

\forall x,z \exists y \leq z ("F^{-1}z \leq x" or A(x,y))

where "F^{-1}z \leq x" is expressed by a bounded formula
in the language L.

So P \neq NP is "essentially" Pi^0_1. What about the
Poincare conjecture and other Pi^0_2 conjectures? I
would expect them to be "essential" Pi^0_1, but I could
of course be mistaken.

Søren Riis