COMPLEXITY THEORY ON REAL NUMBERS AND FUNCTIONS
by C h r i s t o p h Kreitz and Klaus W e i h r a u c h
F E R N U N I V E R S I T ~ T Hagen
F a c h b e r e i c h M a t h e m a t i k und I n f o r m a t i k
P o s t f a c h 940, D - 5 8 O O Hagen I
I. I n t r o d u c t i o n
Since Turing
[17] i n t r o d u c e d the concept of computable real numbers in
1937, m a n y authors have s t u d i e d c o m p u t a b i l i t y of real numbers and
functions
(see for example,
[1,2,6,10,11,12,13,14,16]).
The next subject
to be s t u d i e d is now c o m p u t a t i o n a l complexity.
The goal of our paper is to present a natural concept for c o m p u t a t i o n a l
c o m p l e x i t y of real numbers and functions.
F i r s t we d e v e l o p a c o m p l e x i t y theory on IR c o r r e s p o n d i n g to Blum's
c o m p l e x i t y theory on the r e c u r s i v e
functions.
[3]
(A g e n e r a l i z a t i o n of his
theory to cpo's has already b e e n f o r m u l a t e d by W e i h r a u c h
[18]). In order
to abtain a s u c c e s s f u l d e f i n i t i o n of c o m p l e x i t y we r e p r e s e n t real numbers
by C a u c h y - s e q u e n c e s of dyadic numbers h a v i n g a normed convergence•
define the set of c o m p u t a b l e real numbers
(IR)
We
and an e f f e c t i v e number-
C
ing q of IR
is shown,
C o m p l e x i t y classes of real numbers are i n t r o d u c e d and it
c
that the m o s t i n t e r e s t i n g statements of a b s t r a c t c o m p l e x i t y
theory h o l d a n a l o g u o u s l y in IR
C
Our i n v e s t i g a t i o n of c o m p u t a t i o n a l c o m p l e x i t y of real functions is b a s e d
on the c o m p u t a b i l i t y concept of G r z e g o r c z y k
ing c o m p u t a t i o n m o d e l of Ko & F r i e d m a n
[5]. Improving a correspond-
[7], that uses a rather r e s t r i c t -
ive requirement, we i n t r o d u c e a c o m p u t a t i o n m e c h a n i s m for real functions,
w h i c h enables us to define complexity.
computable
S t u d y i n g t o p o l o g i c a l p r o p e r t i e s of
functions we observe an important r e l a t i o n b e t w e e n continuity
and c o m p l e x i t y of a function.
E x p l o r i n g p r o b l e m s c o n c e r n i n g a r e c u r s i o n - t h e o r e t i c c o m p l e x i t y theory on
real functions we show,
that some c o m p u t a b l e functions cannot have a
c o m p l e x i t y bound b e c a u s e of their c o m p l i c a t e d domains.
a c o m p l e x i t y b o u n d of a function f does exist,
e f f e c t i v e K -set. We prove,
On the other hand
if the domain of f is an
that a h i e r a r c h y - t h e o r e m and a speedup-
theorem h o l d for c o m p u t a b l e functions too. F i n a l l y it is shown,
computable
that a
function may have a high c o m p l e x i t y even if its domain is
simple and it has simple values at simple arguments.
We shall denote the set of all total
by R (k)
(partial)
r e c u r s i v e k-ary functions
(p(k)). Let ~ be a s t a n d a r d n u m b e r i n g of p(1)
and let
W i := dom(~ i) for all i. Let <,> be the s t a n d a r d p a i r i n g function on
166
and D i denote
the
2. C o m p l e x i t y
theory
L e t ~D
where
i-th
finite
on r e a l
:----U {~n I n E IN } be
~n
:=
subset
by
VD<k,l,n>
partial
normed
:=
the set
(k-l)
Cauchy-sequences
and denote
the
s e t of all
6
dora(6) := NC
Sometimes
we
call
a
A sequence
(i)
f 6 P
~ E PNC
Definition
A real
--->
and
An effective
There
(partial)
is s o m e
x
Then
define
i,n E ~
~i
<i(n)
For
total
recursive
the c o m p l e x i t y
The
following
Every
some
be
t
for
6(~).
there
for
some
is a c o m p u t a b l e
real numbers.
can d e f i n e d
c
and
all
as
follows.
(Vn< k) I~D~i(n)-al <2 -n
i E ~
and
(time,
following
IR(t)
:=
{DiI<i(n
and
:= 6(~i)"
'
For
any
for c o m p u t i n g
hold:
is r e c u r s i v e .
) ~ t(n) for a l m o s t
numbers
determined
all n} be
by
t .
immediately:
of c o m p u t a b l e
complexity
qi
tape)
axioms
real
can be p r o v e d
class
real
class
~
numbers
(t)
is c o n t a i n e d
is w e a k l y
in
q-r.e.,
large.
[4] G a p - t h e o r e m :
(Vg,hC R (I), g increasing)
Some
iff
{(i,n,t) I Ki(n) = t}
is s u f f i c i e n t l y
Borodin's
is e a s y
q of IR
the r e s o u r c e
the
let
class,
be
of all
b y NC.
= ~Df(n)
computable
of c o m p u t a b l e
q-r.e,
complexity
for
and
t
set PNC
With
properties
weakly
whenever
Also
class
~(n)
is c o m p u t a b l e
numbering
that
(Vi) dom(< i) = dom(~i)
an o r a c l e
iff
of all
:= ~D~p(i)
assume
: ]lq ~ ~D
the
Cauchy-sequences
@ E NC
, if a := VD~i(k) E ~k
div , otherwise .
let
~p(i) (n). W e
normed
la
VD~p(i) (k) :=
We d e f i n e
numbers,
¢ E 6-1{x}
set
p E R (I)
n E IN , and v D
" 2 -n°
is c o m p u t a b l e ,
is the
c
fixed
(rational)
~(~) := lira ~(n) .
Cauchy-sequence
IR
of all d y a d i c
IR can be d e f i n e d b y
sequence
number
[15]).
(~(k) E ~ k A (Vn<k) I ~(n)- ~(k) I <2-n)}
total
: PNC
(see R o g e r s
in QD by
P N C := {~ : I~ ---> ~D I (VkEDef(~))
A representation
~
numbers
{m - 2 -n I m E ~ } for
defined
of
(3fE R (I)) [(Vn) (h(n) <_ f(n)) A ~ (g-f) = R(f)]
to prove.
of the m o s t
important
results
of the r e c u r s i o n - t h e o r e t i c
complexity-
167
theory
can be p r o v e d
i z a t i o n on ]R
o n l y by e f f e c t i v e
w e s h a l l use p a t t e r n s
c
These patterns
diameter.
diagonalization.
For a d i a g o n a l -
of i n t e r v a l l s w i t h d e c r e a s i n g
are i n s p i r e d
by the g e o m e t r i c a l
approach
to the
Cantor-discontinuoum.
Example ' Define
and
T(wO)
if
w6
Note
3 l+r
:= [i,--~---]
{0,I] ~
t h a t for w C
2 -(n+2)
Using
and
T(w)
or
a weakly-
=
r.e.
set
Let
w
0
Step
n+1 F i n d a w o r d
diameter
{x]
:=
N
n£
T(Wn)
Y
W n + 1 E {WnO,Wnl ]
~f(n) (2n+3)
2 -(2n+2)
theorem:
~h(m) 6 I R c
(2)
(Vi)
Especially
x ~ Yo
theorem
The proofs
the t h e o r e m s
report
as w e l l
as
are v a r i a t i o n s
here.
Details
can
[8].
is a f u n c t i o n
[qi = nh(m) => (Ki(n) > ~m (n)
~h(m) E IR(<h(m))
theorem:
(Vie ~-l{x})
3. C o m p u t a b l e
Computable
I of
h E R (I)
such t h a t for all
w i t h ~ m E R (I)-
(I)
Speedup
the c o m p r e s s i o n
We only formulate
There
an i n t e r v a l l
real number
[3]) c a n be p r o v e d .
technical
~f(n) ~ ~ ( W n + l ) "
nf(n ) E I.]
w e get a c o m p u t a b l e
be f o u n d in the a u t h o r ' s
m E IN
such that
determines
with
(Blum
C0mpression
effect-
f 6 R (I)
diagonalization
ones.
less or e q u a l
of real n u m b e r s :
the s p e e d u p
of the o r i g i n a l
I with diameter
holds.
w e are able to d i a g o n a l i z e
By use of e f f e c t i v e
theorem
= [O,1]
:= s
o
[Note t h a t
By
T
for some
Step
T(e)
= [l,r]
like
{~f(n)I n E IN}
by
:= [i, l___~r ]
T(wl) n I = ~
an i n t e r v a l l - p a t t e r n
Y
T(wl)
{0,I} n and any i n t e r v a l l
T(wO) ~ I = ~
ively over
Let
T : {0,1} ~ ~ {[r,s] I r , s E ~ }
r E R (2)
all
n) ]
holds.
there
(3j~ q-l{x}) [r(n,Kj(n)) ~ K(n)l
real
real
One approach
For e v e r y
\]R(~m)
for a l m o s t
is some
xE ~
for almost all
c
with
n].
functions
functions
is to d e f i n e
are d e f i n e d
a computable
in two d i f f e r e n t ways.
function
as e f f e c t i v e
operator
188
on the set IR
(Mazur []3], Aberth
[]]) using an effective n u m b e r i n g of
c
]R
(like N). The other approach considers
all real continuous
functions
C
on all real numbers
(Grzegorczyk
[6], Lacombe
[10])
and defines a comput-
able function by a r e c u r s i v e functional on all p o s s i b l e names
Cauchy-sequences)
these concepts
for real numbers.
(see Ceitin
(e.g.
There is a strong r e l a t i o n between
[5]), but it isn't fully u n d e r s t o o d till now.
In this paper we shall follow the second approach,
since the first one
p r o v i d e s some results in formal c o n t r a d i c t i o n to c l a s s i c a l analysis and
d o e s n ' t seem to allow a s u c c e s s f u l d e f i n i t i o n of complexity.
(function-)
(partial)
Oracle-Turing-machines
(OTM's)
c o m p u t a b l e real functions.
m e a s u r e of T u r i n g - m a c h i n e s
real functions.
We use
as computing m o d e l for
Thus we can apply a c o m p l e x i t y
for d e f i n i n g the c o m p u t a t i o n a l
complexity of
Our m o d e l is inspired by one of Ko & F r i e d m a n
[7], b u t
our d e f i n i t i o n of c o m p u t a b i l i t y
is m o r e general and more natural.
A
function-Oracle-Turing-machine
M
nE
computes
(if it exists)
the
n-th
term
M~(n)
w i t h oracle
~ E NC
and input
of a sequence
asking the oracle for a finite number of values
M g C PNC
~ ( n l ) , . . . , ~ ( n k) during
the computation.
Definition
A function
f : IR - - ~ ~
Oracle-Turing-machine
(VCeNC)
f(6(¢))
M
is computable,
iff there is an
such that
= 6(M })
(~)
Let OTM(f) be the set of all OTM's s a t i s f y i n g
Condition
(i)
(e) is e q u i v a l e n t to the following ones:
(Vx£ dora(f)) (V~£ 6-I~])
(ii) (Vx% dora(f)) (V~£ 6-I~])
Condition
(~).
(ii) makes
requires M~(n)
( S ~ NC
and
lira M~(n) = f(x)),
(M~(n) diverges for almost all
our d e f i n i t i o n differ from Ko's
to diverge
for all
[7], because be
if
lim
~(n) ~ dom(f).
T h e r e f o r e his m a c h i n e s have to decide w e t h e r
lim
~(n) E dom(f)
b e f o r e c o m p u t i n g any value.
Many c o n t i n u o u s
n6 ~
n).
This r e q u i r e m e n t is too restrictive.
total functions are c o m p u t a b l e e.g. p o l y n o m i a l s with
r e c u r s i v e coefficients,
e l e m e n t a r y functions,
A l s o p a r t i a l functions may be computable.
f : ~
or not
---> ~
defined by
the function
E x a m p l e s are
dom(f) := ~ \ {o}
and
f(x) := I/x
x~
JxT e t c . .
169
and for
y~
:
gy
~
c
--->~
B u t gy h a s
no extension
{x I x ~ y}
For
is n o t
any O T M
M ~. The
Lemma
defined by
M
3.1
For
recursive
topological
Theorem
[i.e.
The
proof
oracles
of p a r t i a l
cannot
Theorem
set
of
Lemma).
, x C dom(f)
x, TMX : ~ - - - > ~
, n@
is a
of
x
that
and
This
on m e t r i c
f
f
at
bound
x.
and
There
the
is c o n t i n u o u s
is b o u n d e d
Ix-yl < 2 -k
first
y
k
in
the
are
so t h a t
steps.
(Weihrauch
of the d o m a i n s
has b e e n
[19])
extended
an h o l d s
functions.
of some
computable
real
function
G -set.
is a c o m p u t a b l e
13
there
k values,
characterization
spaces
real
if
a characterization
is an e f f e c t i v e
0
and
by
ff(x)-f(y)I ~ 2-n) ]
at the
is a d o m a i n
where
then
x E dom(f)
we k n o w
computable
S = N U 0
..
13
13
bound
complexity
M E OTM(f)
functions.
S
this
coincide
[9])
S ~ ~
if
function
(K6nigs
exists.
(Ix-yI < 2 - T M X ( n + l ) =>
distinguish
for p a r t i a l
A
M E OTM(f)
is c o m p u t a b l e
, which
functions
if and o n l y
(i.e.
~
the a r g u m e n t ,
y
continuous
3.3
--~,
at any p o i n t
(Kuratowsky
to c o m p u t a b l e
Therefore,
counting
fan t h e o r e m
as c o m p l e x i t y
between
(Vn)
from
and
topology
function.
the s t e p
by the
computable
for any m a c h i n e
x
computable
f.
f : IR--~
follows
for
analogous
of
( V y 6 dom(f))
the m a c h i n e
From
for any
of c o n t i n u i t y
ln. T M X ( n + 1 )
x= y
function.
relationship
If
x> y
, if
I ~ E 6-i {x}}
TM x can be v i e w e d
3.2
, if
be
f : IR
{TM~(n)
properties
modulus
O
div
can be p r o v e d
computable
Forthermore
function
a total
TM~ : IN - - ~ ~
lemma
any
is an i m p o r t a n t
x<y
gy(x) :=
into
TMX(n) : = m a x
The
, if
'decidable'.
let
following
1
double-sequence
of
open basisintervalls.)
The p r o o f
is t e c h n i c a l
in the
author's
We get
as c o r o l l a r y
f : IR--~IR
only
set.
technical
whose
the n u m b e r
and will
from
domain
xK:=
report
theorem
be omitted.
Further
details
can be
found
[8].
3.4 t h a t
contains
only
~
2 -i , w h e r e
i ~ K
there
is a c o m p u t a b l e
noncomputable
K c ~
real
function
numbers,
is a n o n r e c u r s i v e
but
e.g.
r.e.
170
4. C o m p l e x i t y
In s e c t i o n
M E OTM(f)
3 we h a v e
at
interested
is a
of p a r t i a l
c a s e we
But not
every
Example
Let
say
that
R\ ~
v~
of
for e v e r y
is an
n
statement
an i n c r e a s i n g
the
infinitely
in
many
t(n k)
k
Using
the r e l a t i o n
has
f
such
of
M
that
there
(respectively
a complexity
such
that
dom(f) = ~
G~-set) . T h e n
f
\ ~
(Note,
has no c o m p l e x i t y
M E OTM(f)
and
xE IR\~
such
TMX(n) > t(n)
that
of f).
bound.
machine
an a r b i t r a r y
M E OTM(f)
any
f
k E ~N.
So
and
(increasing)
and
is u n d e f i n e d
(cf.
section
function
for
the m a c h i n e
on
~ ,we can
a point
such
cannot
T M X ( n k ) _< t(nk)
function
a standardnumbering
2)
of n a t u r a l n u m b e r s
nk
V~ (k)
and
TM
(nk) > t(n k)
steps
many
construct
x E IR\~
and
that
distinguish
doesn't
hold
x
and
for i n f i n i t e l y
.
topological
guarantee
function
choose
machine
function
sequence
function
We are n o w
]
diagonalization
Ix- 9 (k) I -< 2 -t(nk)
its
we
an a r b i t r a r y
of a m a c h i n e
n
be c o m p u t a b l e
there
this
complexity
is c o m p u t a b l e .
bound
function
many
~. S i n c e
vg(k)
real
t : iN ~ ~
an e f f e c t i v e
for
for almost all
infinitely
t : iN ~ iN
a real
is an e f f e c t i v e
[i.e.
, the
t : IN ~ iN w i t h
f : IR---> IR
bound.
~
: t is a c o m p l e x i t y
computable
functions
f : IR--->IR
computing
function
TMX(n) -< t(n)
In t h a t
by
M
real
TM x : ~ - - - >
, where
in m a c h i n e s
(VxE dom(f))
To p r o v e
defined
x 6 dom(f)
(computable)
computable
the
properties
existence
defined
"effective
between
complexity
we'll
now
of a c o m p u t a b l e
characterize
of a c o m p l e x i t y
on it.
For
compactness"
this
(cf.
bound
purpose
Bishop
[2],
we
real
domains,
function
and
which
for any
computable
real
formulate
the n o t i o n
of
Martin-L~f
[12])
and
"effective
K -set" .
Definition:
A
set
U(S)
S ~IR
:=
{j E ~
(the set of
A
set
is e f f e c t i v e
I S ~<i~k>~n
all
T ~]R
effective
finite
that
(di -2-k"
union
of e f f e c t i v e
sets
Wh(j) = U(Sj)
iff
of
K -set
compact
S
is c o m p a c t
and
is an r.e.
set.
di +2-k)}
coverings
is an e f f e c t i v e
effective-compact
such
3
open
-compact
S)
if
sets
T
(i.e.
is an
there
S.
and a r e c u r s i v e f u n c t i o n
3
for all
j and
T = j ~ m Sj).
are
h
171
Lemma
4.1
a)
Let
If
such
b)
If
that
S
a)
that
in a f i n i t e
iveness
of
S
= max
S
Let
~. By the
construct
. Then
there
n
can a r r a n g e
with
M~(n)
fan
theorem
a recursive
all
needs
t 6 R (I)
n.
the o r a c l e s
degree.
only
Given
a finite
prefix
the set
and hence
function
for e v e r y
,
all
a finite
of
t E R (I)
is a f u n c t i o n
for a l m o s t
for all n E IN
I ~ E 6-I(S)}
S =
~ t(n)
we
, M C OTM(f)
is a f u n c t i o n
for e v e r y
then
of t r e e s
is f i n i t e
can
[TM~(n)
b)
sets
number
S &dom(f)
there
~ t(n)
TMX(n)
the c o m p u t a t i o n
of the o r a c l e
then
K -set,
is b o u n d e d ,
[OM$(n) i ~ E 6-I(s)}
t(n)
TMX(n)
(Vx e S)
S
~ E 6-I(s)
OM~(n)
( Y x C S)
is an e f f e c t i v e
Since
E 6-I(S)
nE ~,
be c o m p u t a b l e ,
is e f f e c t i v e - c o m p a c t ,
such
Proof
f : IR--->IR
S
t
using
such
the e f f e c t -
that
(Vn)
n.
U
S.
be an e f f e c t i v e u n i o n of e f f e c t i v e - c o m p a c t
j E ~
4.1 a) the f u n c t i o n
t E R (I)
with
using
3
(Yn)
has
the
We g e t
t(n)
required
as a c o r o l l a r y
computable
function
continuous
on
Since
S
f
f'(as
of a u n i f o r m
Here we
f(x)
too,
Definition
Let
is b a s e d
be
A function
t @ R (I)
("f 6 Cs(t)")
( V ~ E 6 -I (S))
Speedup-theorem
For
every
(%q~£ OTM(f))
(3M6OTM(f))
theorems
and using
in
bound
that
set
S
is u n i f o r m l y
is r e c u r s i v e .
for a c o m p u t a b l e
of r e a l
[7] -
4.1
every
the u n i f o r m
seems
applicable
on L e m m a
a)
continuity
requires
real
continuity
functions
by
to us to be too
to f u n c t i o n s
like
b).
K -set,
f : IR--~
IR
be comput-
f.
is a c o m p l e x i t y
there
t E R (I)
which
r E R (2)
bound
is a m a c h i n e
(TM ~(n)) ~ t(n)
function,
such
These
iff
For every
f : R ~ IR
4.1
complexity
an e f f e c t i v e
S ~ dom
real
a))
a definition
able w h e r e
computable
4.1
- as K o u s e d
give
S G ]R
and L e m m a
of u n i f o r m
of c o m p u t a t i o n a l
which
Hierarchy-theorem
3.2
complexity
in L e m m a
bounds
restrictive.
:= I/x
~-i (Sj )}
on an e f f e c t i v e - c o m p a c t
the m o d u l u s
defined
complexity
theorem
defined
, a definition
uniform
from
and
the e x i s t e n c e
function
of
max {TM~(n) I (3jS n) ~
=
properties.
for a l m o s t
one
doesn't
there
of
f
M E OTM(f)
all
to
S
that
n)
can e f f e c t i v e l y
belong
on
such
construct
a
C~(t)
is a c o m p u t a b l e
function
that
(V~,~'e NC) Jr(n, TM~(n))
c a n be p r o v e d
compression-
and
by
considering
speedup-theorem
~ TM~'(n) for almost all
constant
of s e c t i o n
constant
2.
n]
functions
172
So far we h a v e
domains
seen,
functions which
complicated
simple.
simple values"
Theorem
4.2
(¥yeX)
that a notion
in p o l y n o m i a l
Let
X -c [0,1]
We s h a l l
E ~D
enumerate
sequence
of p o l y g o n s
'first'
i elements
p,q
(Vi)
but
q(i) A 2i+3
like
f
their domain
maps
be an u n f i n i t e ,
function
weakly
t E R (I)
such that
X
injective
and c o n s t r u c t
a converging
are c o n s t a n t
functions,
q
on n e i g h b o u r h o o d s
strictly
increasing
of the
such t h a t
and
(Vi_>2) (Vj< i) I 9p(i) (q(i)) -~p(j) (q(i)) I > 2 " 2 -q(i)
~ji
instead
to s h o r t e n
~p(j ) (q(i))
and
(~ji-2-q(i))
' if
j _> 2
of
: [O,1] ~ IR
, otherwise.
(i_> I)
s h a l l be d e f i n e d by t h e i r v a l u e s
z E {~+ij I j-< i] U {~ij t j < i} . Let
fl (O)
:= O
,
fl (I)
:= I
i ~ 2
fi_i(~i±)+ (-i)g(~)-2 -q(~),if
fl (z)
g E R (I)
specified
Estimating
z~ {~[~,~}
:=
fi_l(z)
where
set,
f : [O,I]~IR
~ji
and if
~-r.e.
(3)
l
into
of c o m p l e x i t y .
(Vn) (~p(O)(n) = O ^ ~p(1)(n) = I)
f
is e a s y
arguments
(2)
The p o l y g o n s
is
f ~ C[o,l](t)
~ j i + 2 - q(i)
points
f : R ~ IR
simple
{np(i) I i~ IN} = {O,i] U X
~+ji (~'i)
is n o t
if the a r g u m e n t
(i)
We shall write
if t h e i r
t h e r e are c o m p u t a b l e
"A f u n c t i o n
if
for a d e f i n i t i o n
f., w h i c h
l
of
X .
be r e c u r s i v e
although
time)
is a c o m p u t a b l e
f(y)
are h a r d to c o m p u t e
However
are e a s y to c o m p u t e
isn't suitable
Then there
Let
complexity,
and t h e i r v a l u e s
(e.g.
functions
are c o m p l i c a t e d .
have high
We c o n c l u d e
computable
Proof
that real
or t h e i r v a l u e s
is a
O/l-valued
, otherwise,
function,
whose
properties
afterwards.
the g r a d i e n t s
( V x E [0,1])
of the
fi's
(Vi) ~ f i ( x ) - fi_l(X)
(~ 2 i- I)
i ~ 2 -i
we
can p r o v e
w i l l be
at
173
Hence
the f u n c t i o n
(cf. K o
lim f : [0,1] ~ ]R
i~ ~
i
W e n o w show,
complexity
machine
(3i)
(y = qp(i)
of the f u n c t i o n
complexity
of
f
is v i r t u a l l y
g E R (I) . T h e r e f o r e w e c o m p u t e
g
the
using a
M E OTM(f) .
i > 2
Find
k,j E {0, .... i-I}
and
(~ki
-
less
)
if
F
F'
is less or e q u a l
To see t h a t the a l g o r i t h m w o r k s ,
N o w let
q).
to c o m p u t e
Then
[3]),
for any
complexity
theorem
t E R (I)
is h i g h e r
h a v i n g the d e s i r e d
k
the f u n c t i o n
t'(i) := 3 • t(q(i) + 3) + T ( i )
compression
of
with error
f i i ( ¢ i i ) = f(~p(i))
I
]+
]_
bound
and
for
j
f
O/l-valued
t'
properties.
, and h e n c e
and
|_
~ki
T E R (I)
for g i v e n
t' E R (I)
one can c o n s t r u c t
than
¢'ki-1
i
describe
(only d e p e n d i n g
defined by
is an u p p e r b o u n d
for
s u c h that
21q(~)
~
~ii ~ii ~ii
be a c o m p l e x i t y
Using
fi-i (~ii)
I.
I_
~ji-I
the n u m b e r of s t e p s
and
of
look at the f o l l o w i n g d r a w i n g .
|%
t E R (I)
f(~p(k)) = fi_l (<~k i_l)
to 2 - [ q ( i ) + l ]
." "
I÷
and
2 -[q(i)+l]
F-F'> 0 , otherwise
~ji
+
+
_
~ji < ~'~i~ < ~ii < ~ii < ~kj
that
is m i n i m a l .
an i n t e r p o l a t i o n
the e r r o r
O
+
~ji
an i n t e r p o l a t i o n
than
Compute
OUTPUT
-
such
f(~p(j)) = fi_l (~3i_I)
construct
p
and is c o m p u t a b l e
^ f(Y) = fi(¢ii ) E ~D )
that the c o m p u t a t i o n a l
Using
on
exists
[7]). M o r e o v e r
(VyEX)
INPUT
f :=
for the c o m p l e x i t y
of
recursive
functions
(Blum
a function
g 6 R (I)
whose
a computable
real f u n c t i o n
g.
f
174
In this paper we have i n t r o d u c e d a concept for a theory of c o m p u t a t i o n
and c o m p l e x i t y on IR. As a next step the r e c u r s i o n - t h e o r e t i c
complexity
theory for functions should be put forewards and e x t e n d e d to sequences
of functions and functionals.
References
[ I]
O. Aberth, C o m p u t a b l e Analysis, M c G r a w - H i l l ,
1980.
[ 2]
E. Bishop, F o u n d a t i o n s of c o n s t r u c t i v e analysisr McGraw-Hill,
[ 3]
M. Blum, A m a c h i n e - i n d e p e n d a n t theory of the c o m p l e x i t y of
r e c u r s i v e functions, J. ACM 14 (1967) 332 - 336.
[ 4]
A. Borodin, C o m p l e x i t y classes of recursive functions and the
e x i s t e n c e of c o m p l e x i t y gaps, Conf. Rec. ACM Symp. on Theory of
C o m p u t i n g (1969), 67 - 78.
[ 5]
G.S. Ceitin, A l g o r i t h m i c operators in constructive c o m p l e t e
s e p a r a b l e m e t r i c spaces (Russian), Doklady Akad. N a u k 128 (1959),
49 - 52.
[ 6]
A~ Grzegorczyk, On the d e f i n i t i o n s of c o m p u t a b l e real continuous
functions, F u n d a m e n t a m a t h e m a t i c a e 44 (1957~ 61 - 71.
[ 7]
K. Ko & M. Friedman, C o m p u t a t i o n a l complexity of real functions,
TCS 20 (1982), 323 - 352.
[ 8]
C. Kreitz & K. Weihrauch, K o m p l e x i t [ t s t h e o r i e auf reellen Zahlen
und Funktionen, I n f o r m a t i k B e r i c h t Nr. 28 (1982), F e r n u n i v e r s i t ~ t
Hagen.
Topology
(Vol.
I) , A c a d e m i c Press,
1967.
[ 9]
K. Kuratowski,
[10 ]
D. Lacombe, E x t e n s i o n de la n o t i o n de function recursive aux
functions d'une au p l u s i e u r s v a r i a b l e s r&elles, Comptes Rendus
Acad. Sci. Paris 240 (1955), 2478 - 2480, Vol. 241 (1955), 13 - 14,
151 - 153, 1250 - 1252.
[11 ]
A.A. Markov, On the c o n t i n u i t y of c o n s t r u c t i v e functions, Uspehi,
Math. Nauk. 9 (1954), 226 - 230.
[12 ]
P. M a r t i n - L ~ f , Notes on c o n s t r u c t i v e m a t h e m a t i c s ,
Wiksell, 1970.
C o m p u t a b l e Analysis, W a r s c h a u
1966.
Almquist &
[13 ]
S. Mazur,
[14 ]
H.G. Rice, R e c u r s i v e real numbers,
784 - 791.
1963
[15 ]
H. Rogers, Theory of r e c u r s i v e functions and e f f e c t i v e computability, McGraw-Hill, 1967.
Proc. Amer.
Soc.
5 (1954),
[16 ] E. Specker, N i c h t k o n s t r u k t i v b e w e i s b a r e S[tze der Analysis,
J. Symb. Logic 14 (1949), 145 - 158.
[17 ] A.M. Turing, On c o m p u t a b l e numbers w i t h an a p p l i c a t i o n to the
e n t s c h e i d u n g s p r o b l e m , Proc. L o n d o n Math. Soc. 42 (1937), 230 - 265.
[18 ] K. W e i h r a u c h , R e c u r s i o n and c o m p l e x i t y theory on cpo's, Proc.
5th GI-Conf. (P. D e u s s e n ed.), LNCS 104, S p r i n g e r (1981),
195 - 202.
[19 ] K. W e i h r a u c h , C o m p l e x i t y on m e t r i c spaces,
Nr. 21 (1981), F e r n u n i v e r s i t [ t Hagen.
I n f o r m a t i k Bericht