For note that although this statement originates in theoretical computer science, it may be easily formulated as statements about natural numbers. In particular, \(\textbf{P} \neq \textbf{NP}\) is equivalent to the statement that for all indices \(e\) and exponents \(k\), there exists a propositional formula \(\phi\) such that the deterministic Turing machine \(T_e\) does not correctly decide \(\phi\)’s membership in \(\sc{SAT}\) in \(\lvert \phi\rvert^k\) steps. e. a statement \(\Theta\) of the