On the power of real Turing machines over binary inputs Felipe Cucker, Dima Grigoriev
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Felipe Cucker, Dima Grigoriev. On the power of real Turing machines over binary inputs. SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 1997. hal-03049470
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On the p ower of real Turing machines
over binary inputs
Felipe Cucker
Universitat Pompeu Fabra
Balmes Barcelona
SPAIN
e mail cucker upf es
y
Dima Grigoriev
Depts of Comp Science and Mathematics
Penn State University
University Park PA
USA
e mail dima cse psu edu
In recent years the study of the complexity of computational problems
involving real numbers has b een an increasing research area A founda
tional pap er has b een where a computational mo del the real Turing
machine for dealing with the ab ove problems was developed
One research direction that has b een studied intensively during the last
two years is the computational p ower of real Turing machines over binary
inputs The general problem can b e roughly stated in the following way Let
us consider a class C of real Turing machines that work under some resource
b ound for instance p olynomial time branching only on equality etc If
we restrict these machines to work on binary inputs i e nite words over
f g they de ne a class of binary languages D The question is what can
we say ab out D dep ending on C
More formally let us denote by IR the direct sum of countably many
copies of IR and let P IR b e the set of its subsets Also let us denote by
the subset f g of IR and as usual by the subset of IR consisting
Partially supp orted by DGICyT PB the ESPRIT BRA Program of the EC
under contracts no and pro jects ALCOM I I and NeuroCOLT
y
Partially supp orted by Volkswagen Stiftung
of those vectors whose comp onents are in Given any complexity class
C P IR we de ne its Bo olean part to b e the class of binary languages
BP C fX X C g
Our problem now can b e stated as given a complexity class of real sets C
characterize BP C
A p ossible origin of the problem is the recent interest in the computa
tional p ower of neural networks The rst results characterized the p ower of
nets with rational weights working within p olynomial time by showing that
they compute exactly the sets in P cf The same problem was then
considered for neural networks with real weights and it was shown that the
p ower of these nets working within p olynomial time is exactly P poly cf
and
This latter problem considers in a natural way a setting in which an
algebraic mo del having real constants op erates over binary inputs A next
step was then taken by P Koiran who passed from a structured mo del
the neural net to a general one the real Turing machine However
he did not deal with the real Turing machine as it was introduced in
but with a restricted version of it that can do only a mo derate use of mul
tiplication namely all rational functions intermediately computed in the
input variables as well as in the machine s constants must have degree and
co e cient size b ounded by the running time For this weak mo del he con
sidered the class P of sets accepted in p olynomial time and he proved that
W
BP P P poly see
W
Subsequently several pap ers exhibited new results on Bo olean parts In
it was shown that BP PAR PSPACE poly where PAR is the class
W W
of subsets of IR decided in weak parallel p olynomial time Also for additive
machines i e real Turing machines that do not p erform multiplications at
all it was shown in that BP P P poly and that BP NP
add add
NP poly Here P and NP denote the obvious classes but we recall that
add add
the nondeterministic guesses in this mo del are real numbers Moreover if
the machines are order free i e they are required to branch only on equality
NP P and that BP NP tests we now have that BP P
add add
These results were subsequently generalized in to all the levels of the
p olynomial hierarchy constructed up on NP or NP
add
add
None of the mentioned results was done for the unrestricted real Turing
machine In fact for this case it was even asked whether it existed a subset
of not b elonging to the BP P cf First steps in this direction
IR
were done in where it is shown that if we consider order free machines
then we have the inclusion BP P BPP the class of sets decided by
IR
randomized machines in p olynomial time with b ounded probability error
see ch as well as a p ositive answer to the question ab ove In fact if
PH is the p olynomial hierarchy constructed up on NP the existence of
IR IR
binary languages not b elonging to BP PH and a fortiori nor to BP P
IR IR
was also proved in
The aim of this pap er is to prove that BP PAR PSPACE poly
IR
where PAR is the class of sets computed in parallel p olynomial time by
IR
ordinary real Turing machines As a consequence we obtain the existence
of binary sets that do not b elong to the Bo olean part of PAR an extension
IR
of the result in since PH PAR and a separation of complexity
IR IR
classes in the real setting
Some geometrical background
In the rest of the pap er IN ZZ Q IR and C denote the sets of natural integer
rational real and complex numbers resp ectively By IR we denote the
alg
real closure of Q i e the eld of all real algebraic numbers Also for
any p olynomial f with integer co e cients we shall denote by jco e f j the
maximal absolute value of its co e cients
The aim of this section is to show how to nd real algebraic p oints in
the connected comp onents of non empty op en sets We closelly follow
Thus let g g ZZ X X and let
N k
k
V fx IR g x g x g
N
b e an op en non empty semialgebraic set For the rest of this section we
consider d a b ound on the degree of each g and L b e a b ound for all
i
jco e g j
i
Q
N
g Let also d be the degree of Lemma Lemma Let g
i
i
g and L jco e g j Then there exists a positive integer such that any
connected component of V has a non empty intersection with the ball B R
where
k
1
d
1
R L
k
Let us recall now see section that a p oint a IR is a critical
k
p oint for a function f IR IR when it satis es
f f
a a
X X
k
In this case the value b f a is said to b e a critical value of f In the case
when f is a p olynomial function Sard s lemma th eor eme or
implies that there are only a nite number of critical values of f
This last fact was used in and in several subsequent pap ers to re
duce the dimension of non empty semialgebraic sets to zero avoiding thus
cascading of pro jections in the algorithm for deciding emptiness of semial
gebraic sets
Let us now consider the p olynomials
k
X
g R X
i
i
and
N
Y
G g g
i
i
O (k )
d k O N
2
We have that deg G d N d and L jco e G j L Ld
due to lemma
The following result gives a b ound on the small critical values of G
Lemma There exists a positive integer such that for every non zero
critical value a of G we have jaj C where
k
2
d
2
C L
Proof Let us consider the system of equations in the variables X X Z
k
G G
G Z
X X
k
k
as well as its set of solutions S IR On any connected comp onent
of S the co ordinate Z b eing the critical value of G is constant since G
is continuous and due to Sard s lemma Now since the degrees and the
co e cients of the p olynomials app earing in this system are b ounded by d
and O L d resp ectively if we apply the quanti er elimination algorithm
given in or along X X onto Z we get a nite set of p oints in
k
IR just the critical values such that each non zero one has absolute value
greater than
k
2
d
2
L
2
Remark In the preceding pro of the use of quanti er elimination is not
strictly necessary One can use instead the b ounds for the representative
p oints from the connected comp onents of S given in the main theorem of
Because of the preceding lemma we have that the algebraic set
k
W fx IR G x C g
is a non singular closed hypersurface with the prop erty that each con
nected comp onent of V B R contains at least one b ounded connected
comp onent of W cf Note that W do not intersect the b oundary of
B R
Now lemma of asserts the existence of integers v v
n
k
d such that the system
G v v G
k
G C
k k
X d k X d k
k
P
k
G
k
where has a nite number of solutions in IR Moreover
i
X
i
each of these solutions is an absolutely irreducible dimensional comp onent
k
of the variety in C given by this system of equations Due to Bezout s
k
inequality the number of real solutions is b ounded by d Besides cf
lemma in each b ounded connected comp onent of W contains a p oint
satisfying the system
We can summarize the preceding results in the following theorem that
will b e our main technical to ol in the next section
Theorem Let g g ZZ X X satisfy for every i N the
N k
bounds deg g d and jco e g j L Then with the notations introduced
i i
k k
above there are integers v v d such that the set W IR
k
de ned by
G v v G
k
G C
k k
X d k X d k
k
k
is nite Moreover the number of its points does not exceed d and every
connected component of
k
V fx IR g x g x g
N
contains at least one point of W
Computing with binary inputs
In this section we shall deal with real Turing machines RTM for short as
they were introduced in The reader is referred to this pap er for the
de nition and main prop erties of the ab ove quoted computational mo del
L
We just recall that we denote IR the direct sum IR and by P the
IR
i IN
class of subsets of IR that can b e decided by a RTM in p olynomial time
The goal of this section is to prove that the Bo olean part of P is
IR
included in PSPACE poly i e that any subset of that can b e decided
in p olynomial time by a real Turing machine can b e decided by a classical
Turing machine in p olynomial space using a p olynomial advice In the
next section we will prove a more general result namely a similar inclusion
for parallel real Turing machines Because of clarity of exp osition we will
however rst show the inclusion for the Bo olean part of P
IR
We b egin by recalling the de nition of non uniform classes as given in
that we extend to complexity classes over the reals
De nition Let C resp C IR be a class of sets and F be any
class of functions from IN to S ig ma resp from IN to IR The class
C F is de ned to be the class of al l subsets B resp B IR for
which there exists a set A C and a function f F such that B fx
hx f jxj i Ag
We will b e mainly interested in the case F poly the class of functions
f such that for some p olynomial p we have jf n j p n for each n IN
For the b o olean case one can nd the main prop erties and characterizations
of classes like P poly or PSPACE poly as well as of some other non uniform
complexity classes in chapter of
Theorem The inclusion BP P PSPACE poly holds
IR
q
Proof Let M b e a RTM working in p olynomial time say n and let
b e its real constants
k
For any n IN the machine M has an asso ciated algebraic computation
q
q n
tree T having depth n and size b ounded by To each branching no de
M n
i of this tree corresp onds a rational function g Q X X
i k n
such that the branching is done according to whether the actual input x
n
IR satis es g x or g x
i i
The idea of the pro of is to nd IR such that for every
p alg
n
e
x the path followed by x in the tree T obtained by replacing the
M n
constants by the is the same as the one followed in T This ensures
j j M n
n
e
that the tree T accepts the same subset of than T On the other
M n M n
hand we will require some co di cation of the s that allows us to p erform
e
the op erations in T in PSPACE together with a short way of writing this
M n
co di cation that make p ossible to give it as a p olynomial advice
Before obtaining a description of the s let us do a last mo di cation on
T that was rst used in Let I b e the set of branching no des of T
M n M n
n
For every i I and every x we consider the rational functions
g Q Z Z
i x k
obtained by replacing the real constants by indeterminates
k
Z Z and the indeterminates X X by the binary values
k n
n
x x We obviously have for every x that g x g
n i i x k
n
Let b e these values The accepted subset of can b e characterized by
i x
the set of signs
sign
i x i x
where sign z if z and otherwise Now for some i x the element
can b e zero However since the set of values
i x
n
f i I x g
i x
is nite there exist an such that all the negative values in the ab ove
set are strictly smaller than and for this the following equivalences hold
g i g
i x k i x k
g i g
i x k i x k
Thus replacing the tests g X by g X we have that the new
i i
computation tree having real constants satis es the following
k
n
prop erty for every x all the test values are di erent from zero
Assuming that the rational functions g Z Z are p olynomials
i x k
something that we can do by just replacing it by the pro duct of its numer
ator and denominator we can resume the ab ove remarks in the following
way the elements satisfy a system of p olynomial inequations
k
of the form
n
g i I x
i x
we changed the sign of some g in order to have all the inequalities in the
i x
same form and any other real numbers satisfying this system
k
will pro duce when used as constants in the tree T the same outcome
M n
n
for every x
We can now describ e how to obtain such numbers
First we construct the g and the G of the preceding section for the set
n
of p olynomials fg ZZ Z Z Y i I x g Then applying
i x k
theorem we deduce the existence of integer vectors v v v v
k k
such that the set W describ ed there is nite and non empty Let v b e the
k
rst such vector for the lexicographical ordering in IN and let W b e its
corresp onding set of solutions We then have that any connected comp o
nent of the semialgebraic set V given by the system contains a p oint of
W Thus we take b e any p oint of W b elonging to V and
k
we distinguish it among the other p oints of W by its p osition p for the
k
lexicographical ordering in IR Note that from the equations de ning
W and this p we can co de cf or each co ordinate of this p oint
see complexity analysis b elow
The following non uniform parallel algorithm decides then the same lan
guage as M when restricted to binary inputs
input a a
n
get the advice p corresp onding to n
n
for all x in parallel do s
for all path in parallel do
for all i branch no de in do
compute the p olynomial g
i x
o d
o d
o d
compute g and G s
compute C s
compute v s
th
co de the co ordinates of the p p oint of s
the set W given by v C and G
simulate the computation of M over a a s
n
replacing the by the p oint co ded in s
k
Let us estimate the complexity of the algorithm ab ove As we have
q
n
seen the number of no des of the tree T is b ounded by Therefore
M n
the number of p olynomials g is b ounded by the same quantity Each
i x
q
of these p olynomials is computed by a straight line program of length n
q
q n
n
and thus we get again a b ound of for their degrees and of for the
absolute value of their co e cients The degree d of G is then b ounded by
q q q
O n n n
and the absolute value of its co e cients L by
q q q
2n n n
2q
n
in bit length We can then according to theorem and thus by
b ound by
q
k O n
d
the integers v v v and by
k k
q
k O n
d
the number of p oints in W A rst consequence of these two last upp er
b ounds is the fact that the advice ab ove has p olynomial size
Concerning the running time it is clear that step s can b e done in
p olynomial time using an exp onential number of pro cessors b ecause given
n q
an x and a path the at most n p olynomials that app ear in that
path have exp onential degree in a constant number k in fact of vari
ables and therefore an exp onential number of monomials Any arithmetical
op eration b etween two such p olynomials can b e done within these resources
and we have a p olynomial number of such op erations
The pro duct G is computed with a binary tree of pro ducts having p oly
nomial depth Since each pro duct can b e done in parallel p olynomial time
the same applies for the whole tree and then for step s A similar remark
holds for the constant C and thus for step s
The determination of v can b e done in parallel p olynomial time since
we check for all p ossible vectors V whether the dimension of the resulting
W is zero and then we select the rst v that gives a p ositive answer Note
that the determination of the dimension of each W can b e done in PSPACE
just combining the main idea of section with the parallel algorithms
given in or
For step s one can apply the algorithms given in or However
we remark here that a cylindrical algebraic decomp osition together with the
co ding a la Thom see for the algorithms and for complexity
analysis su ces b ecause the double exp onential b ehaviour of this algorithm
is only in the number of variables which is constant in our case b eing NC
in the rest of the parameters This results on a pro cedure for s working
in parallel p olynomial time
Finally note that each arithmetical op eration of M is translated in step
s into an op eration of elements in ZZ Z Z and it is done then also
k
in parallel p olynomial time On the other hand at each test of the form
g Z Z we use the same algorithm of step s for determining
k
the sign of g Z Z Y on the p oint co ded in s
k
The ab ove considerations show that the describ ed algorithm runs in par
allel p olynomial time Since this is equivalent to p olynomial space
ch we have shown that the set decided by the algorithm ab ove b elongs to
PSPACE poly 2
Binary inputs for parallel real Turing machines
Our next goal is to extend our previous result to the class PAR of sets
IR
decided in parallel p olynomial time We recall from the de nition of
a computational mo del for parallelism in the real Turing machine setting
together with the complexity class it de nes when restricted to p olynomial
time
De nition An algebraic circuit C over IR is a directed acyclic graph where
each node has indegree or Nodes with indegree are either labeled as
inputs or with elements of IR we shal l call the last ones constant nodes
Nodes with indegree are labeled with or Final ly nodes
with indegree are of a unique kind and are called sign nodes There is one
node with outdegree called output node In the sequel the nodes of a circuit
wil l be called gates
To each gate we inductively asso ciate a function of the input variables in
the usual way In particular we shall refer to the function asso ciated to the
output gate as the function computed by the circuit Note that sign gates
return if their input is greater or equal to and otherwise
De nition For an algebraic circuit C we de ne its size to be the number
of gates in C and its depth to be the length of the longest path from some
input or constant gate to the output gate
De nition Given an algebraic circuit C the canonical enco ding of C is
a sequence of tuples of the form g op g g IR where g represents the
l r
gate label op is the operation performed by the gate g is the gate which
l
provides the left input to g and g its right input By convention g and g
r l r
are if gate g is an input gate and g is if gate g is a sign gate whose
r
input is then given by g or a constant one the associated constant being
l
then stored in g Also we shal l suppose that the rst n gates are the input
l
ones and the last one the output gate
De nition Let fC g be a family of algebraic circuits We shal l say
n n IN
that the family is P uniform if there exists a real Turing machine M that
th
generates the encoding of the i gate of C with input i n in time poly
n
nomial in n
De nition We shal l say that a set S can be decided in parallel p oly
nomial time S PAR for short when there is a P uniform family of
IR
circuits fC g having depth polynomial in n and such that C computes the
n n
characteristic function of S restricted to inputs of size n
Remark It is p ossible to de ne parallel p olynomial time in a di er
ent way namely by putting an exp onential number of RTM to work together
in p olynomial time One can prove however that this mo del de nes the same
class PAR we just introduced
IR
Before going into the next theorem we will recall a result concerning the
number of satis able sign conditions of a p olynomial system
Lemma Lemma of Let f f IR X X be a nite
s k
s
P
degree f Then the number of satis able family of polynomials and D
i
i
systems of the form
f X X f X X
k s k s
O k
where belongs to f g for i s is bounded by D
i
Theorem The equality BP PAR PSPACE poly holds
IR
Proof Let S b e a set in PAR and fC g b e the family of circuits deciding
IR n
S Let also M b e the RTM that generates these circuits and b e
k
its real constants
Given any n IN we consider for any sign gate i of C and any binary
n
n
string x the rational function g Q X X that
i x k n
the gate receives as input Note that the co e cients of g dep end on x
i x
and since they dep end on the output of previous sign gates Since the
number of p ossible answers to previous sign gates is doubly exp onential we
obtain a doubly exp onential number of rational functions and therefore we
can not apply directly the construction of theorem However we can use
lemma to reduce this number
n
Let us x x and plug x in the input gates of C This will make us
n
q
to consider rational functions in Q Z Z Let also n b e a b ound on
k
q
n
sign gates whose input the depth of C At depth there are at most
n
functions have degree b ounded by By lemma the number of p ossible
outputs of these sign gates is b ounded by
q q
O k O k n n
For each set of outputs at depth we consider the sign gates at depth
q
n
There are at most of them and their asso ciated functions have degree
b ounded by Thus again by lemma we b ound by
q q
O k O k n n
the number of p ossible outputs for Multiplying b oth expresions we deduce
that the total number of p ossible outputs at depths and is b ounded by
q
O k n
Inductively we prove that the number of p ossible outputs over all the sign
gates is b ounded by
q q 2q
O k n n O k n
a number which is singly exp onential in n
n
Let us then consider for any x the set of all rational functions
g Z Z obtained by varying i over all sign gates of C and over
i x k n
all p ossible outputs of the set of sign gates If we now consider this set for
n
any x we will have that the set decided by the circuit C is determined
n
by the signs that the functions in this set take when evaluated at
k
As in theorem we can assume the functions g to b e p olynomials
i x
and we can also assume that they do not vanish on by adding a
k
new real number
Since the number of p olynomials g is singly exp onential in n we can
i x
apply the same metho d of theorem Note however that the corresp onding
n
step s will now b e required to select for any x and any depth
l the p ossible sign conditions for the test gates at depth l This is done
sequentially in l in order to avoid dealing with a doubly exp onential number
of sign conditions Once these p ossible sign consitions are known the rest
of the algorithm works like the one in theorem simulating the circuit C
n
instead of the tree This shows that the binary elements of S are a language
in PSPACE poly
On the other hand the inclusion of PSPACE poly in BP PAR is triv
IR
ial 2
An immediate corollary of the theorem ab ove is the following separation
left op en in Recall that EXP is the class of subsets of IR accepted
W
by RTM in weak exp onential time i e in exp onential time but such that for
all intermediately computed rational function g deg g and the bit length of
jco e g j are exp onentialy b ounded see or for a formal de nition
of the weak mo del
Corollary The inclusion PAR EXP is strict
IR W
Proof The Bo olean part of EXP is the class of all subsets of Therefore
W
it strictly contains PSPACE poly 2
Remark The corollary ab ove improves the separation PAR EXP
IR IR
shown in In this latter case the fact that a real Turing machine working
in exp onential time can pro duce p olynomials of doubly exp onential degree
while a circuit of p olynomial depth can not togheter with an irreducibil ity
argument su ced to show the separation The arguments used now are
much more delicate and somehow surprisingly pass throught the b o olean
part of these classes
One can still improve a bit theorem by allowing the real machine to
take advice
Theorem The equality BP PAR poly PSPACE poly holds
IR
Proof The p olynomial advice in PAR poly introduces a p olynomial num
IR
h
b er of real constants let s say n for each input size n One can now simply
h
check that replacing the constant value k in the pro of of theorem by n
do es not afect the exp onential character of the b ounds there and thus the
same arguments apply The only limitation is that in steps s and s one
can not use cylindrical algebraic decomp osition b ecause of the exp onential
dep endence it has in the number of variables for its parallel running time
and it is restricted to use the faster algorithms given in and 2
Remark Theorems and are rather surprising since they show that
multiplication or non uniformity under the form of a p olynomial advice
function do not help in the presence of parallelism to decide binary sets
Note that results weaker than theorem namely that the Bo olean part of
PAR where no multiplications are allowed or of PAR where few of
add W
them are allowed coincide b oth with PSPACE poly were proved in and
On the other hand a main question that remains op en is whether
BP P PSPACE poly We know that this Bo olean part contains P poly
IR
but its exact p ower is still to b e determined Note that for the integer RAM s
it is known that the computational p ower of this mo del in p olynomial time
is exactly PSPACE for several sets of primitive op erations However in
all these cases there is a primitive op eration that can not b e e ciently
simulated by a real Turing machine Thus for instance it is shown in
that integer RAM s with op erations have the p ower of PSPACE
However the simulation of the integer division by a real Turing machine over
integers of exp onential length take exp onential time and therefore the argu
ments of can not b e used to show the inclusion PSPACE poly BP P
IR
Ackowledgement Thanks are due to Pascal Koiran for p ointing to us the
p ossibility of allowing advice in the real complexity classes that lead from
theorem to theo erem
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