  We introduce a novel variant of BSS machines called Separate Branching BSS
machines (S-BSS in short) and develop a Fagin-type logical characterisation for
languages decidable in non-deterministic polynomial time by S-BSS machines. We
show that NP on S-BSS machines is strictly included in NP on BSS machines and
that every NP language on S-BSS machines is a countable union of closed sets in
the usual topology of R^n. Moreover, we establish that on Boolean inputs NP on
S-BSS machines without real constants characterises a natural fragment of the
complexity class existsR (a class of problems polynomial time reducible to the
true existential theory of the reals) and hence lies between NP and PSPACE.
Finally we apply our results to determine the data complexity of probabilistic
independence logic.
