# Theory of complexity – Is there a reduction in polynomial time of a problem NP-difficult to complete (tautology)?

The statement is: If a problem $$P$$ is $$mathbf {NP}$$-hard, then there is a reduction of $$text {FALSIFIABLE}$$ at $$P$$. ($$text {FALSIFIABLE}$$ being the set of formulas for which there is an assignment that makes the formula false; this is trivially equal to the complement of $$text {TAUT}$$.)

This statement is correct.

Why? Simply because $$text {FALSIFIABLE}$$ is $$mathbf {NP}$$-Achevée. You can prove it exactly the same way $$text {SAT}$$ is proven to be $$mathbf {NP}$$-complete, by reversing only the truth values ​​"true" and "false" (that is, instead of looking for an assignment that makes the formula true, you are looking for an assignment that makes the false formula).