Re: Improved ℙ≠ℕℙ proof

On Mon, 2024-06-03 at 22:26 +0800, wij wrote:

The P!=NP proof should be completed.
The updated proof may even be shorter, intuitive and robust !!!
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This file is intended a proof that ℙ≠ℕℙ. The contents may be updated anytime.
https://sourceforge.net/projects/cscall/files/MisFiles/PNP-proof-en.txt/download
 
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Algorithmic problem::= Problems that can be processed by asymptotic analysis.
 
ANP::= Another NP is a set defined as {q| q is a description of the algorithmic
   decision problem that provides 1. A set of certificate data C 2. A Ptime
   (Polynomial time) verification method v 3. The answer of q is 'yes' iff
   there exists a certificate c, c∈C, such that v(c) is true 4. q can be
   computed in at most O(2^|q|) steps. }.
   More precisely, ANP is the set of problems that can be solved by the
   following pseudo-C/C++ program temp_anp(q):
 
   bool temp_anp(Problem q) {           // Problem: Description of the problem
     Certificate c,begin,end;           // Certificate data can be accessed by
     begin= get_begin_certificate(q);   //   iteration, at least.
     end  = get_end_certificate(q);
     for(c=begin; c!=end; c=next(c)) {  // O(2^|n|) loop (see Note2)
       if(v(c)) return true;            // v:Certificate->{true,false}, Ptime
                                        //      verification function.
     }
     return false;
   }
 
   Note1: Relative to the Turing Machine language for ℕℙ, the reason using
         pseudo-C/C++ is that 1.C-code (almost all high level programming
         language not involving special hardware features) and TM language are
         computationally interchangeable. 2.It describes the problem more
         clearly (but not always) 3.The result 'false' is visible 4. ℕℙ
         definition does not say the certificate C and the verication v are
         given, fixed arguments. In ANP, v(c) is explicitly spedified a Ptime
         function 5.Easier for machine aided verification.
 
   Note2: The semantics of the for loop in temp_anp(q) includes nested loops for
         sets like C=C1×C2×C3×...
 
Polynomial time procedure::= (or polynomial time function) A procedure composed
   of sequential execution of O(P) number of fixed-time procedures.
   (Therefore, O(P) number of sequential Ptime procedure is equivalent to a
   single Ptime procedure)
 
Size-1-subproblem::= The problem whose size of input is less by one than that
   of the orginal problem.
 
Theorem: ANP problem can be divided into two size-1-subproblems.
   Proof: By spliting the certificate as follow:
          bool temp_anp(Problem q) {
            if(q.certificate().size()<Thresh) { // Thresh is a small constant
              return solve_thresh_case(q);
            }
            Problem q1,q2;
            split_certificate(q,q1,q2);        // split the certificate in q
            return temp_anp(q1) || temp_anp(q2); // to form q1,q2
          }
 
Assume solving some ANP problem by temp_anp(q) whose size-1-subproblem
temp_anp(q1) is solved, then the remaining task has one more information I
(i.e. whatever the computaion of temp_anp(q1) can provide) to reduce the
workload of solving the remaining task, and defined as solve_remain(q2,I)..
If ℙ=ℕℙ, which means the remaining task can be completed independently in Ptime
without I. In this sitution, solve_remain(q2,I) is equivalent to temp_anp(q2).
But the complexity of computation is W(|q|)=W(|q|-1)+ W(|q|-1)= 2^(|q|-1)*W(1),
a O(2^N) level of complexity contradicting he assumed Ptime. Therefore, ℙ≠ℕℙ.
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A typo in the last paragraph:


Assume solving some ℕℙℂ problem by temp_anp(q) whose size-1-subproblem
temp_anp(q1) is solved, then the remaining task has one more information I
(i.e. whatever the computaion of temp_anp(q1) can provide) to reduce the
workload of solving the remaining task, and defined as solve_remain(q2,I).
If ℙ=ℕℙ, which means the remaining task can be completed independently in Ptime
without I. In this sitution, solve_remain(q2,I) is equivalent to temp_anp(q2).
But the complexity of computation is W(|q|)=W(|q|-1)+ W(|q|-1)= 2^(|q|-1)*W(1),
a O(2^N) level of complexity contradicting the assumed Ptime. Therefore, ℙ≠ℕℙ.