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erties, and u will always denote one of the bit size norms. G¨odel’s numbering (2)
| |
for N N shows that that such is still closed with respect to finite products. (No-
× C
tice however that the beautiful numbering (3) of N∗ using primes is not polynomial
time computable; it may be replaced by another one which is in PF).
1.6. P/NP problem. By definition, a subset E U belongs to the class P
⊂
iff its characteristic function χ (equal to 1 on E and 0 outside) belongs to the
E
class PF. Furthermore, E U belongs to the class NP iff there exists a subset
∈
E′ U V belonging to P and a polynomial G such that
⊂ ×
′
u E (u,v) E with v G( u ).
∈ ⇐⇒ ∃ ∈ | | ≤ | |
Here V is another world (which may coincide with U). We will say that E is
obtained from E′ by a polynomially truncated projection.
The discussion above establishes in what sense this definition is model indepen-
dent.
Clearly, P NP. The inverse inclusion is highly problematic. A naive algorithm
⊂
calculating χ E from χ E′ by searching for v with v G( u ) and χ E′(u,v) = 1
| | ≤ | |
will take exponential time e.g. when there is no such v (because u is a bit size
| |
function). Of course, if one can treat all such v in parallell, the required timewill be
polynomial. Or else, if an oracle tells you that u E and supplies an appropriate v,
∈
you can convince yourself that this is indeed so in polynomial time, by computing
χ (u,v) = 1.
E′
Notice that the enumerable sets can be alternatively described as projections of
decidable ones, and that in this context projection does create undecidable sets.
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Nobody was able to translate the diagonalization argument used to establish this
to the P/NP domain. M. Freedman ([Fr2]) suggested an exciting new approach
to the problem P = NP(?), based upon a modification of Gromov’s strategy for
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describing groups of polynomial growth.
It has long been known that this problem can be reduced to checking whether
some very particular sets – NP–complete ones – belong to P. The set E U is
⊂
called NP–complete if, for any other set D V,D NP, there exists a function
⊂ ∈
f : V U,f PF, such that D = f−1(E), that is, χ (v) = χ (f(v)). We will
D E
→ ∈
sketch the classical argument (due to S. Cooke, L. Levin, R. Karp) showing the
existence of NP–complete sets. In fact, the reasoning is constructive: it furnishes
a polynomially computable map producing f from the descriptions of χ E′ and of
the truncating polynomial G.
In order to describe one NP–complete problem, we will define an infinite family
of Boolean polynomials b indexed by the following data, constituting objects u of
u
the constructive world U. One u is a collection
m N; (S ,T ),...,(S ,T ), (10)
1 1 N N
∈
where S , T 1,...,m , and b is defined as
i i u
⊂ { }
N
b (x ,...,x ) = 1+ (1+x ) x . (11)
u 1 m k j
iY=1 kY∈Si jY∈Ti
 
The size of (10) is by definition u = mN.
| |
Put
E = u U v Fm, b (v) = 1 .
{ ∈ |∃ ∈ 2 u }
Using the language of Boolean truth values, one says that v satisfies b if b (v) = 1,
u u
and E is called the satisfiability problem, or SAT.
1.6.1. Claim. E NP.
∈
In fact, let
E′ = (u,v) b (v) = 1 U ( ∞ F ). (12)
{ | u } ⊂ × ⊕i=1 2
Clearly, E isthe full projection of E′. A contemplation will convince the reader that
E′ P. In fact, we can calculate b (v) performing O(Nm) Boolean multiplications
u
∈
and additions. The projection to E can be replaced by a polynomially truncated
projection, because we have to check only v of size v m.
| | ≤
1.6.2. Claim. E is NP–complete.
In fact, let D NP, D A where A is some universe. Take a representation of
∈ ⊂
D as a polynomially truncated projection of some set D′ A B,D′ P. Choose
⊂ × ∈
a normal, say Turing, model of computation and consider the Turing protocols of
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computation of χ D′(a,b) with fixed a and variable polynomially bounded b. As we
have explained above, for a given a, any such protocol can be imagined as a table
of a fixed polynomially bounded size whose rows are the consecutive states of the
computation. In the “microscopic” description, the positions in this table can be
filled only by 0 or 1. In addition, each row is supplied by the specification of the