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This paper solves a long standing open problem of whether NP-completeproblems could be solved in polynomial time on a deterministic Turing machineby showing that the indistinguishable binomial decision tree can be formed in a3-SAT instance. This paper describes how to construct the decision tree andexplains why 3-SAT has no polynomial-time algorithm when the decision tree isformed in the 3-SAT instance. The indistinguishable binomial decision treeconsists of polynomial numbers of nodes containing an indistinguishablevariable pair but generates exponentially many paths connecting the clauses tobe used for sequences of resolution steps. The number of paths starting fromthe root node and arriving at a child node forms a binomial coefficient. Inaddition, each path has an indistinguishable property from one another. Due tothe exponential number of paths and their indistinguishability, if anindistinguishable binomial decision tree is constructed in which there existone or more paths generating an empty clause, the number of calculation stepsneeded to extract the empty clause is not polynomially bounded. This resultleads to the conclusion that class P is a proper subset of class NP.

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