126 K. Meer – C. Michaux
4.2 Linear and additive machines
Considering the order-free linear resp. additive BSS-model the P versus NP ques- tion can be answered.
Theorem 4.3 [80], [81]
P
= lin
6= NP
= lin
and P
= add
6= NP
= add
. The proofs again use non-uniform methods; to show that deciding integer-solvability
of linear equational systems separates the according classes. The result was inde- pendently proven by Smale, see [107]. Koiran [64] mentioned the Knapsack-problem
to establish the same separation. For the ordered case things seem to be much more difficult. Non-uniform approaches
so far have not suffice to show P
≤ lin
6= NP
≤ lin
. In fact, Meyer auf der Heide [88] has solved the Knapsack-problem non-uniformly in polynomial time; however, the no-
tion of non-uniformity used in the latter work is more general than that of families of circuits with polynomial size
The concept of digital nondeterminism, which turned out to be important for both the weak and the full BSS-model, is of no special meaning in linear and additive
settings. As proved by Koiran we have Theorem 4.4
[64] DNP
≤ lin
= NP
≤ lin
, DNP
= lin
= NP
= lin
, DNP
≤ add
= NP
≤ add
and DNP
= add
= NP
= add
. The basic idea to switch from arbitrary to digital nondeterminism is guessing the
coding of an accepting computation path instead of guessing real numbers which force the according machine to take that path. Then it has to be checked whether
the guessed path can appear during a computation the same idea is also used to gain the above mentioned result DNP
w
= NP
1 w
. Additionally let us mention that all the above NP -classes are decidable in the cor-
responding framework. For NP
= lin
, NP
= add
cf. [81], for NP
≤ lin
see [41] and for NP
≤ add
cf. [43] as well as [28] and [116]. The existence of complete problems in the additive models is settled in [30]. For linear machines a positive answer is given in [47], even
though in that case no universal machine exists see [81] and section 6 below. This seems to be related to the non-existence of so called universal machines cf. [81] and
section 6 below.
4.3 Full BSS-model
Kategori