  By using of analytical multi-logic expresses in conjunction with
non-deterministic Turing machine the proposition was proved that algorithm of
deterministic Turing counter machine of polynomial time complexity can be
decreased to the algorithm of linear time complexity in non-deterministic
Turing counter machine. Furthermore, it was shown that existence of reduction
of polynomial time complexity to the linear time complexity by switching from
deterministic to non-deterministic Turing machine for string recognition imply
P equals to NP. Analytical generation functions of higher order logic were used
for finding of k value in Fagin's R. Theorem 24.
