The Real Dimension Problem is NPR-complete.

We show that computing the dimension of a semi-algebraic set of R^n is an NP-complete problem in the Blum-Shub-Smale model of computation over the reals. Since this problem is easily seen to be NPR-hard, the main ingredient of the proof is an NPR algorithm for computing the dimension.On montre que le calcul de la dimension d'un ensemble semi-algébrique de R^n est un problème NP-complet dans le modèle de Blum-Shub-Smale de calcul sur les nombres réels. Puisqu'il est facile de voir que ce problème est NPR-dur, le principal ingrédient de la preuve est un algorithme NPR de calcul de la dimension

Satisfiability of cross product terms is complete for real nondeterministic polytime Blum-Shub-Smale machines

Nondeterministic polynomial-time Blum-Shub-Smale Machines over the reals give rise to a discrete complexity class between NP and PSPACE. Several problems, mostly from real algebraic geometry / polynomial systems, have been shown complete (under many-one reduction by polynomial-time Turing machines) for this class. We exhibit a new one based on questions about expressions built from cross products only.Comment: In Proceedings MCU 2013, arXiv:1309.104

A Measure of Space for Computing over the Reals

We propose a new complexity measure of space for the BSS model of computation. We define LOGSPACE\_W and PSPACE\_W complexity classes over the reals. We prove that LOGSPACE\_W is included in NC^2\_R and in P\_W, i.e. is small enough for being relevant. We prove that the Real Circuit Decision Problem is P\_R-complete under LOGSPACE\_W reductions, i.e. that LOGSPACE\_W is large enough for containing natural algorithms. We also prove that PSPACE\_W is included in PAR\_R