# $P \neq NP$ and determinism

Suppose $P \neq NP$. Does it imply that there exists some superpolynomial time bound, such that any $NP$-complete problem, like SAT, can be used to simulate an arbitrary deterministc Turing Machine working in that time bound?

Rephrasing does $P \neq NP$ imply that there exists some class $D$ of languages solvable by a deterministic Turing Machine, such $P \subsetneq\ D \subseteq NP$ and SAT is $D$-hard?

• $D$-hard under what kind of reduction? Since SAT is $NP$-hard, doesn't that automatically make SAT $D$-hard? Any language in $NP$ (and thus any language in $D$) can be reduced to SAT in polytime. – Tom van der Zanden May 4 '15 at 14:54
• Your rephrasing sounds like a question that Ladner's Theorem answers. Or do you mean something else? – Kyle Jones May 4 '15 at 16:14

On the other hand, I suspect there is no known proof that if unambiguousGC(polylog, NC) ⊈ coNP/poly then there is a superpolynomial time-constructible function $T$ and a superlogarithmic space-constructible function $S$ such that