Type 2 computational complexity of functions on Cantor's space · Fachbereich Mathematik und...

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Theoretical Computer Science 82 (1991) 1-18 Elsevier

Fundamental Study

Type 2 computational complexity of functions on Cantor's space

Klaus Weihrauch Fachbereich Mathematik und Informatik, Fernuniversitiit Hagen, D-5800 Hagen, Germany

Christoph Kreitz Fachbereich Informatik, Technische Hochschule Darmstadt, D-6100 Darmstadt, Germany

Communicated by G. Rozenberg Received November 1988 Revised July 1989

Abstract

Weihrauch, K., and C. Kreitz, Type 2 computational complexity of functions on Cantor's space, Theoretical Computer Science 82 (1991) 1-18.

Continuity and computability on Cantor's space C has turned out to be a very natural basis for a Type 2 theory of effectivity (TIE). In particular, the investigation of Type 2 computational complexity (e.g. complexity in analysis) requires the study of oracle machines which (w.l.g) operate on Cantor's space. However, no general Type 2 complexity theory has been developed so far. In this paper we lay a foundation for this theory by investigating computational complexity of functions r: C··> {O, 1}* and 1:: C --> C and the related concepts of dependence and inputlookahead. In the former case results of Gordon and Shamir are extended and generalized. For continuous functions 1:: C··. C the relation between the three concepts is studied in detail. Compact sets are proved to be natural domains of resource bounded functions on C. Finally, we demonstrate that an optimization of the input-lookahead used by a machine may result in a rapid increase of computation time.

Contents

1. Background, informal summary and explanation ...................................... 2 2. Type 2 theory of effectivity ......................................................... 2 3. Continuous functions r: C --> Wd ................................................... 4 4. Continuous functions r: C -- > C ..................................................... 8 5. Relating dependence, input-lookahead, and computation time. . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6. Conclusion....................................................................... 18

References ....................................................................... 18

0304-3975/91/$03.50 © 1991-Elsevier Science Publishers B.Y.

2 K. Weihrauch and C. Kreitz

1. Background, informal summary and explanation

The theory of computability and computational complexity for functions on denumerable sets, the "Type 1 theory of effectivity", is well elaborated. Basically, it can be built up in the following way: Take the set Wd:= {O, 1}* of all the finite 0-1 words as a basis and use an appropriate machine model (e.g. Turing machines) to define computability and co~putational complexity for (possibly partial) functions f: Wd --~ Wd. The notion of computability and computational complexity can then be transferred to other sets M (e.g. the natural numbers, rational numbers, finite graphs, etc.) by means of notations P: Wd --~ M. Many comprehensive presentations of parts of this theory are available (see e.g. [4, 6, 13, 14]).

Correspondingly, for functions on "Type 2" sets, i.e. set with a cardinality not greater than that of the continuum (e.g. the subsets of~, the real numbers, or the continuous real functions), a uniform "Type 2 theory of effectivity", TIE, has been built up. Here the sets ~ and 18:= ~1\01 are used as basis sets and continuous and computable functions r: 18 --~ ~ and ~ : 18 --~ 18 are considered. Continuity and computability are transferred to functions on other Type 2 sets M by means of representations l):1B --~ M. A detailed introduction to TIE can be found in [8, 9, 14, 15, 17].

Although some interesting results from Type 2 computational complexity of real functions f:IR--~1R are already known (see e.g. [2, 3, 7,10-12]) a general Type 2 computational complexity theory has not been developed so far. This paper is a contribution towards closing the gap by presenting some new results that do not have counterparts in Type 1 complexity theory.

In the following section we will briefly review the basic concepts of TIE. For reasons of technical adequacy it will be formulated using Wd instead of ~ and Cantor's space C:= {I, 0}1\oI instead of lB. Section 3 then investigates continuous functions r: C --~ Wd. For measuring complexity we define the dependence of functions and the input-Iookahead and computation time of programs and machines calculating them. We prove some effectivity and non-effectivity theorems and study the relation between these concepts. In Section 4 the same notions will be introduced for continuous functions r: C --. C. In this case, however, the subject becomes more difficult due to the complicated domains of the functions. Our results indicate that there are limitations to a meaningful definition of computational complexity of functions on Type 2 sets. The natural domains of resource bounded functions on C are the compact subsets of Cantor's space. In the last section, therefore, we choose C as a representative of compact sets and study the relation between dependence, input-Iookahead and computation time for total functions r: C --. C.

Some additional details can be found in the author's technical report [16].

2. Type 2 theory of effectivity

In this section we will very briefly summarize the basic concepts of TIE. For reasons of technical simplicity TIE has originally been developed with ~ and IB = ~1\01

.....

.f , .. , .... "GIzi

". .

Computational complexity on Cantor's space 3

as basic sets (see [14,15] for details). However, for investigating computational complexity Wd:= {O, 1}* and e:= {O, I}N are more adequate.

As a computation model we introduce Type 2 machines. A Type 2 machine "of kind (k, m)" is an oracle Turing machine having k input tapes for words WE Wd, m one-way input tapes for sequences pEe, finitely many work tapes, and one write-only one-way output tape. For a Type 2 machine M, two functions 1M: Wd k X

em --~ Wd and gM : Wd k X em --~ e are defined. For z:=(Xl> ... ,Xk,Pj, ... ,Pm)EWdkxem , YEWd, and qEe let

IM(Z) = Y iff M with input Z halts with Y on the output tape,

gM(Z) = q iff M with input Z computes forever generating the infinite sequence q on the output tape.

For every Type 2 machine, any finite initial segment of the output tape can already be determined from finite initial segments of the input tapes. Therefore, computable Type 2 functions are continuous w.r.t. the following topologies.

We consider the discrete topology on Wd and Cantor's topology on e which is defined by the basic open sets [w] := {p Eel W is a prefix of p} for WE Wd. For pEe we define the prefix oflength k, p[k]:= p(O) ... p(k -1) E Wd. On cartesian products we consider the usual product topologies.

As in Type 1 theory of effectivity pairing functions which we will generally denote by ( ... ) are useful. For p, q E e we define (p, q):= (p(O)q(O)p(1)q(1) ... ) E C. For x E Wd and pEe we define (x, p):= (t(x)l1p(O)p(1) ... ) E e where t(O):= 00, t(1):=

01 and t(aj ... ak):=t(aj) ... t(ak). Finally for x,YEWd we define (x,Y):=

t(x)l1t(Y)EWd. Tupling with more than two factors is defined inductively: (p, x, q):= (p, (x, q» etc .. All the tupling functions are computable and homeomorphisms between their ranges and their domains. The projections of their inverses (e.g. (p, q) ~ p) are also computable.

Let a : W d -? TIM be a standard notation of TIM, the set of all Type 2 machines of kind (0, 2). A representation X: e ~ [e ~ Wd] of the set of all continuous functions r: e --~ Wd with an open set as domain can be defined as follows.

x(r)(q):= {la(xlP, q) if r = (~, p) for some x E Wd and pEe, undefined otherwise.

Correspondingly, a representation",: e -? [e -? e] of the set of all continuous functions 1:: e --~ e with a Gil-set (i.e. a denumerable intersection of open sets) as domain can be defined by replacing g for I in the above formula. Each of the two representations X and '" has a computable universal function and satisfies a continuous and a computable version of the smn-theorem (see [14, Chapter 3.2]). Since


every continuous function r: C --~ Wd (2': C --~ C) has an extension in [C ~ Wd] (in [C ~ C]), these sets are most reasonable to be studied. Note that XP and I/Ip are computable if p is computable, and that for every Type 2 machine of kind (0,1) there is a word x E Wd with fM = X(x, (00 ... » and gM = I/I(x, (00 ... ».

Let {3: Wd --~ N be the injective binary notation of N. Considering the discrete topology on N, a function 2' : C --~ N is continuous iff 2' = {3r for some continuous r: C ~ Wd. 2' is called computable iff 2' = (3r for some computable function r:c --~ Wd.

3. Continuous functions r: C -+ Wd

The dependence between input and output of computable functions has been investigated in [5]. In this chapter we will extend their results.

For any continuous function r: C --~ Wd, if F(p) = y then the output y depends only on some finite initial part of p. The length of the shortest of these initial parts will be called dependence.

3.1. Definition. Let r: C --~ Wd be continuous and assume x ~ C. Then Dep(r, X) : C --~ N is defined by

Dep(r, X)(p):= {JLk[(V q E X n[p[k]])]F(p) = F(q) if p E dom(r), undefined otherwise.

Thus, provided p E X, Dep(r, X)(p) is the length of the shortest prefix of p which already determines F(p). Note that X ~ Y implies Dep(r, X) ... Dep(r, Y) and that Dep(r, X) is a continuous function with dom(Dep(r, X» = dom(r). Hence Dep(r, X) can be expressed by Dep(r, X) = (32' for some continuous 2' : C --~ Wd with dom(2') = dom(r). However for the cases X = C or X = dom(r), 2' cannot be determined effectively from r, even if we consider only computable functions r.

3.2. Theorem. (1) There is no continuous function ~: C --~ C such that for all compu

table p E C, Dep(xp, C) = (3X.ll(p)' (2) There is no continuous function ~ : C --~ C such that for all computable p E C,

Dep(xp, dom(xp» = (3X.ll(p)'

Proof. (1) Define a computable function 2' :C2~ Wd by

2'(r, p):= {undefined if p = r.= (00 ... ), E otherwIse.

By the smn-theorem for X there is a continuous function r:c~c with 2'(r,p)=

Xr(r)(P)' Then

{o if r;c (00 ... ), Dep(Xr(r) , C)(ll ... ) = 1 h .

ot erwlse.

.. -

,

..... ,.

- I


Assume that .:1 exists. Then XLlJ(r)(ll ... ) = (0 if r,c (00 ... ), 1 otherwise) for all computable functions r. By the utm-theorem for X there is a continuous function n:C··~Wd with n(r)=l iff r=(OO ... ) for all computable rEC. By continuity, n- 1{1} is open, hence a neighbourhood A of (00 ... ) is mapped to 1. But A contains a computable function r,c (00 ... ), a contradiction.

(2) Define a computable function ~ : C2 ~ Wd by

~(r ):={p(O) if(3k)r(k),c0, , p undefined otherwise.

Apply the smn-theorem, consider the case p = (00 ... ), and assume that .:1 exists. Then a contradiction can be derived as in Case (1). 0

Although Dep(r, X) is continuous if r is continuous, the property of computability cannot be transferred from r to Dep(r, X).

3.3. Theorem. There is a computable function r: C ~ W d such that Dep( r, C) and Dep(r, dom(r)) are not computable.

Proof. Let h: f\J ~ f\J be a total recursive function such that A:= range( h) recursive. Define a computable function r: C ~ Wd by

if (3m, n)(h(m) = nand p E [O"lOml]),

if (3m, n) (h(m),c nand p E [O"lOml]), r(p):= {~ undefined otherwise.

is not

Then r(n):= Dep(r, C)(O"llO ... ) = Dep(r, dom(r))(O"110 ... ) exists for all n E f\J, and r(n) = n + 1 iff n ~ A. Since A is not recursive, Dep(r, C) and Dep(r, dom(r)) are not computable. 0

Theorem 3.4 essentially covers Theorems 1 and 7 in [5]. The dependence function Dep(r, X) determines the minimal number of input

symbols which are necessary to define the result r(p) uniquely. A program which computes r on X is called input-optimal on X iff on input p E X it has to read only Dep(r, X)(p) input symbols in order to determine the result. A program which is not input-optimal wastes input information.

We shall now discuss the problem of determining input-optimal programs.

3.4. Definition. (1) Let M be a Type 2 machine of kind (0,1) (i.e. fM:C"~ Wd). The input-Iookahead of M, Ila(M): C .. ~ f\J, is defined by

{The number of input symbols which M with input q

Ila(M)(q):= reads if the computation halts,

undefined otherwise.


(2) The input-Iookahead of the X-program p, Ila(p): e --. N, is defined by

{

The number of input symbols of q which the

Ila( )( ):= machine a(x) with input (r, q) reads, if p=(x, r) p q for some xEWd, rEe and Xp(q) exists,


Obviously Dep(xp, C) ... Ila( p) for all pEe. Contrary to the dependence function, the input-Iookahead function can easily be determined by observing the computations.

3.S. Lemma. There is a computable function .I: e ~ e such that Ila(p) = ,BXl:(p) for all programs pEe.

The proof is obvious. In particular, we can conclude that the input-Iookahead of a computable program p is computable. If, however, the dependence of a computable function .I is not computable (see Theorem 3.3) then there cannot be an inputoptimal Type 2 machine computing .I by Lemma 3.5.

By Theorem 3.2 the dependence of a function Xp cannot be determined continuously from p. However, if for a continuous function r: e --. Wd we have both, a program for r and a program for Dep(r, C), then we are able to compute an input-optimal program for r. We will show this in the following theorem which is a generalization of Corollary 2 in [5]. The proof presented here is based on the same idea but considerably shorter.

3.6. Theorem. There is a computable function .:i : e 2 ••• e such that for all p, q E e with ,Bxq = Dep(xp, C):

(1) X<1(p,q) = XP' (2) Ila(.:i(p, q)) = Dep(xp, C).

Proof. We define a Type 2 machine M of kind (0,2) which for input (p, q) E e and r E e works as follows.

(1) For any n EN let

r\(q, r, n):= the first word WE Wd such that xir[n1wOO . .. ) exists,

r 2(q, r, n):= xq(r[n1r\(q, r, n)OO .. . ).

(2) For n = 0,1,2, ... determine r2(q, r, n) until a number k is found with ,Br2(q, r, k) = k.

(3) Let Xp(lk100 ... ) be the output of the machine M. Since xq(r[n100 . .. ) does not exist in general, the "first" word w in (1) must be determined by a step counting argument for the (computable) universal function of X. Let x be the name of the above machine, i.e. assume a(x) = M. Then define .:i(p, q):= (x, (p, q». Assume ,BXq = Dep(xp, C).

, -

. t

..


Case r E dom(xp): Then r E dom(Xq) and r 1 (q, r, n) and rz( q, r, n) exist for all n (by continuity of Xq)' We have

n = Dep(xp, C)(r) ~ n = Dep(xp, C)(r[nlr1(q, r, n)OO ... )

~ n = prz(q, r, n).

Therefore the number k determined in (2) satisfies k = Dep(xp, C)(r), Xp(r[klOO ... ) exists and is equal to Xp(r) and X<I(p,q)(r) = X(X,(p,q»(r) = fct(x)«p, q), r) = Xp(r) .

Caseredom(xp): Then for no pair (w,n), PXq(r[nlwOO ... )=n and re

dom(X<I(p,q»' Thus we have provedXp = X;!(p,q)' Letj:= Dep(xp, C)(r). Thenj = rz(q, r,j), hence

j is the number k determined in (2). Obviously the machine a(x) uses only the first j symbols of p. This proves (2). 0

As a result of Theorem 3.6, there is in fact an input-optimal program for every r E [C -+ Wd] and, if rand Dep(r, C) are computable, then there is even an input-optimal Type 2 machine for r (cf. [5, Corollary 2]). However, it is not possible to continuously determine the input-optimal program for r without having the program for Dep(r, C).

By slightly modifying the definition of input-Iookahead we can easily introduce the computation time of machines and programs.

3.7. Definition. (1) Let M be a Type 2 machine of kind (0,1). The computation time of M, Time(M): C --~ 1'1, is defined by

Time( M)( q) := if the computation halts, {The number of steps which M on input q needs


(2) The computation time of the x-program p, Time(p): C --~ 1'1, is defined by

{

The number of steps which the machine a(x) with . input (r, q) needs if p = (x, r) for some

Tlme(p)(q) := x E W d, r E C and Xp (q) exists,


Trivially, Ila(M)';;;Time(M), Ila(p),;;;Time(p), and Lemma 3.5 also holds with

time instead of input-Iookahead. Since the computation time of a machine or a program is a function from C to 1'1, the usual Type 1 definitions for "small" complexity like "polynomial time" cannot be applied. Other reasonable definitions have not

been proposed yet. However, it makes sense to define complexity classes by means of computable bounds r: C --~ N. If r is a total function or dom(r) is compact,

then - because of Konig's Lemma (the "fan theorem") - r is bound by a constant and the complexity class is particularly simple. In other cases quite complicated situations might arise as happens for partial complexity bounds in Type 1 recursion theory.


Type 1 complexity theory can easily be embedded into Type 2 theory. Define a representation 8 : I[: --~ N by 8(x, p) = n iff f3 (x) = n for x E Wd and p E IC and associate with each function f:N --~ N the function r: I[: --~ Wd defined by r:= f3-lf8. Then theorems from Type 1 theory, e.g. hierarchy theorems or the speedup theorem (cf. [6]), can immediately be transferred to Type 2 theory. At present no interesting new results or even interesting questions for this kind of Type 2 complexity on [I[: ~ Wd] are in sight. Instead of measuring the computation time of a machine absolutely, time (and input-Iookahead) could be measured in terms of dependence. This, however, will not be discussed here any further.

On compact subsets of their domain continuous functions r: I[: --~ Wd behave particularly simply. By definition, a subset of a metric space is compact iff for every open covering of it there is a finite subcovering. Each compact set is closed and complete and a closed subset of a compact set is compact again. As a fact, Cantor's space I[: is compact and therefore subsets of I[: are compact iff they are closed.

By the following theorem a continuous function r: I[: --~ Wd is "finite" on every compact subset X of its domain, i.e. the behaviour of r on X is determined by some finite "Graph" Y <;; Wd x Wd.

3.8. Theorem. Let r: I[: --~ Wd be continuous and let X <;; dom(r) be compact. Then there is a finite subset Y <;; Wd x Wd such that (1), (2), and (3) hold.

(1) X<;;U{[w]I(3x)(w,X)E Y}, (2) (V'PEX)(V'(W,X)E Y) (pE[W]~r(p)=x), (3) (V'(W,X)E Y)[w]nX,c0.

By condition (3), the set Y has no dummy elements.

Proof. By continuity of r, for any pEdom(r) there is some k with r(q)=r(p) for all qE[p[kl ]. Let Z:={(w,x)l[w]nX,c01\(V'qE[W])r(q)=x}. Then X<;; U{[w]I(3x) (W,X)EZ}. Since X is compact, there is a finite subset Y<;;Z with X<;;U{[w]I(3x) (W,X)E Y}. Properties (1), (2) and (3) hold trivially. D

Fully effective versions of Theorem 3.8 can be proved by using representations of the set of compact subsets of I[: (see [8]). Note that from the finite set Yabove, one can construct a machine for r which is input-optimal on the set X.

4. Continuous functions r: IC --~ IC

A continuous function r: I[: --~ I[: can be considered as a sequence of continuous functions rj:I[:--~Wd where rj(p)=r(p)(i) for all iEN. This connection leads immediately to the definitions of dependence, input-Iookahead, and computation time for functions r: I[: --~ IC. Let (N --~ N) be the set of partial functions f: N --~ N.

. -

, -

..

• !


4.1. Definition. Let r: C --~ C be a continuous function and assume X ~ C. The dependence DEP(r, X): C ~ (N ~ N) is defined by

DEP(r, X)(p)(n):= {JLk[(V q E X n[p[k]]) r(p)(n) = r(q)(n)] if p E dom(r), undefined otherwise.

Note, that due to the similarities between Definitions 3.1 and 4.1 the negative results for Dep in the previous chapter hold similarly for DEP.

As the following example shows, DEP(r, C) becomes trivial if r has a very complicated domain and is not worth studying any further. Let 8 be a denumerable dense subset ofC. Then C\8=n{C\{s}lsE8} is a Ga-subset ofC, hence C\8= dom(r) for some rE[C~C]. Obviously, DEP(r,C)(p)(n) is undefined for all p E C and n E N in this case.

Below, we will therefore consider only the case X ~ dom(r) where X is compact, particularly X = dom(r) = C.

4.2. Definition. The input-Iookahead of the !/I-program p, ILA(p): C ~ (N ~ N), is defined by

The number of input symbols of q which the machine a(X) with input (r, q) reads

ILA( p)( q)( n) := until the nth output symbol is determined, if p = (x, r) for some x E Wd, rE C,


The input-Iookahead ILA(M): C ~ (N ~ N) for a Type 2 machine M of kind (0, 1) is defined accordingly.

Notice that ILA(M)(q)(n) may be defined for some nEN even if qtdom(gM)'

Therefore DEP(gM, X)(q)(n):so; ILA(M, X)(q)(n) holds if gM is a total function, but is not generally true. As in the case of dependence, a complicated domain of !/Iq has a very strong effect on ILA(q).

The following theorem which has originally been proved by the authors in [10] for the case of real functions gives examples for programs q with arbitrarily increasing functions ILA(q)(p).

4.3. Theorem. Let 8 be a denumerable dense subset of C and assume dom( !/Iq) = C\8. Then for every t:N~N there is some pEdom(!/Iq) with ILA(q)(p)(n»t(n) for infinitely many n.

Proof. Let II be a numbering of 8 and assume (w.l.g.) that t is increasing. Define a sequence (Wi> nil E Wd x N (i = 0,1,2, ... ) as follows.

8tep 0: wo:= e, no:= O.


Step k+ 1: If Wk is not a prefix of Ilk, then Wk+I:= Wk; nk+I := nk else nk+I:= min{n> nk I not ILA(q)(llk)(n)..; t(n)}

Wk+I:= {llk[t(nk+I)]O if Ilk(t(.nk+I)) = 1, Ilk [t (nk+ I)] 1 otherwIse.

Since Ilkedom(!/Iq), ILA(q)(lld(n) is undefined for almost all n, hence min{ ... } exists in any case. Since S is dense, the case" Wk is not a prefix of Ilk" cannot occur infinitely often. Hence the sequence Wo, WI, . .. converges to a function p E C. The construction guarantees p ,p Ilk for all k, therefore peS. From the construction we know that there are infinitely many kEN such that the else case occurs. In such a case y:= Ilk[t(nk+I)] is a common prefix of p and Ilk' Assume ILA(q)(p)(nk+I)"; t(nk+I)' Then obviously ILA(q)(llk)(nk+I)"; t(nk+l) which is false by construction. Therefore, the assumption is false. This proves the theorem. 0

By observing the computations, the input-Iookahead of a program can effectively be determined (cf. Lemma 3.5). There is a computable function r: C2 x Wd --~ Wd such that ILA(q)(p)({3(y)) = {3r(q, p, y). Contrary to the results of the previous section, however, for functions r E [C ~ C] an input-optimal program does not always exist. This is true simply because the domains are much more complicated now, which may cause DEP(r, X)(p) to be completely undefined. We therefore have an additional reason to restrict our investigations to functions with a reasonably simple domain.

The definition of computation time is a slight modification of Definition 3.7.

4.4. Definition. The computation time of the !/I-program p, TIME(p): C ~ (N~N), is defined by

The number of steps which the machine a(x) with input (r, q) needs until the nth output symbol

TIME(p)(q)(n):= is determined, if p=(x, r} for some xEWd, rEC,


The computation time TIME( M) : C ~ (N ~ N) for a Type 2 machine M of kind (0,1) is defined accordingly.

Obviously, ILA is always a lower bound for TIME and Theorem 4.3 therefore holds correspondingly for computation time instead of input-Iookahead. This means that both ILA(q)(p) and TIME(q)(p) may vary considerably with the argument p. As we will see, the arguments for which the input-Iookahead or the computation time is bound by some function t: N ~ N form a compact set. Compact subsets of C, therefore, are the most reasonable ones on which to study the behaviour of computation time or input-Iookahead.

We first prove a non-effective version.

, .

, '

",

Computational complexity on Cantor's space

4.5. Theorem. Let q E C and t : 1\1-;. 1\1 (1) The set M 1 := {p E C I (Vn) ILA(q)(p )(n),;:; t(n)} is compact. (2) The set M 2:= {p E C I (Vn) TIME(q)(p)(n),;:; t(n)} is compact.

Proof. (1) For n E 1\1 define

An:= U {[ w] Ilg( w),;:; t(n) and ILA(q)« wOO . . . »(n),;:; 19( w)}.

11

As a finite union of closed sets, An is closed. Obviously, An = {pEClILA(q)(p)(n),;:; t(n)}. Hence MI =n An is closed, thus a compact set.

(2) similar to (1). D

Of course, Theorem 4.5 also holds for Type 2 machines instead of programs. Furthermore, appropriate names for the compact sets MI and M2 can be determined effectively from the program q as we will show next. For this purpose we introduce two representations of K(C), the set of all compact subsets of Cantor's space, which are based on the enumeration representation M : C -;. 2 Wd.

4.6. Definition. (1) The enumeration representation M: C -;. 2 Wd is defined by M1(p):={wEWdlllt(w)l1 appears as a subword inp}. (Remember t(al ... an)= Oa 10a2' . . Oan.)

(2) The weak representation "w: C -;. K (IC) and the strong representation ,,:C --~K(IC) of compact sets in C are defined by

"w(r)=X iff C\X=U{[W]lwEM(r)},

,,(r,s)=X iff ["w(r)=X andMl(s)={wIXn[w];60}].

M is the counterpart to the enumeration representation of t>J in [9, 14]. Representations of the compact subsets of the real numbers similar to "wand " have been discussed by the authors in [8]. Furthermore, "w corresponds to the concept of "metric complement" and" to "locatedness" in [1].

4.7. Theorem. There is a computable function r: C2 -;. C such that for all q, r E C and t : 1\1-;. 1\1 with t = (3r{3-I:

"wr(q, r) = {p E C I (Vn) TIME(q)(p)(n),;:; t(n)}.

Proof. Let q, r E C and t: 1\1-;. 1\1 with t = {3r{3 -I. For each n E 1\1 define

Bn := {WE WdlIg(w) = t(n) and TIME(q)(wOO ... )(n)';:; t(n)},

en := {w E Wdllg(w) = t(n) and W ~ Bn },

An:= U {[w]1 WE Bn},

Dn := U {[ W] I WEen},

Since An = {pITIME(q)(p)(n)';:; t(n)}, A:= nAn = {p E C I (Vn) TIME(q)(p)(n)';:; t(n)}. Since Dn = C\An, U {Dn I n} = C\A. There is a computable function which from any q and r determines a function SEC such that M(s)={wl(3n) WE en}. Then "w(s) = A. D


Since in general no (finite) prefixes of q and r contain enough information to guarantee [w]nA:;e0 for weWd and A={peCI('v'n)TIME(q)(p)(n)'" (,8r,8-I)(n)}, there is no chance to continuously enumerate all the words weWd with [w] n A :;e 0, i.e. there is no continuous function ~ : C2 --~ C such that M~ (q, r) = {w I [w] n A:;e 0} for all q, r e Co Therefore, in the above theorem "w cannot be replaced by". We omit a formal proof. For similar reasons, TIME cannot be replaced by ILA even if we only require r to be continuous. However, if only total functions are considered then there is an effective version for ILA.

4.8. Theorem. There is a computable function r : C2 --~ C such that for all q, r e C for which I/Iq is a total function and for t = ,8r,8-I:

"wr(q, r) = {p e C I ('v'n) ILA(q)(p)(n)'" t(n)}.

The proof of this theorem is similar to the one of Theorem 4.7 (use Konig's Lemma). Again, a version with" instead of "w is not true. Theorems 4.7 and 4.8 hold correspondingly for Type 2 machines, e.g. "wL1(r) = {p e C I ('v'n) TIME(M)(p )(n) ... t(n)} for all r e C and t = ,8r,8-1 where L1 is a computable function.

By the next theorem, computation time and input-Iookahead are bounded on compact sets. We prove this in an effective version.

4.9. Theorem. (1) There is a computable function r:WdxC2--~C such that

ILA(q)(p)(n)"',8r(,8-I(n), q, r)

for all n eN, p e "w(r) whenever "w(r) £; dom(I/Iq). (2) Correspondingly for TIME instead of ILA. (3) Correspondingly for machines M instead of programs q.

Proof. (1) Let q, r e C and n eN. Generate simultaneously two lists of words X Io X2,." and YIoY2,'" as follows: xeWd appears in the first list iff xeM(r); Y

appears in the second list iff ILA(q)(yOO ... )(n) ... lg(y). Since C=U[Xi]UU[y;] and C is compact, there are numbers i, k with C = [XI] U .•• U [Xi] U [YI] U ••• U [Yk]' Let m:= max{lg(Yj)ll ... j ... k} where m is undefined if k= O. There is a computable function r which determines ,8-I(m) from (,8-I(n), q, r). r has the desired properties.

(2) Similar. (3) Similar. 0

If in Theorem 4.9 "w is replaced by " then even minimal upper bounds can be determined effectively. We will not discuss more details in this paper.

Putting Theorems 4.5 and 4.9 together we see that a Type 2 machine computing a function r: C --~ C has bounded complexity (either in time or in input-Iookahead) on a set S iff S is contained in a compact subset of dom(r).

, .

, }


4.10. Corollary. (1) For all q e C and S s;;; C the following properties are equivalent: (a) (Vp e S)(Vn) ILA(q)(p )(n):so; t(n) for some bound t: l'1li ~ l'1li, (b) S s;;; A ~ dom( I/Iq ) for some compact As;;; C. (2) Correspondingly for TIME instead of ILA. (3) Correspondingly for Type 2 machines.

Thus, the compact subsets of C are the natural domains of resource bounded functions. In the same way, complexity classes defined by denumerable sets T = {ti lie l'1li} of bounds correspond to denumerable unions of compact sets, called Ku-sets. Typical examples for Tare

{k+ nkl kel'lll}

{k+k· t(n)lkel'lll}

(polynomially bounded)

(O(t))

{t'l t'(n) = t(n) for almost all n} «3no)(Vn;;:. no) ... ..;; t(n))

Effective versions and non-effectivity properties corresponding to the above theorems can be proved for complexity classes and Ku-sets as well. (see [16, Corollary 24]).

Let M be a Type 2 machine with gM: C --~ C. By Theorem 4.9(c) there is a computable function t :I'III~I'III such that (Vp, n) TIME(M)(p)(n)..;; t(n). This bound t is "global" for all p e C. It might be possible to present (the Ku-set) C as a union of smaller compact sets Ki (i e l'1li) such that on each Ki the machine M has a time bound ti considerably smaller than t. The following theorem is an example for such a situation.

4.11. Theorem. Let t: l'1li ~ l'1li be a computable function. Then there is a computable function r:c~c with (1) and (2).

(1) There is no Type 2 machine M with r = gM and

(VpeC)(Vn) TIME(M)(p)(n)..;; t(n).

(2) There are a Type 2 machine M with r = gM and a constant c> 0 such that

(Vn) TIME(M)(11. . . )(n)..;;c· n+c

and for all compact K s;;; C with (11 ... ) ~ K there is a constant k e l'1li with

(VpeK)(Vn) TIME(M)(p)(n)..;;c· n+k.

Thus the machine M in (2) computes locally in linear time, however, the constants k give rise to a very large global time bound. We only sketch the proof. From Type 1 complexity theory (e.g. [6]) we know that there is a computable function Pt : l'1li ~ {O, I} such that TIME(M) ~ O(t) for any Turing machine computing Pt. Define r by

r(p )(n):= {Pt(n) if min{! Ip(i) = O} = n, o otherwise.

Then r satisfies (1), and straightforward programming yields a machine M for r satisfying (2).


In a corresponding context for real functions (see e.g. [11]) it turns out that inversion, i.e. the function f: x ~ 1/ x, is easily computable on each compact subset of its domain, but not globally easily computable.

S. Relating dependence, input-Iookabead, and computation time

In Section 3 we have investigated input-optimal machines for continuous functions .l: : C --~ Wd. We have shown that there are input-optimal programs for all functions. This is no longer true for continuous functions r: C --~ C since the domains may be much more complicated now. We also have shown in the previous section that compact sets are the natural domains for investigating computational complexity and input-Iookahead of functions r: C --~ C. For simplicity, we shall now consider Cantor's space C itself as a representative ofthe compact sets and study the relation between dependence (i.e. DEP(r):= DEP(r, C», input-Iookahead, and computation time for total functions r: C ~ C.

For a total function r, a ",-program p, and a Type 2 machine M for r we have for all qEC, nEN

DEP(r)(q)(n).;;; ILA(p)(q)(n).;;; TIME(p )(q)(n),

DEP(r)(q)(n).;;; ILA(M)(q)(n).;;; TIME(M)(q)(n).

The dependence of a continuous function may become arbitrarily large. For instance, define r:c ~ C by F(q)(n) = q(t(n» where t:N~N. Then for all q, DEP(r)(q) = t. However, for total continuous functions there are input-optimal ",-programs, the computation time of which is not too bad.

5.1. Theorem. Let r : C ~ C be continuous. Then there is a ",-program p for r such that (1) ILA(p) = DEP(r); (2) if (Vr, n) DEP(r)( r)( n) .;;; t( n) for some increasing function t: N ~ N then

(Vr, n) TIME(p)(r)(n)';;;c' t(n)· 2 t(n)+c for some CEN.

Proof. Let n E N. Define

An :={q[m]lp EC, mEN, DEP(r)(q)(n) = m}.

Obviously, C = U {[x] I x E An} and An is prefixfree (i.e. x is not a prefix of Y if x, YEAn and X'" y). By compactness of C there is a finite subset B ~ An with C = U {[x] Ix E B}. Since An is prefixfree, no proper subset B ~ An satisfies C = U {[x] I x E B}. Therefore, An is finite. The set An consists of the minimal prefixes which must be known for determining the values r(q)(n) for all q E C. Let Xl>"" Xk be pairwise different words with An = {Xl' ... , Xk} and 19(xj) .;;;lg(xi+l) for I.;;; i < k. Define Yj:= IF(xjOO ... )( n) for I.;;; i.;;; k and

Wn := '(X I )Yi'(X2)Y2 ... '(Xk)Yk 11.

. . -, ~

o 1

...


Then

s := Wo WI W2 ••• E C

encodes the function table of r. It is not difficult to design a Type 2 machine M of kind (0,2) which with input (s, r) determines r(r), reads not more input digits from r than necessary and determines r(r)(n) in almost c+ c2'{t(i) . 2t (i) Ii:;;;; n} steps. Notice that t is increasing and I{i·ili:;;;;n}:;;;;2n·2n. Let xEWd with a(x)=M and define p := (x, s). Then p has the required properties. 0

An effective version of Theorem 5.1 can also be proved. There is a computable function I: C --~ C such that for all q E C where r:=.pq is total .pI(q) = rand p:=I(q) and r satisfies (1) and (2). Especially p=I(q) is computable if q is computable. Note that, depending on TIME( q) and the complexity of q : N -+ {O, I}, the computational complexity of the program function p:N-+{O, I} may become very large whereas the computation time TIME(p) depends only on DEP(.pq) but not on the complexity of p. For obtaining results which also include the complexity of the programs we have to consider machines. Remember that TIME( p) is measured with p already given as an oracle on one input tape while in the definition of TIME(M) (cf. Definition 3.7) all data of p would have been constructed during the calculation which means that the complexity of p does appear in TIME(M).

S.2. Theorem. Let M be a Type 2 machine with gM : C -+ C. Then there is a Type 2 machine M' with:

(1) gM =gM', (2) DEP(gM) = ILA(M'), (3) if t:N-+N is increasing and ('tIr, n) TIME(M)(r)(n):;;;; ten), then ('tIr, n)

TIME(M')(r)(n):;;;; c· ten) . 2t(n)+cforsomecE N. Furthermore there isa computable function for determining M' from M.

Proof. Let gM(w)(n) E {O, I} be the value which the machine M with won the input tape writes on position n of the output tape. If w is not sufficiently long, gM ( w) (n)

does not exist. Let M' with input r determine the nth output symbol according to the following informal algorithm:

k:= -1; bl := false; REPEAT

k:= k+1; i:= -1; b2 := false; REPEAT

i:= i+1; IF ('tIwEWd, 19(w) = i)gM(r[k]w)(n) exists THEN b2 :=true

UNTIL b2 = true; IF ('tIw E Wd, 19( w) = i)gM(r[k]Oi)(n) = gM(r[k]w)(n) THEN bl := true

UNTIL bl = true; output:= gM(r[k]Oi)(n)


Since gM (p) is defined for all p, t( n) - k is an upper bound of the index i of the inner loop. The index k of the outer loop finally obtains its maximal value DEP(gM) (r)(n) which is bounded by t(n). Obviously M' satisfies (1) and (2) of the theorem. We estimate the computation time TlME(M')(r)(n) if the above algorithm is refined reasonably. (Constants are to be chosen appropriately). The inner loop requires at most

steps. The outer loop requires at most

steps. Finally for the determining the nth symbol at most

steps are needed since t is increasing and 2' {i· 2; I i ~ n} ~ 2 . n . 2n. This proves (3). Finally the construction of M' from M is effective. 0

Comparing M and M' we may say that M possibly wastes input information while M' is optimal in that sense.

If input information is very expensive (e.g. as a result of a costly physical measurement or a preceding computation) then M' should be used instead of M. However, not wasting any input information might have to be paid for by a high computation time as property (3) indicates. In fact, the exponential growth of time may be disastrous. Therefore the question arises whether the estimation given in Theorem 5.2(3) can be substantially improved. In general this is not possible provided P;c NP.

5.3. Theorem. There is a computable function r: c ~ C with the following properties: (1) n+ 1 ~ DEP(r)(p)(n)~2n + 1 for all pEe, n EI\J.

(2) There are a Type 2 machine M and a constant c with gM = r such that for all pEe, n EN:

ILA(M)(p)(n)~2n+l and TlME(M)(p)(n)~c· n+c.

(3) There is an input-optimal machine M for r which computes in polynomial time

iff P= NP.

" ..


Proof. There are an NP-complete set Y~Wd and a function h:WdxWd~{O, I} such that

(a) Y ~ A for some regular prefixfree set A ~ Wd, (b) h(y, w) is computable in time c'(lyl+lwi)+c', (c) (\fy) (YE Y~(3w) h(y, w)=O), (d) (\fy, w) (h(y, w) =O~(lg(y) =lg(w) and 0 is the last symbol of w)). Define r by

r(p )(n):= H(p(n), h(p(O) ... p(n -1), p(n + 1) ... p(2n)))

where H(XI' X2) = (0 if (Xl = 0" X2 = 0) or (Xl ¥- 0" X 2 ¥- 0), 1 otherwise) for all pEe and n E "I. r( p)( n) depends on p( n) in any case and at most on p(2n), hence (1) holds.

(2) A machine satisfying (2) may operate as follows. Assume the input pEe and assume that p[n] is on an auxiliary tape and that the output symbol q(n -1) has been produced. Determine the nth output symbol q(n) as follows:

append p(n) to the word on the auxiliary tape; if p[n] t A then q(n):= H(p(n), 1) else: q(n):= H(p(n), h(p(O) ... p(n -1), p(n + 1) ... p(2n))).

Notice that the third step is executed for at most one number n since A is prefix-free. Since A is regular, the second step requires constant time. Thus the total time for determining q(O) ... q(n) is linearly bounded in n. This proves (2).

(3) Assume P = NP. We describe a machine for r with optimal input-Iookahead. Assume the input is p, assume that p[n] is on an auxiliary tape and that the output symbol q(n -1) has been produced. Determine q(n) as follows:

Define

append p(n) to the word on the auxiliary tape; if p[n] t A then q(n):= H(p(n), 1) else: Let m be the smallest number, n < m ..; 2n such that for all WE Wd with 19( w) = n - m:

h(p[nl, p(n + 1) ... p(m)w) = h(p[nl, p(n + 1) ... p(m)On-m)).

q(n):= H(p(n), h(p[nl, p(n + 1) ... p(m)on-m)).

Again the third step is executed for at most one number n. Since DEP(r)(p)(n)..; 2n + 1, the machine works correctly in step 3. The tests in step 3 are in P if P = NP. Hence step 3 can be performed in polynomial time. Clearly the above algorithm has optimal input-Iookahead. Now assume on the other hand that there is an input-optimal machine M for r which operates in polynomial time. For given y E Wd we observe the computation of M with inputs p = (yOO . .. ) and n:= 19(y). If y t Ythen ILA(M)(p)(n) = DEP(M)(p)(n) = n+1. If yE Ythen there is a word X with 19(x) = n -1 and h(y, xO) = 0 and h(y, x1) = 1. Therefore, ILA(M)(p)(n) = DEP(M)(p )(n) > n + 1. We obtain y E Y iff M with input p = (yOO ... ) for determining the nth output symbol reads the symbol p(n + 1) from the input tape. This can be decided in polynomial time by assumption on M. D


6. Conclusion

In this paper we have laid a foundation for a theory of computational complexity on Type 2 sets. We have proposed three notions suitable for investigating the complexity of Type 2 functions: dependence, input-Iookahead and computation time. We have shown that compact sets play an important role in Type 2 complexity and presented some results relating the different notions. In a next step, this foundation should be used to investigate concrete complexity classes for various kinds of Type 2 sets.

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