Indistinguishable binomial decision tree of 3-SAT: Proof of class P is a proper subset of class NP

Abstract: This paper solves a long standing open problem of whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine by showing that the indistinguishable binomial decision tree can be formed in a 3-SAT instance. This paper describes how to construct the decision tree and explains why 3-SAT has no polynomial-time algorithm when the decision tree is formed in the 3-SAT instance. The indistinguishable binomial decision tree consists of polynomial numbers of nodes containing an indistinguishable variable pair but generates exponentially many paths connecting the clauses to be used for sequences of resolution steps. The number of paths starting from the root node and arriving at a child node forms a binomial coefficient. In addition, each path has an indistinguishable property from one another. Due to the exponential number of paths and their indistinguishability, if an indistinguishable binomial decision tree is constructed in which there exist one or more paths generating an empty clause, the number of calculation steps needed to extract the empty clause is not polynomially bounded. This result leads to the conclusion that class P is a proper subset of class NP.