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Re: P!=NP proof, revised

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The claim is changed to "NPC is O(2^N)" which is stronger than "P!=NP".
The idea of reduction is removed.

The following is form
https://sourceforge.net/projects/cscall/files/MisFiles/PNP-proof-en.txt/download

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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.

ANP::= {q| q is a statement of a decision problem that a computer can solve in
    O(2^|q|) steps using the following fnp algorithmic 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<v, C>, where v = Ptime verification
    function, and C = certification dataset (description).

    [Note] Also testified by 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: ℕℙ ⊆ ANP and ANP ⊆ ℕℙ, therefore ANP=ℕℙ
          (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. Although the definition of ANP does
          not depend on NDTM theory, there are thousands of real-world ℕℙℂ
          problems available for verification and reference)

Prop2: An ANP problem q<v,C> can be arbitrarily partitioned into two subproblems
       q1<v,C1>, q2<v,C2), where C = C1∪C2.
   Proof: The certification dataset can be recursively partitioned as follows:

          bool banp(Problem q) {
            if(certificate_size(q)<Thresh) { // Thresh is a small constant
              return solve_thresh_case(q); // Solve q in constant time
            }
            Problem q1,q2;
            split_certificate(q,q1,q2); // Split the certification dataset C
                                        // to form q1,q2, in which the number of
                                        // certificates are roughly the same.
            return banp(q1) || banp(q2); // Compute the subproblems respectively
          }

Prop3: Any two ANP problems q1 and q2 can be synthesized into another ANP
       problem q, denoted as q = q1 ⊕ q2. The certification dataset C and the
       verification function v of q are defined as follows:

       C = C1 ∪ C2     // C1 and C2 are the certification datasets of q1 and q2
                       // respectively

       bool v(x) {
         return v1(x) || v2(x);  // v1 and v2 are the verification functions of
                                 // q1 and q2 respectively
       }

       Therefore, we have the identity: q1<v1,C1>⊕ q2<v2,C2> = q<v1||v2, C1∪C2>
       (The algebraic operation rules for '⊕'  are the same as those for
        addition in arithmetic).

Prop4: ℙ=ℕℙ iff the algorithm fnp in the ℕℙ definition (or ℕℙℂ algorithm) can be
       replaced by a Ptime algorithm.
   Proof: Omitted (A very direct semantic of 'equivalence')

Prop5: For ℕℙℂ subproblems q1<v,C1> and q2<v,C2> (same v, but not required), 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. I.e. the certification data of the ℕℙℂ problem are each, in the
       sense of computation, equivalently independent.
   Proof: Since ℕℙℂ is the most complex class of ℕℙ problems, by the definition
       of ANP, the order of card(C) of the ℕℙℂ problem must be of order O(2^|q|)
       ,no lower.
       We can add an AuxInfo object called 'auxiliary information' to banp to
       store the results after calculating a certain ℕℙℂ problem q, and rewrite
       it as banp2.
       Depending on the content of the auxiliary information, banp2 can
       represent possible algorithms for ANP/ℕℙℂ problems with complexities
       ranging from O(N) to O(2^N).

        bool banp2(Problem q, AuxInfo* ibuf) { // Calculate q and write the
                                      // obtained auxiliary information to *ibuf
        // Check and initialize *ibuf
        if(certificate_size(q)<Thresh) {
          return solve_thresh_case(q,ibuf);
        }
        Problem q1,q2;
        split_certificate(q,q1,q2);// The split function only requres Ptime
                                  // conversion from q to ℕℙℂ subproblems q1,q2,
                                  // others are non-issue.
        AuxInfo I;               // I stores information to help solve problems.
        if(banp2(q1,&I)==true) { // banp2(q1,I) recursively computes subproblem
                                 // q1 and stores any auxiliary information
                                 // that can be derived from q1 into I.
          write_ibuf1(ibuf,q);   // Write auxiliary information
          return true;           // The information obtained from subproblem q1
                                 // is useful? for any problem q.
        }
        bool rv=banp2_i(q2,&I);  // Solve q2 with given information I of which
                                 // any source is effective? for q2.
        write_ibuf2(ibuf,q);
        return rv;
      }

      [Note] From banp2(q1,&I), we know that the auxiliary information I is only
             meaningful to store information related to the certification
             dataset C1 of problem q1. This is because if it is not related, the
             computation of other subproblem can generate the auxiliary
             information by itself.

      The above `banp2_i` does not care which ℕℙℂ subproblem (including trivial
      problems) the given information `I` originates from. The `I` generated can
      also provide auxiliary information for almost all other ANP problems.

      Since the auxiliary information I obtained from computing a fixed
      subproblem (including trivial problem) cannot provide sufficient
      information to accelerate computation to the extent of improving the
      complexity order for any other subproblem (would require infinite space/
      time), the proposition is thus proved.

Conclusion: By Proposition 5, each certification data of the ℕℙℂ problem are
      independent in the sense of Ptime verification, the computation is
      equivalent to testing them one by one. Since the ℕℙℂ problem has O(^|q|)
      independent certification data, the complexity of the ℕℙℂ problem is
      O(2^N) (ℙ≠ℕℙ is proved in the way).
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P!=NP proof, revised wij <wyniijj5@gmail.com> - 2025-11-19 15:23 +0800
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