This paper was motivated by the following two questions which arise in the
theory of complexity for computation over ordered rings in the now famous c
omputational model introduced by Blum, Shub and Smale:
(i) is the answer to the question P = ?NP the same in every real-closed fie
ld?
(ii) if P not equal NP for (FS, does there exist a problem of R which is NP
but neither P nor NP-complete?
Some unclassical complexity classes arise naturally in the study of these q
uestions. They are still open, but we could obtain unconditional results of
independent interest.
Michaux introduced /const complexity classes in an effort to attack questio
n (i). We show that A(R)/const = A(R), answering a question of his. Here A
is the class of real problems which are algorithmic in bounded time. We als
o prove the stronger result: PAR(R)/const = PAR(R), where PAR stands for pa
rallel polynomial time. In our terminology, we say that R is A-saturated an
d PAR-saturated. We also prove, at the nonuniform level, the above results
for every real-closed field. It is not known whether R is P-saturated. In t
he case of the reals with addition and order we obtain P-saturation land a
positive answer to question (ii)). More generally, we show that an ordered
Q-vector space is P-saturated at the nonuniform level (this almost implies
a positive answer to the analogue of question (i)).
We also study a stronger notion that we call P-stability. Blum, Cucker, Shu
b and Smale have (essentially) shown that fields of characteristic 0 are P-
stable. We show that the reals with addition and order are P-stable, but re
al-closed fields are not.
Questions (i) and (ii) and the /const complexity classes have some model th
eoretic flavor. This leads to the theory of complexity over "arbitrary" str
uctures as introduced by Poizat. We give a summary of this theory with a sp
ecial emphasis on the connections with model theory and we study /const com
plexity classes from this point of view. Note also that our proof of the PA
R-saturation of R shows that an o-minimal structure which admits quantifier
elimination is A-saturated at the nonuniform level. (C) 1999 Elsevier Scie
nce B.V. All rights reserved.