Re: Proof of P and NP

On 05/23/2025 04:43 PM, Suresh Devanathan wrote:

Suppose you are trying to solve a Boolean satisfiability problem C[x]. Now
convert the problem into probabilistic domain.
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For probability vector x
C[x] = p
      = p + 0*K*(B - p)
      = lim K->infinity p + 1/K*K(B- p)
      = B
>
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For friend
   in case a satisfiability, B = 1
   in case untestable, assume x_n=  0.5, B = 0
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For enemy,
  in case a satisfiability, B = 0
   in case untestable, assume x_n = 0.5, B = 1
>
>
since by cook's thesis, even in 1 problem in NP proven to be P , every
problem proven P. If
reverse result is always true of enemy, the number of permutation is at
least as great as 2^N
where N is number of inputs, showing problem is be NP.
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complete proof
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-suresh

There's a great book from the 90's about polynomial-time
approximations, not solutions yet approximations, to
non-polynomial problems, that usually employ binary search
and interpolation and extrapolation among ordered relations
and computational terms under addition formulae and product
rules, and scalar products.
Then, approximations of course, numerical methods, they
have nominally non-zero error-terms about their modeled
and asymptotic error-bounds, and whether they're so in
the limit variously is or isn't so.
In the probabilistic setting then, there are also issues
about the counting-arguments and measure-problems involved
in discrete and continuous distributions, about them being
so or not being so in the inductive-limit, then as with
regards to various infinite-limits, in some sorts continuum-limits,
since there are various law(s), plural, of large numbers what
get involved about at least three different definitions of
continuous domains, and their relations to each other in
structure of inductive-limits, various infinite-limits,
and whether continuum-limits, what's variously called
"non-standard" or "super-classical".
So, about convergence and its transcendental form emergence,
has that otherwise you're just positing a successful guessing.
Approximations have error terms, and statistics' random
variables largely have unknown distributions.