From: wyniijj5@gmail.com   
      
   On Wed, 2025-11-19 at 15:25 +0800, wij wrote:   
   > The following is a snipet of the revised proof   
   > https://sourceforge.net/projects/cscall/files/MisFiles/PNP-pro   
   f-en.txt/download   
   >    
   > I think the idea of the proof should be valid and easy to understand. The   
   rest   
   > technical apart should be straightforward (could take pages or dozens of   
   pages,   
   > so ignored). But, anyway, something like the C/C++ description is still   
   needed.   
   > Can you find any defects?   
   >    
   > OTOH, C/C++ can be the language for for proving math. theorems, lots easier   
   than   
   > TM language to handle and understand. Opinions?   
      
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   Algorithmic Problem::= A computer problem in which the computational steps are   
   a   
       function of the problem statement's length (size). This problem can be   
       described asymptotically as the relationship between the problem size and   
       the computational steps.   
      
   Polynomial-time procedure (or Ptime procedure)::= O(P) number of consecutive,   
       fixed-sized basic operations (because the procedure is a deterministic   
       process, it is sometimes called a "function" or "operation"). Therefore, as   
       defined by O(P), O(P) number of Ptime procedures executed consecutively can   
       also be considered as a single Ptime procedure.   
      
   Reduction::= The algorithm for computation problem A can be transformed into   
       computation problem B by a Ptime procedure, denoted as A≤B (because the   
       Ptime transformation itself includes the computation of problem A, any two   
       Ptime problems can be reduced to each other).   
      
   ANP::= {q| q is a statement of a decision problem that a computer can solve in   
       O(2^|q|) steps using the following fnp algorithm template. q contains a   
       certification dataset C, card(C)∈O(2^|q|), and a Ptime verification   
   function   
       v:C->{true,false}. If ∃c,v(c)=true, then the answer to problem q is true;   
       otherwise, it is false.}   
      
       // begin_certificate is a Ptime function that retrieves the first   
       // Certificate element from the problem statement q. If this element does   
       // not exist, it returns a unique and virtual EndC element.   
       Certificate begin_certificate(Problem q);   
      
       // end_certificate is a Ptime function that retrieves the element EndC from   
       // the problem statement q.   
       Certificate end_certificate(Problem q);   
      
       // next_certificate is a Ptime function that retrieves the next element of   
       // c from the certification dataset C. If this element does not exist,   
       // return the EndC element.   
       Certificate next_certificate(Problem q, Certificate c);   
      
       // v is a Ptime function. v(c)==true if c is the element expected by the   
       // problem.   
       bool v(Certificate c);   
      
       bool fnp(Problem q) {   
         Certificate c, begin, end;   // Declare the certification data variable   
         begin= begin_certificate(q); // begin is the first certification data   
         end= end_certificate(q);     // end is the false data EndC used to   
                                      // indicate the end.   
         for(c = begin; c != end;   
             c = next_certificate(q, c)) { // At most O(2^|q|) steps.   
                                           // next_certificate(c) is a Ptime   
                                           // function to get the next   
                                           // certification data of c   
             if(v(c) == true) return true; // v: C->{true, false} is a polynomial   
                                           // time verification function.   
         }   
         return false;   
       }   
      
       Since a continuous O(P) number of Ptime functions (or instructions) can be   
       combined into a single Ptime function, at roughly this complexity analysis,   
       any Ptime function can be added, deleted, merged/split in the algorithm   
       steps without affecting the algorithm's complexity. Perhaps in the end,   
   only   
       the number of decision branches needs to be considered.   
      
       An ANP problem q can be expressed as q, where v = Ptime verification   
       function, and C = certification dataset (description).   
      
       [Note] According to Church-Turing conjecture, no formal language can   
              surpass the expressive power of a Turing machine (or algorithm,   
   i.e.,   
              the decisive operational process from parts to the whole). C   
              language can be regarded as a high-level language of Turing   
   machines,   
              and as a formal language for knowledge or proof.   
                                                                   
   Prop1: ANP = ℕℙ   
      Proof: Omitted (The proof of the equivalence of ANP and the traditional   
             Turing machine definition of ℕℙ is straightforward and lengthy,   
   and   
             not very important to most people. Since there are already thousands   
             of real-world ℕℙℂ problems available for verification, the   
   definition   
             of ANP does not rely on NDTM theory)   
      
   Prop2: An ANP problem q can be arbitrarily partitioned into two   
   subproblems   
          q1, q2⊕ q2 = q.   
      
   Prop4: ℙ=ℕℙ iff the algorithm fnp in the ℕℙ definition (or ℕℙℂ   
   algorithm) can be   
          replaced by a Ptime algorithm.   
      Proof: Omitted   
      
   Prop5: For subproblems q1 and q2 (same v), if C1 ∩ C2 = ∅, then   
          there is no information in problem q1 that is sufficient in terms of   
          order of complexity to speed up the computation of q2.   
      
   [continued in next message]   
      
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