# Implications of $\mathsf{P}\neq\mathsf{NP}$ in $\mathsf{BSS}$ model

What are implications of $\mathsf{P}\neq\mathsf{NP}$ in $\mathsf{BSS}$ model to $\mathsf{Turing}$ model and $\mathsf{Valiant's}$ counting complexity model?

In opposite direction what are implications of $\mathsf{P}=\mathsf{NP}$ in $\mathsf{BSS}$ model to $\mathsf{Turing}$ model and $\mathsf{Valiant's}$ counting complexity model?

• I think this has been asked before. Have tried searching for it? – Kaveh Aug 30 '15 at 0:05
• I got $VNP$ versus $VP$ implication in search not this. – user34945 Aug 30 '15 at 0:06
• AFAIK, the Boolean part of $P_{\mathbb{R}}$ lies in the counting hierarchy (by reduction to PosSLP) and $NP_{\mathbb{R}}$ is contained in PSPACE (See "On the complexity of numerical analysis" by Allender, Burgisser, Pedersen and Miltersen). So this should imply a separation between P and PSPACE? – Nikhil Aug 30 '15 at 12:01

$\newcommand\Ptime{\mathsf P} \newcommand\NP{\mathsf{NP}} \newcommand\poly{\mathsf{poly}}$ It is known that $\Ptime/\poly \neq \NP/\poly \implies \Ptime_{\mathbb C}\neq \NP_\mathbb{C}$ [1] where the latter two represent the classes $\Ptime$ and $\NP$ in the BSS model over the complex numbers. (You can take the contrapositive to obtain a consequence of $\Ptime_\mathbb C = \NP_\mathbb C$.) To the best of my knowledge, this is the only known result, though results of the same kind are known for other classes (in particular in Koiran's weak model).
• where in $[1]$ does it say $\mathsf{P/Poly}\neq\mathsf{NP/poly}\implies \mathsf{P_{\Bbb C}}\neq\mathsf{NP_{\Bbb C}}$? – Turbo Jul 11 '16 at 4:16
• also does failure of $\tau$ conjecture imply $\mathsf{P_{\Bbb C}}\neq\mathsf{NP_{\Bbb C}}$? or does $\mathsf{P_{\Bbb C}}\neq\mathsf{NP_{\Bbb C}}\iff \mathsf{P_{\Bbb R}}\neq\mathsf{NP_{\Bbb R}}$ hold? – Turbo Jul 11 '16 at 4:18