CLASSES OF COMPUTABLE FUNCTIONS
DEFINED BY BOUNDS ON COMPUTATION:
PRELIMINARY
REPORT*
by
E. M. McCreight
and A. R. Meyer
Carnegie-Mel lon University
Department of Computer Science
Pittsburgh, Pa.
April, 1969
The results reported here represent a portion of the first
author's doctoral dissertation at Carnegie-Mellon
University
under the supervision
of the second author.
This research
was supported in part by the Advanced Research Projects
Agency of the Office of the Secretary of Defense under Contract No. F 44620-67-C-0058
and is monitored by the Air
Force Office of Scientific Research, and in part by the
Fannie and John Hertz Foundation.
This paper is a slightly
revised version of a report which will appear in the Proceeding
of the ACM Symposium on Formal Languages and Computability,
Marina Del Ray, California
(]969).
ABSTRACT
The structure
space bounded
of the functions
by t is investigated
The t-computable
classes
ing recursively
enumerable
recursive
functions
tions for a certain
classes
partially
recursive
t.
embedded
ordered
equal
running
the t'-computable
Any countable
by set inclusion.
time such
t.
increas-
the primitive
partial
funcorder
of t-computable
For any recursive
is (approximately)
that the t-computable
functions.
under
the t-computable
in the family
t' which
functions
as a corollary
to equal
t, there is a recursive
an actual
to be closed
are shown
can be isomorphically
in time or
for recursive
are shown
unions;
computable
equal
functions
to
].
INTRODUCTION
A rich
structure
ing functions
is imposed
according
to the amount
pute them.
Following
investigate
this structure
or space arising
the axiomatic
three
that any countable
in the ordering
computation.
The Union Theorem
within
under increasing
recursively
the classes
classes
for appropriate
recursive
machine
time or space.
Finally,
as the class
running
of functions
time of a program.
t' can equal a running
chosen.
The proof
to a theorem
Other
(5.2) estab-
on
of func-
t, are closed
As a consequence
the multiply
cases
similar
of t-computable
measure
of Turing
that the class
for any recursive
t' is almost
6 show
time measure
is of independent
interest
computational
using non-trivial
complexity.
the
that whether
on the particular
of a construction
of
t, also be defined
in time t', where
in section
we con-
recursive
and several
(6.9) shows
results
embedded
by bounds
functions
hierarchy,
theorem
(6.9)
to
of time
that the classes
t and the familiar
time depends
about
Theorem
are in fact special
computable
of theorem
of the first examples
applied
functions,
in time t can,
notions
determined
unions.
of the Grzegorczyk
functions
to com-
it is possible
can be isomorphically
indicates
enumerable
of recursive
computable
results.
functions
(5.5)
classes
functions
order
recursive
by classify-
of automata.
time t, for recursive
that the primitive
functions,
of Blum,
of particular
principal
of the computable
tions computable
clude
approach
models
partial
functions
of time or space required
independently
from specific
This paper contains
lishes
on the computable
as one
priorities
-2-
2.
PRELIMINARIES
_
is the set of nonnegative
recursive
functions
sive functions
of n variables,
of n variables•
from /_xi_x--.X/_
/ to
sively enumerable",
If P(x)
is a predicate
functions•
P(x)
"almost
The statement
b E _1,
_Pi _
enumeration
of functions
in
_I
1.
domain
2.
%i,x,Y[_i(x)
Intuitively,
number
of
_].
or if y is undefined•
the basic
recur-
function
such
respec-
x then %x[P(x)]
(i.o.)"
"for
used for
is true for
means
that
sufficiently
that for all
large functions
is an f >- b (a.e.).
f..•"
"
recursive
function
_ = [_0,_],o..]
in a
is a sequence
two axioms:
(_i) for all i E/_,
= y] is a recursive
the amount
halts
after
and
predicate•
of time or space
receiving
input x.
"_ i (x) e y" is true by convention
We assume
paper of Blum
often",
is also
The phrase
is a b E _I
for "recur-
that P(x)
"xx[P(x)]
A Blum measure
(_0
i) = domain
it finally
means
is the ith partial
satisfying
the statement
(a.e.)"
"for arbitrarily
_i(x) represents
i when
defined,
"there
there
_]
The %-notation
Similarly,
Similarly,
The function
standard
on _.
the variable
many x E _.
f..." means
"for every
means
and "infinitely
containing
"%x[P(x)]
is true for infinitely
means
with
everywhere",
many x C _.
f-> b (a.e.)...".
in this paper
(total)
"r.e • ", "a.e ." , and " i.o. " are used
of one variable
large functions
and _ n is the set of
"Function"
is a statement
all but finitely
_ n is the set of partial
l_/.
The abbreviations
tively.
integers•
the reader
[1], and with
used by program
If _.(x)
i
for any y E
is at least cursorily
the papers
is un-
of Hartmanis
familiar
and
-3-
Stearns
[7, 16] on time and space
is taken
arate
to be the tape square
input, output,
visited
input x.
tape by Turing
tape.
occur on the storage
As an aid
functions
in exposition,
(determined
that t <_ %x[g(t'(x),x)]
3.
machine
machines.
machines
i when
tape,
and that T satisfies
by context
sep-
squares
halts
computations
axioms.
assert
that two
This means
and independent
on
and the output
the Blum
informally
equal".
have
of tape
it finally
that non-trivial
we sometimes
T = _T0,T],... ]
which
= the number
to be a read-only
t and t' are "approximately
a fixed g E _2
Ti(x)
This ensures
tape,
Turing
on Turing
tapes;
The input tape is taken
tape is a write-only
must
measure
and storage
on the storage
bounded
that there
is
of t and t') such
(a.e.) and t' _< _x[g (t (x),x) ] (a.e.).
RECURS IVE ENUMERABIL ITY
The basic
objects
bounded
by functions
grams.
Whether
(3.1)
E /N
with
respect
Remark.
and output
computes
(rather
the function
than almost
example,
much
all) inputs
measure
larger
indices)
machine
uses
x.
than
t(0) cannot
respect
to
The set of functions
separate
input
is a program
which
tape squares
for all
the Turing
that a constant
be computed
with
T with
On the other hand,
has the property
pro-
I_i< t (a •e •)_=_ili
iff there
t(x)
by these
are
on the measure.
t-bounded
tape-measure
at most
measures
and _ a Blum measure.
i E_]
,t)def'_0
=
is t-computable
and which
computed
and _i _ t (a.e.)_.
to _. is_
For the Turing
whose
depends
be a function,
I _Pi E _1
tapes, a function
number-of-steps
is t-bounded
(or more precisely,
def'fi
t-computable
and the functions
Let t:_-_/_
The set of programs
(3.2)
t:_-+_,
are the set of programs
a program's measure
Definition.
is F(_,t)
of study
machine
function,
in t steps at all
for
E F(_,t)_.
-4-
arguments,
because
zero requires
condition
printing
more
than t(0)
is included
(3.3) Remark.
redundant,
finitely
often.
in place
recursive
anticipate
any difficulties
Hartmanis
recursive
More
function
strongly,
halting
(3.4)
The results
of F(_,t),
problem
Fact.
(3.1) is not
(and hence
_i
of this paper
not attempted
also
apply with
to restrict
to extend
domains,
_i ) to
our results
though we do not
so.
[7] observe
is t-computable
Young
in Definition
co-infinite
in doing
everywhere"
but it is convenient
We have
with
at argument
e-of the F and _ classes.
_i-< t (a.e.) allows
functions
and Stearns
the "almost
that _0i C _I
to total functions.
to partial
of the function
Hence,
in the definition
the condition
fiI_ i <_ t (a.e.)]
attention
steps.
The requirement
since
be undefined
out the value
that it is undecidable
for any given
[17] has pointed
sufficiently
out that a standard
whether
large
reduction
a
t E
I"
to the
implies
For any Blum measure
_ and any function
t E _I'
_"_(_,t) _ _ = _iI_0i E _"
_(_, t)] is not r.e
Essentially
the same reduction
(3.5)
Fact.
For any Blum measure
F(_,t)
is not r.e.
An elegant
generalization
The observation
ently
when
large t C _I
argument
_, and all sufficiently
of fact
in Hartmanis
t = _i for some _i E _ I' their
_(_,t)
by programs
asks whether
this
(3.5) appears
that _i_(_,t) is an r.e.
is implicit
which
yields
proof
and Stearns
yields
recursive
for suffici-
[7].
Moreover,
an enumeration
bound
t.
t E _I'
in Blum [2].
set of functions
run in (approximately)
is true for arbitrary
large
t.
Young
of
[17]
The following
theorem
-5-
gives an affirmative
(3.6)
Definition.
sively
enumerable
r.e.
iff _
partial
A sequence
of (partiaO
functions
(r.e.) iff %i,x[fi(x) ] E _2"
= _filiE _]
The proof
answer.
for some r.e.
of the following
A set _
sequence
useful
f0,f],..,
is recur-
of functions
of functions
is
f0,f],...,
fact is left as an amusing
exercise
for the reader.
(3.7)
Fact.
A set _
_
_]
= _0iii
is r.e. =
E W_ for some r.e.
set W c_
= _q0ili E R] for some recursive
If _
c
_I
is r.e.,
large t E _1,
(3.8)
_
all sufficiently
_, there
large t E _1,
is an r.e. set W c F(_,Xx[g(t(x)
there
is a g E _2
such that:
for
,x) ])
_0iii E W].
of the proof:
grams enumerated
sufficiently
For any Blum measure
such that _(_,t)=
grams computing
_, and all
c _(_,t).
Theorem.
Sketch
then for any Blum measure
set R c_/.
The object
all and only
actually
is to enumerate
the t-computable
have measures
a sequence
functions
not much
larger
of pro-
such that the prothan t.
We
shall
th
indicate
how to do this for the tape measure
enumeration
to compute
compute
is defined
t(0) within
_0i(0) within
possible,
T.
The i
program
by the following
instructions,
x tape squares.
If this is possible,
any of the x tape
try to compute
until the x tape squares
t(]),
squares
remaining.
then _0i(]), etc.,
are exhausted.
"Given
input x, try
try to
If this is
continuing
If at any point
in the
in this manner
this process
-6-
turns up an argument
more
than t(y)
tape squares,
simultaneously
tape squares
y at which
computing
the computation
of q0i(y) loops
halt and give output
q0i(x) and t(x) without
for _0i(x) than have
been required
output _0i(x).
(if _0i(x) loops or requires
Clearly,
immediately
halt
more
instructions
Ti(Y) > t(y) for some y, this will
be discovered
large
Observing
using
and give
than t(x)
compute
q0i.
by the program
compute
a function
the functions
equal
to zero
(a.e.) can be computed
(3.2)
that the programs
from remark
t-computable
functions.
Moreover,
tape squares
for almost
all inputs
(3.9) Corollary.
x.
(Hartmanis-Stearns,
all sufficiently
(3.]0) Remark.
each program
large t C _1,
Given
(a.e.).
in space
enumerate
uses at most
the
t(x) + x
F2
Young)
_(_,t)
are equal
For any Blum measure
is an r.e.
any _, it is certainly
functions
above
to zero
set of functions.
not the case
to _(_,t)
_, and
that all r.e.
for some t E <_1.
For
I
sets of recursive
equal
if
on all
will
zero, it follows
tape
Also,
and the program
that
more
zero."
if T.l _ t, the preceding
inputs,
try
for t(x) up to that point.
in the alloted
squares) ,halt and give output
Otherwise
at any point
If _0i(x) can be computed
Otherwise
space
zero.
or requires
example,
let f,g E _1
g E _(_,t)
_ f _ _(_,t).
of some program
and hence cannot
functions,
Then
for g, the r.e.
be obtained.
the Blum axioms
(3.]]) Fact.
measure
be such that g is harder
Let _¢-_
_ such that _=
_1
for any t E _1
to compute
larger
are weak
given
enough
g but not f,
any r.e. set of recursive
that we have
be an r.e. set of functions.
_(_,Xx[0]).
f, viz.,
than the measure
set _ (_,t) - (f] contains
However,
than
There
is a Blum
-7-
(3.12)
Remark.
corollary
measure
hand,
Thus far we have
(3.9) it follows
_ or t E _1'
if we choose
functions
bounded
predicate.
Define
increasing,
and
tions
the
an
for
For
roughly
sets
and
b E _
of
some
of
f.(x)
1
= huge(x)
f.(x)
1
requires
is
1
sequence
d(x)
sets
all recursive
functions
if we require
_, there
t can
that t be bounded
is a b E _I
as t ranges
by
1•
and
much
Fact
Let
program
over the
such that
(non-recursive)
A (x) where
i
characteristic
(3.7)
d 6 _1
for
sequence
contained
be
a
huge
function
CA'I (again
of
E A i] is a recursive
E _I
function
is
A0,AI,...
is very rapidly
of
in
the
The
sequence
b-computable
which
d exists
A.1
for
since
each
funci exceeds
CA0,CA.I,...
is
functions).
we have
steps
the
an infinite
such that Xi,x[x
f.(x) = _(x).C
l
CA • E _1
x _ At,
_(_,t)
Construct
recursive
r.e.,
some
measure
r.e.
On the other
=
true even
Blum measure
many
of the proof:
is
for any Blum
not to be r.e.
majorizes
From
above by b.
infinite
f0,fl,..,
_I
_ _(_,t)
that non-recursive
remains
For every
there are uncountably
disjoint,
t which
to t C _I"
function.
Theorem.
Sketch
that _1
then
indicates
This
attention
is well-known
function),
remark
lead to new classes.
functions
_I
a (non-recursive)
The preceding
(3.13)
immediately
since
(e.g. Rado's
by a recursive
restricted
for
by
f. 1 (x)
almost
choosing
more
than
= 0,
all
and
such
_
d(x)
moreover
f. 1 (x)
x.
On the
other
large
enough,
it
steps
to
compute
can
hand,
can
for
be
be
almost
computed
for
x E A i'
guaranteed
all
in
such
that
x.
(a.e.)
-8-
Now choose
any S c_,
and let
i
ts(X) =
It follows
cisely
that
[fiii E S].
are uncountably
(3.]4)
is not an r.e.
Proof:
4.
there
many distinct
t which
otherwise.
if x E A° for some i E S
those functions
Since
Corollary.
function
(x)
b(X)
are ts-computable
are uncountably
classes
For every
is bounded
f.
l which
sets S c_,
of ts-computable
Blum measure
above
many
_, there
by a recursive
are pre-
functions.
there
[_
is a (non-recursive)
function
such that _(_,t)
set of functions.
There
are only countably
many
r.e.
sets of functions.
[_
INVARIANCE
The Blum
axioms
are intentionally
theorem
following
arising
from any conceivable
whether
an r.e.
functions
the axioms
with
Fact
of time or space
(3.]]) indicates
is a class
The following
that any
that
of t-computable
definition
is useful
in
_' is a refinement
of
large
t E _1,
there
is a t' E _I
such that
(_' t')
same classes
which
of t-computable
is obviously
allows
model.
Let _ and _' be Blum measures.
In short, measures
ment
being
on the measure.
iff for all sufficiently
_: (_,t)
the point
to all notions
functions
the measure.
dependence
Definition.
applies
machine
set of recursive
varies
considering
(4.])
from
weak,
a transitive
one to ignore
are refinements
functions
relation
of each
other define
for large enough
on measures.
some of the pathologies
t E _].
the
Refine-
This definition
illustrated
by fact
(3.]]),
-9-
and it is not obvious
of each
(4.2)
other.
This question
Theorem
measure
that there are measures
(Blum).
was answered
Given
of the proof:
compression
theorem,
arbitrarily
complicated
range(d)
= [2,3],
Define
_ nor _' is a refinement
By a minor modification
pairs
of functions
and d is slightly
of the proof
neither
Theorem
(independently
classes
(4.2) indicates
more
can be obtained.
accurately
machine
models
However,
putable
(4.3)
reflect
classes
Theorem "
before
properties
further
or if
sufficiently
than d in the _' measure.
of the other.
axioms
restrictions
which
by Borodin
measures
For any Blum measures
of t-computable
restrict
measures
arising
[3] and Young
we can assert
interlace
must be placed
notion
of time or space
restructions,
t:_-+_,
to grow
a measure-invariant
Additional
of different
that for all functions
= _0,I],
than c.
otherwise.
if _i(x) is undefined,
to compute
that additional
have been considered
even without
are
such that range(c)
of c and d) h C _2
that c is harder
of measure
_ there
(x)
_ nor _' is a refinement
on the notion
of Blum's
_' as follows:
(_i (x),x)
_hi(X)
Hence
measure
to compute
'(x)=
fast, one can guarantee
a Blum
of the other.
c,d E _1
harder
Blum.
_, one can construct
one can show that in any given
a new measure
By choosing
are not refinements
for us by Manuel
any Blum measure
_' such that neither
Sketch
which
from familiar
[17, 19].
that the t-com-
in a regular
_ and _' ' there
to
manner.
is a g E _ 2 such
-10-
F(_,t) c F(_',Xx[g(t(x),x)])
and
F(_ ',t) c F(_,%x[g(t(x),x)]).
Proof:
By a theorem
such that for every
(a.e.).
of Blum
i E _'_i
The theorem
[1], for any _,_',
<
_ %x[g(_ i'(x),x)]
follows
if we assume
there
is a g E _2
(a.e.) and _i <- Xx[g(_i(x),x)]
(without
loss of generality)
that
g is non-decreasing.
5.
SET THEORETIC
For
STRUCTURE
f,g E _],
g can be expressed
an almost
(5.0)
Blum's
by saying
everywhere
Remark.
program
the notion
faster
Speed-up
The
theorem,
relation
who observed
equivalence
that for every
program
An apparently
for f is faster
union
there
"as complicated
classes
of equally
that the _
follows
(though
classes
in terms
formulation
program
f, there
might
for g.
_I
was proposed
leads to a partial
computable
as
is
of set inclusion
are actually
the _
closed
be that no
In view
partially,
classes
under
are measure
order
[]3],
partial
the classes
E _(t)
rather
= g E _(t)].
than totally
are not closed
under
intersection).
independent,
for g.
on the
This
among
of
program
by Rabin
functions.
as g iff (Vt E _])[f
from the fact that
The results in this section
mention of _.
and
complex
they are trivially
computing
may not be a "fastest"
as" on
that it is transitive
since f is as complicated
The fact
ordered
more natural
however,
program
(hard to compute)
for g.
than the "fastest"
order can also be considered
_(t)
that f is as complicated
so we suppress
-11-
(5.1)
Theorem.
such
that
The
[7]
of
For
_(t)
proof,
as
(5.2)
Theorem.
of
omit,
for
order
of
The
[11]
we
theorem
order
is clear
of
countable
of
Fischer
[10].
(5.3)
is
_I'
proof,
that
recursion
additional
I) _
range(c)
=
_0,1}],
(5.3.2)
_
(t I) _
the
there
is a
t'
E
_I
f: _/-+/_.
proof
by
Hartmanis
and
Stearns
measures.
classes
is
extremely
compli-
classes
_ _2
E
is
isomorphic
partially
(depending
--_i, the
a
we
is a
to
ordered
only
family
omit,
_
is based
recursive
can
theorem
_(t
order
can
under
the
on
some
be
family
set
such
measure)
chosen
as
a
be
partial
family
that
_
result
of
for
2) =
(_c
2) =
(_c E
which
sub-
theorem
of Mostowski
on /_
embedded,
appears
into
and
in Meyer
which
a generaland
is proved:
Theorem
all
a
order
isomorphically
construction
stronger
on
(5.2)
_(tl),_(t
E _(t]))[_0
i =
c _
can
2)
be
chosen
to
_(t
I)
satisfy
_:
_i >
t2
(a.e.)
and
and
_(t
inclu-
h <_ t <_ %x[g(h(x),x)](a.e.)].
conditions
_(t
h
order
The
(5.3.])
C _1,
function
machine
partial
which
Actually
Corollary.
t
t-computability
a g
there
partial
ization
Turing
the
large
t E
asserting
any
resembles
(t E _i)
arbitrarily
lengthy
for
countable
there
{_(t)I
for
large
from:
Any
Moreover,
that
the
which
t-computability
sion.
any
_ _(f)
partial
cated,
sufficiently
U _(t')
a similar
The
all
- _(t2))[range(c)
=
_0,I}].
-12-
We now establish
classes
(5.4)
which has
Definition.
bounded
(5.5)
There
A set _of
finite
Union Theorem.
Proof:
Say _
of the t-computable
corollaries.
total
subset _0
Let _
c
_1
functions
c_
from /_ to /_
there
be an r.e.
such that _F(f)=
is self-
is a t E _
F(t),
self-bounded
J
and hence
= _f0,f] ,.. .] and %i,x[fi(x)]
f0,ma_._x[f0,f]],max_f0,f],f2},..,
defines
property
such
that
(a.e.).
If E _0]]
is a t E _1
closure
some surprising
iff for every
t > Xx[max[f(x)
a powerful
the same union as _.
Hence,
we assume
set of functions.
__(f)=
fE_
E _2"
is a non-decreasing
!
_t).
Observe
that
r.e. sequence
without
which
loss of generality
that i > j _ f. e f. (everywhere).
i
j
The function
"guess(])",...
ables
are explicitly
wise,
i E A(x),
= x.
variables
from stage
The computation
Find
= _, define
Certain
the computation,
the same values
_i <_ xi_i(x)
E A(x)},
"guess(0)"
and these vari-
to stage
begins
unless
at stage
> fguess(i)(x)];
t(x) = fx (x) and go to stage
t(x) = min{fguess(i)(x)Ii
zero.
call this
x+].
set guess(i)=x
they
Other-
for all
and go to stage x+1."
leave as a difficult
i, _.
<_ t (a.e.) =
i
for some fk E_.
Corollary.
exercise
(_n,k E_)[guess(i)=k
and _i _ fk (a.e.)] _
(5.6)
during
set to new values.
If A(x)
define
We
to have
"Set guess(x)
set A(E).
in stages
are given values
are assumed
Stage x:
t is computed
(_k E/_)[_
the proof
of the claim
at every
i _< fk (a.e.)].
that for all
stage after
Hence
stage
i E F(t)_
n
i E F(f k)
_
There
is a t E _1
such that
the set of primitive
recursive
-13-
functions
with
(of one argument)
respect
machine
to both
the Turing
the set of t-computable
machine
space measure
P E _1
By theorems
is primitive
tive recursive
recursive
space.
convince
recursive
of one argument
the corollary
himself
machine
[4], a function
computing
function.
Since
p uses
the primi-
are an r.e. self-bounded
follows
from theorem
that the same function
(5.5).
t works
set of
We
let
for time and
[_
Corollary•
Same as corollary
(5.6) with
"primitive
11
replaced
by "elementary
Grzegorczyk
Proof:
and "multiply
Each
sets of recursive
[14] observe
property
functions
recursive
of these classes
functions•
that a function
Remark•
Corollary
and the R. W. Ritchie
that only
(5.9)
Remark.
almost
everywhere)
classes
The function
functions
functions
of PSter"
satisfy
t of Corollary
primitive
to be r.e.
recursive
[12]
self-bounded
[8, 9], and D. Ritchie
the R. W. Ritchie-Cobham
in time
_!
to the Grzegorczyk
of predictably
space
of
of PEter
iff it can be computed
applies
machine
functions
recursive
in the class.
(5.6) also
the Turing
every
also
recursive"
_11
_
and D. Ritchie
is in the class
by a function
[8],
are well-known
Meyer
that these classes
or space bounded
providing
of Kalmar"
[8] for each n > 3", "n-fold
for each n > 1"
_2
and the Turing
[15] and Cobham
iff some Turing
by a primitive
functions
functions,
the reader
of R. W. Ritchie
recursive
time or space bounded
(5.8)
functions
time measure.
Proof:
(5.7)
is precisely
computable
measure
class
functions
is used.
(5.6) must
function,
majorize
and hence
(exceed
cannot
-14-
itself be primitive
the Turing
machine
considerably
also
more
majorizes
below,
time or space
slowly
t cannot
measure,
recursive
be t-computable
l_is means
than Ackermann's
the primitive
[3] has observed
[6, 16] for Turing
of Blum measures,
machine
that
function,
functions.
that t must
for example,
By remark
the minimal
but not all Blum measures.
weaker
result which
Definition.
growth
in
grow
which
(5.12)
rate theorems
time and space can be extended
rate be non-decreasing.
a somewhat
(5.]0)
Therefore,
t can be made non-decreasing.
Borodin
growth
recursive.
A weak
He requires
By relaxing
however
minimal
class
that a minimal
this condition
applies
growth
to a wide
we can obtain
to all measures.
rate
such that lim inf t(x) = oo and for all i E/_,
is a function
t: /_/-+/_
_i <- t (a.e.)
X->CO
(_k E _!)[9.
(5.]I)
l
_ k (a.e.)].
Corollary.
Proof:
The constant
sive functions.
t-bounded
There exists
Remark.
recursive
question:
recursive
weak
minimal
if f0,f],..,
functions,
A modification
is
is a t E_]
of the union
do not exist
U
F(f k)
k
= F(t)
of the union
such
that the
That
theorem.
growth
self-bounded
in which
suggests
sequence
shows
non-decreas-
the following
of non-decreasing
for some non-decreasing
theorem
rate.
set of recur-
programs.
of Blum measures
rates
growth
self-bounded
construction
is an r.e.
of the proof
there
minimal
the constant-bounded
from the proof
Borodin's
weak
are an r.e.
theorem,
are precisely
lim inf t = oo follows
(5.]2)
functions
By the union
programs
a recursive
t E
that the answer
l"
is
-15-
yes providing
that any one of f0,fl,..,
counter-example
Another
theorem
cannot
corollary
be extended
machines.
An
languages
important
recognizable
machines
properly
space bounded
bounds
the question
(5.13)
Corollary
a language
machine
observation
problem
languages
by Turing machines.
can be settled
(Blum).
There
is recognizable
within
iff it is recognizable
functions.
(made independently
space-bounded
in automata
by nondeterministic
those
Thus Borodin's
the constant
non-deterministic
open
contain
linear
beyond
is an amusing
(5.5)) by Blun about
is unbounded.
theory
linear
Turing
is whether
space bounded
recognized
of
the
Turing
by deterministic
For certain
peculiar
space
trivially.
are arbitrarily
space
within
large t C _I
such
t on a nondeterministic
space
that
Turing
t on a deterministic
Turing
roachine.
Sketch
that
of the proof:
any language
machine
Then
immediate
bounded
in space bounded
language
g _ _I
self-bounded
with
in the sequence
is also recognizable
to the union theorem
in the sequence.
of the computation
the desired
deterministic
and it is
in space
deterministically
By reformulating
of characteristic
for nondeterministic
to obtain
to simulate
sequence,
nondeterministically
in terms
machine.
space required
machines
such
nondeterministic
on a
recognizable
tions and using a space measure
appeal
f E _I
the extra
is an r.e.
by the next function
recognition
increasing
gOf on a deterministic
nondeterministic
f, gOf, gogof,..,
by a function
space
space
describes
more powerful
that a language
is a strictly
within
within
g essentially
the apparently
ones.
recognizable
can be recognized
The function
There
machines,
t E _I"
[_
func-
one can
-16-
The preceding
linear
corollary
space bounded
computational
Turing
complexity
only
mentioning
that functions
6.
The classes
functions
recursive
becomes
(6.1)
of t-computable
whether
a more
from
such that
Gap Theorem.
theorem
that
it is worth
(5.]3) can be found
(6.]) which
from
those classes
We now ask whether
a class
of bounds,
arises
between
the following
Theorem.
(Blum)
For every
for every
_i E _1,
such
only
of this question
theorems:
Blum measure
_, there
(x),x)]
For every
t E _1
the
and in
by considering
The significance
large
Blum's
(3.]3)).
by non-recursive
times.
(Borodin)
_(_,t)
different
structure
) _ _-"
_(_,Xx[g(_i
F(_,t)
Remark.
i i
,,,0
tractable
the contrast
there are arbitrarily
(6.3)
about
on the grounds
determined
too broad
are running
_(_,_i
(6.2)
significantly
t (cf. Theorem
Weak Compression
is a g E _2
functions
are themselves
actually
obvious
for
BOUNDS
by recursive
bounds which
theorems
computations,
corollary
problem
above by _x[log2(x)].
functions
particular
criticized
time-consuming
t satisfying
have properties
determined
on the original
Hc_wever, since
sometimes
to excessively
are bounded
WELL-BEHAVED
machines.
are
they apply
which
sheds no light
Blum measure
_, and every
g E _2'
that
= F(_,xx[g(t(x),x)])
and hence
=_(_,Xx[g(t(x),x)]).
compression
follows
theorem
as an immediate
is much more
powerful
than
corollary.
Theorem
(6.2)
is
-17-
also weaker
than the original
The following
preceding
(6.4)
there
Corollary.
a _i
These
like
the
results
observation
One
the
of
is
the
(6.5)
in
the
that
Definition.
with
respect
Definition.
forms a measured
functions
graph
in
be
making
only
be
The
hence
_(_,t)
namely,
same
the
remark
the
struc-
properties
applies
running
times
on which
time
o_._f a
running
time
approximately
are
"honest"
_ be
(_,_i).
theorems
about
of
reflect
-
that
statements
indicating
t E _I'
to
Blum's
automata.
thed.r
a Blum
(cf.
Meyer
computational
measure
and
theorem
(6.1)
equals
and
D.
Ritchie
complexity.
g E_ _2"
A function
_ E 0_)1
to _ iff (_i E _)[_0 i = _ and _i _ %x[g(_(x),x)](a.e.)].
can also be formulated
using
Blum's
[|] defini-
set.
A sequence
set iff hi,x,Y[_i(x)
Any Blum measure
is recursive.
the two
features
functions
(Blum)
_I"
and
point;
nondeterministic
of honesty
tion of a measured
Fact.
classes.
running
Let
to
the
with
_ such that for every
important
may
values
together
= F(_,_i)
classes
for
S,.,ch
their
The notion
(6.7)
appear
about
time.
is g-honest
(6.6)
which
essential
that
running
[9])
an
names
(5.13)
F(_,t)
illustrate
theorem
chosen
that
by Borodin.
trilogy.
is a Blum measure
such
t-computability
badly
depends
There
proved
of (6.13) below,
form a surprising
E _1
gap
tur__._eof
of
corollary
theorems,
is
gap theorem
A function
_ = _0,_i...
] of
= y] is a recursive
is a measured
can be a member
(partial)
functions
predicate.
set, as is any r.e.
of a measured
set of
set iff its
-18-
(6.8)
form
Fact.
The set of functions
a measured
set for any g E _2
given any measured
functions
(6.9)
Theorem.
For every
that:
for all t E _1,
with
9, there
respect
Blum measure
there is a t' E _ N _1
F(9 ,t') and hence
__
of a priority
(see also, Young
Detailed
argument
of
the
9.
respect
Conversely,
is a g E _2
such
that
the measured
is a measured
such
to 9
set.
set _ such
that
_,t ) = _(9 ,t ').
interest
in the theory
[]8], and Borodin
sketch
with
to _ contains
9, there
The proof of (6.9) is of independent
examples
g-honest
and Blum measure
set and Blum measure
the set of g-honest
F(_ , t)=
which are
as one of the first
of computational
complexity
[3]).
proof:
Given
t E _1,
our aim is to construct
I
t'
E _1
such that for all i E_,
means F(_,t)
the number
= F(_,t'),
and also
of steps required
be in a measured
(Detailed
_.
l > t (i.o) _ 9 i > t' (i "o.), which
such that t' is approximately
to compute
t'
which
ensures
equal
that t' will
set.
sketch
of the proof
continued
to
on the following
page)
-19-
In order
to guarantee
a dovetailing
will
procedure
generally
will
not be time to compute
be chosen
larger
which
programs
to satisfy
The computation
Programs
0,],...,x
priority
ordering
unless
grams
of steps
are under
which remains
also remain
infinitely
program
than y.
The value
first,
y by those
of t'(y)
it must be
programs
those
t at some arguments.
The
beginning
below.
at stage
zero.
at any stage x and have a
Similarly,
means
on the value
than
the same as it was
the same from
depend
described
in stages
running
This difficulty
it must be smaller
mechanism
generally
at the previous
at each
stage
their running
times
stage
certain
times
to pop above
to stage
stage
pro-
exceeded
t' somewhere.
unless
they are
changed.
Let _(x)
Stage x:
their
which
to exceed
a "pop", which
and they want
Pop requests
explicitly
changed.
t'(y).
at argument
consideration
by
arguments.
is that there
conditions:
and second,
of t' proceeds
are requesting
t somewhere
taken
by a priority
it is explicitly
conditions
two conflicting
have been discovered
are resolved
condition
other
be defined
of t' on smaller
defining
argument
to be t-bounded,
which
conflicts
t(y) when
t' will
of t' on large arguments
large values
the first
computed
than the number
appear
small values
t'(y) to satisfy
t of some easily
must
before
in satisfying
is met by defining
of
so that
be defined
The difficulty
the second condition,
be some recursive
function
which
takes
all integer
values
often.
Part
I.
Assign
x not be popped.
lowest
priority
Devote
x steps
to program
x, and request
to a dovetailed
t, and see if there is a z such that t(z) can be computed
but not at Part
the alloted
I of any earlier
time, go to Part
II.
stage.
computation
at this
If such a z cannot
that
of
stage,
be found
in
-20-
Otherwise,
and request
Part
simulate
programs
that those programs
0,1,...,x
which
on input
did not halt
z for t(z) steps,
be popped.
Go to
II.
Part
stage;
II.
Let y = _(x).
See if t'(y) has been defined
if so go to stage x+].
requests
of programs
Try to find
0,],...,y
the program
that P was requesting
programs
which
determine
y were
at the end of stage y.
P with highest
requesting
had higher
take as many
steps
priority
priority
define
priority
and
t'(y)
on input y.
to be the largest
satisfying
(at this stage x) to P, request
number
(]) and
of
(2),
that P not be
and go to stage x+].
However,
the preceding
attempt
(3)
it requires
more
(4)
it requires
more steps
If condition
occurs
y), such
than P,
steps taken on input y by any of the programs
popped,
(at stage
no____tt
to be popped,
on input y as P takes
If such a P can be found,
lowest
and
a pop at stage y and such that none of those
(2) at sta_e y
assign
the priorities
both
(]) at _
actually
Otherwise,
at an earlier
steps required
Stopping
every y.
y = _(x)
t'(y)
to compute
condition
Moreover,
than x steps,
(3) occurs
first, define
to find P should
go to stage
to compute
x+].
t(y).
If condition
to be the larger of t(y) and the number
t(y);
go to stage x+].
(4) guarantees
that
t'(y)
is too large
(4)
of
END.
t'(y)will
t'(y) can fail to be defined
only because
if either
or
than are required
first,
be stopped
be defined
for
at a stage x such that
to compute
in the time alloted
-21-
to stage x.
Since
(independent
of t) recursive
the number
in the Turing
by at worst
The proof
is based
that
of x, it can now be proved
to compute
machine
by a fixed
t' is approximately
case
equal
the time to compute
an elementary
function
composed
the t-bounded
and t'-bounded
with
at infinitely
the claim requires
version
many
a delicate
stages _ _i > t'
programs
argument
which
(i.o.).
that
to t'.
t' will
be
t'.)
are the same
on the claim that _i > t (i.o.) _ the pop request
is changed
of program
Verification
we postpone
i
of
for the final
of this paper.
The following
corollary
(6 ]0)
•
to stage x is bounded
function
of steps required
(For example,
bounded
the time alloted
rather
artificial
(6.4) in a stronger
Definition
definition
us to conclude
form.
Two Blum measures,
.
allows
_ and _'
, are similar
iff _i = _i
for all i such that range(_0i) _ _0].
(6.1])
Remark.
In particular,
%x[0]
E
(6.]2)
(_,t) A _(_',t)
Fact.
We have
measures
are trivially
if _ and _' are similar,
Given
a Blum measure
(6.]3)
Similar
_
then for all t: _-+
(_,t) =
any measured
Corollary.
from
Every
set _ and Blum measure
to _'
t E _I
_,
_, there
is
(6.9) and (6.]2):
Blum measure
there
other.
and _ c _'
is similar
to a measure
!
that for every
of each
_',t).
_' such that _ is similar
immediately
refinements
is a _i E _1
(_' 't) =_(_'
'_i')"
such
that
_' such
-22-
Our final result
familiar
time and space measures
automaton
models.
functions
(6.14)
Definition.
sufficiently
Fact.
the function
mally,
i_i
(6.16)
since many
such that
t E _I
the important
to the
machines
property
and other
that the
are "proper":
_ is proper
iff _i E _ (_,_i) for all
E _1"
The running
Theorem.
from Turing
and hence
A Blum measure
time of a function
steps
are required
let _ be any Blum measure.
g E _2
large
have
are Xx,y[x]-honest
large
(6.4) does not apply
arising
Such measures
measure
(6.15)
is that corollary
There
(Vj E_)[q0j
to print
be much
smaller
a large value.
than
For-
is a
<-Xx[g(_j(x),x)](a.e.)].
Let _ be a proper
such that for every
cannot
Blum measure.
There
exist
arbitrarily
_i E _i,
(_,t) _ __,_( _i ).
Proof:
of
(6.1) and
increasing.
Let h C _2
(6.15), and assume
For any f E _I'
The sequence
functions.
F(_,t)
By
= _F(_,f
n
Suppose
f0,f],...,
the union
without
define
of the functions
loss of generality
g E _2
that h is
f0 = f' and fn+1 = Xx[h(fn(X),X)].
is an r.e.
theorem,
maximum
there
increasing
is a t E _1
sequence
of recursive
such that
n ).
_"
_(_,t)
since _ is a proper
By choice
be the pointwise
= __(_,_i
measure,
of t, we have
tonicity
of h, we have
(a.e.).
Hence,
) for some _i E
i.e.,
there
I.
Then
_ i E _ (_, t)
is a q0j = _. such that 9o <- t (a.e.).
_. <_ f
(a.e.) for some n.
j
n
_i = _0j < Xx[h(_j(x),x)]
By
(6.15) and mono-
< _x[h(f n (x),x)]
= f n+]
-23-
%x_(_i(x)'x)]
Therefore,
by
of
ing
names
still
main
implication
names
for
avoid
compression
from
tion
for
large
it happens
measured
The
such
phenomenon
gap
Another
set,
namely,
(i.e.,
sizes
the
We
negative
is
that
is
) c
that
defined
members
_
(_ ,t) ,
as
it
there
there
is
t E
_1"
by
of
computability
of
there
theorem
are
no
By
a measured
classes,
appears
a well-behaved
in
and
(6.2),
gaps
above
restrict-
set,
at
we
the
can
same
since
Blum's
bounds
chosen
postpone
got
of
a measured
set
of
(6.]6)
for
any
t'
exceeds
Xx[g(t(x),x)]
is
an
question.
t' can
g
occurs
even
are
gaps
them.
Although
at
a more
We
phenomenon
the
consequences
in
gap
(6.2)
when
function
thoroughly.
non-decreasing
_(_,fn+2
(6.9)
to be
that
sort
gap
set.
open
(a.e.)
set.
gap
investigated
same
implies
Borodin's
t'
t)
the
that
that
C
classes
the
Remark.
t in
Theorem
functions
theorem
in a measured
of
exactly
a measured
(6.17)
I.(x) ,x)])
t-computability
(bounding
can
fn+2
I-]
describe
time
a_"
_(_,%x[h(_
_
a contradiction.
set
=
(6.])
O-I (_,_i)
The
_ %x[h(fn+](x)'x)]
can
E _2'
(i.o.).
below
canno____t
be kept
function
the
g of
top
complete
of
and
show
there
F(_,t)
by
is also
open
be
constructed.
too
gap
ca____n
be
kept
the
of
arbitrary
t'
whether
_1
must
when
this
t E
have
similar
such
exceed
that
funcset
grows
argument
t E
the
(6.2)
= F(_,t')
an
is a
Whether
It
an
functions
in a measured
exposition
replacing
for
rapidly),
in a
point.
_1
yet
to
any
by
to
a
t'
be
the
proof
corresponding
Xx[g (t (x) ,x) ] (a.e.)
t is non-decreasing,
a
- 24 -
ACKNOWLEDGMENTS
The authors
the interest
would
shown by Manuel
(4.2), by Patrick
Michael
Blum, who
C. Fischer, who
in the proof of Theorem
sharpened
like to express
and simplifying
sincere
provided
suggested
(4.]).
for his invaluable
a significant
this material.
We would
help
appreciation
the proof
(6.9), and by Paul Young,
form of definition
J. Fischer
their
of Theorem
simplification
who proposed
especially
for
the
like to thank
over many months
in revising
-25-
References
].
Blum, M., A machine independent
JACM, v. ]4 (]967), 322-336.
theory
of computational
2.
Blum, M., Recursive function
Bull. of Math., v. 9 (]966),
3.
Borodin, A., Complexity classes of recursive functions and the existence of complexity gaps, to appear in Proceedings of ACM Symposium
on Computability
and Formal Languages (]969).
4.
Cobham, A., The intrinsic computational
difficulty
Proc. of ]964 Congress for Logic, Math., and Phil.
Holland (]964).
5.
Grzegorczyk,
A., Some classes of recursive
(]953), ]-46.
6.
Hennie, F., One-tape, off-line Turing
and Control, v. 8 (]965), 553-578.
7.
Hartmanis, J. and R. E. Stearns, On the computational
algorithms, TAM_____S,
v. ]]7 (]965), 285-306.
8.
Meyer, A. R. and D. M. Ritchie, The complexity of loop programs, Proc.
22 National ACM Conf., Thompson, Washington,
D. C. (]967), 465-470.
9.
Meyer, A. R. and D. M. Ritchie, A classification
of functions by
computational
complexity,
Proc. Hawaii International
Conf. on System
Sciences, Univ. of Hawaii (]968), ]7-]9.
theory and speed
745-750.
of computation,
Ca___nn.
of functions,
of Science, North
functions,
machine
complexity,
Rozprawy
computations,
Matematyczne,
Information
complexity
of
]0. Meyer, A, and P. C. Fischer, On computational
speed-up, Conf. Record
9th Annual Syrup. on Switching and Automata, IEEE Computer group (]968),
35]-355.
.o
]]. Mostowski, A., Uber gewisse universelle
Math., v. ]7 (]938), ]]7-]]8.
]2. P_ter,
R., Rekursive
funktionen,
relationen,
Acad6miai
Kiad_,
Ann.
Soc. Polon.
Budapest
(]95]).
]3. Rabin, M. O., Degree of difficulty of computing a function and a
partial ordering of recursive sets, Technical Report No. 2, Hebrew
Univ., Israel (]960).
]4. Ritchie,
Doctoral
D. M., program structure and computational
dissertation, Harvard Univ. (]969).
]5. Ritchie, R. W., Classes
v. ]06 (]963), ]39-]73.
of predictably
computable
complexity,
functions,
TAMS,
-26-
16. Stearns, R. E., J. Harmanis and P. M Lewis, Hierarchies
of memorylimited computations,
Proc. 6th Annual Symp. on Switching Circuits
and Logical Design, IEEE Computer group (1965), ]79-]90.
17. Young, P. R., Toward a theory of enumerations,
Conf. Rec. of 9th
Annual Syrup. on Switching and Automata, IEEE Computer group (]968),
334-350.
18. Young, P. R., Speed-ups by changing the order
to appear in Proceedings
of the ACM Symposium
Languages
(]969).
in which sets are enumerated,
on Computability
and Formal
19. Young,
Purdue
look-ups,
P. R., Computational
University
(1968).
speed-up
by table
Report
CSD TR 30,
Securit]r
Classification
i ii
i
(Security
ii
ii
¢leaelflcetion
1. ORIGINATING
ACTIVITY
ol title,
(Corporate
DOCUMENTCONTROLDATA
i iiii ii i i i
. R _ll O
i
body of ebmlract
end Indexln d annotation
author)
[2a.
CARNEGIE-MELLON
UNIVERSITY
Department
of Computer
Science
Pittsburgh,
Penns.F!vani_
I _21 3
3. REPORT
TITL. I[
CLASSIS
OF COMPUTABLE
PRELIMINARY
REPORT
4. DESCRIPTIVE
NOTES
E.
A.
M.
R.
t5. REPORT
ii
the overall
REPORT
report
SECURITY
le clasellled_
CLASSIFICATION
I UNCLASSTFIED
rb.GROUp
I
DEFIED
BY BOUNDS ON COMPUTATIONS:
"
Interim
name, mldd/e
Inlfl41,
last
name)
McCreight
Meyer
DATE
April
84.
(First
when
(Type o/report and In©lue/ve dates)
_clentific
S. AU THOR(a)
FUNCTIONS
i
mum/ be entered
?4.
TOTAL
NO.
IUI,
ORIGINATOR'S
1969
CONTRACT
OR
OF
PAGES
|?b.
]
28
GRANT
NO.
NO.
REPORT
OF
REFI
19
NUMBER(S)
_44620-67-¢-0058
b. PROJECT
NO.
9178
c.
61 54 501R
d.
681 304
i0.
.1
II.
DISTRIBUTION
has been approved
is unlimited.
NOTES
TECH,
13.
_4, ..y b. oeel...d
for
sale;
STATEMENT
This document
distribution
liUPPLEMENTARY
lib.,hleOTHER.port)REPORT
NO,S)
(_.y o_., ..._,.
la.
public
SPONliORING
release
MILITARY
and
ACTIVITY
Air Force Office
of Scientific
(SRMA) 1400 Wilson
Boulevard
Arlington,
Virginia
22209
OTHER
its
Rese
ABSTRACT
The structure
of the functions
computable
in time or space
bounded
by t is investigated
for recursive
functions
t.
The
t-computable
classes
are shown to be closed under increasing
recursively
enumerable
unions;
as a corollary
the primitive
recursive
functions
are shown to equal the t-computable
.functions
for a certain
recursive
t.
Any countable
partial
order can be
isomorphically
embedded
in the family
of t-computable
classes
partially
ordered
by set inclusion.
For any recursive
t, there
ms a recursive
t' which
is (approximately)
equal to an actual
running
time such that the t-computable
functions
equal the t'computable
functions.
\
DD "
I NO V ell
1 347 '
"
'
''
'
'
Security"
'"
Classification
rch
Security
C!a s siflcs
tion
........
,,
LINK
14.
K [Y
.,.,
,
A
LINK
B
•
LINK
¢
WOIqOI
,
i
,
,
,
,
,,,
MOLl[
J
WT
Security
ROLl[
,
,......
Classification
WT
,
NOt.[
WT
,