Computable Invariance
Computable Invariance Vasco Brattka Theoretische Informatik I FernUniversitat Hagen D-58084 Hagen Germany
e-mail: [email protected] October 1, 1996
Abstract In Computable Analysis each computable function is continuous and computably invariant, i.e. it maps computable points to computable points. On the other hand, discontinuity is a sucient condition for non-computability, but a discontinuous function might still be computably invariant. We investigate algebraic conditions which guarantee that a discontinuous function is suciently discontinuous and suciently e ective such that it is not computably invariant. Our main theorem generalizes the First Main Theorem ouf Pour-El & Richards (cf. [20]). We apply our theorem to prove that several set-valued operators are not computably invariant.
1 Introduction In the model of computability of Computable Analysis, as introduced by Grzegorczyk and Lacombe (cf. [9, 17]) and further developed by Pour-El & Richards, Friedman & Ko, Kreitz & Weihrauch and others (cf. [20, 11, 26]) each computable function is continuous. Consequently, a lot of operations fail to be computable, simply because they are discontinuous. For instance, the operator of di erentiation d : C [0; 1] ! C [0; 1], f 7! f 0 is discontinuous w.r.t. the usual topology of uniform convergence on C [0; 1] and hence non-computable, i.e. there is no uniform algorithm which, given a program of a continuously di erentiable function f as input, computes a program of f 0 . In this case it makes sense to ask, whether at least f 0 is computable for each computable and continuously di erentiable f . A negative answer to this question has been given by Myhill (cf. [18]). Let us formalize this situation: we will call an operator1 F : X ! Y computably invariant, if and only if F maps the subset Xc X of computable points to the subset Yc Y of computable points, i.e. F (Xc ) Yc. We can immediately conclude the following implications:
F continuous (= F computable =) F computably invariant. 1 We use the notation f : X ! Y for partial functions with domain dom(f ) X .
1
Now, the question arises, how continuity and computable invariance is related. On the one hand, a constant function f : IR ! IR with a non-computable value shows that continuity does not imply computable invariance. On the other hand, the characteristic function f : IR ! f0; 1g of a non-trivial subset A IR shows that computable invariance does not imply continuity. Nevertheless, in almost all natural cases it appears that non-computable operators are also discontinuous and not computably invariant. Typically, it is very easy to prove that an operator is discontinuous and hence non-computable, but it is more complicated to prove that it is not computably invariant. The classical example of the operator of di erentiation illustrates this situation. In this paper we will show that an operator which is suciently discontinuous and suciently e ective in a certain sense is not computably invariant. In Section 2 we will express this sucient condition for computable non-invariance in the following way:
C F =) F is not computably invariant; where C is a certain discontinuous operator and is a computable reducibility for functions. More precisely, C : ININ ! ININ is de ned by ( C (p)(n) := 01 ifelse(9k)p(k) = n + 1 for all p 2 ININ ; n 2 IN, i.e. C is the operator which translates an enumeration p of a set A IN into a characteristic function of A.2 So, on the one hand the topological part of the reduction C F implies that F is at least as discontinuous as C and on the other hand the computable part of the reduction C F implies that F is suciently e ective such that it can be used to translate enumerations into characteristic functions. We claim, that in almost all non-trivial natural situations, where it can be shown that an operator F is not computably invariant, the proof implicitly contains a proof of C F . For instance, this is the case with Myhill's proof of the computable non-invariance of the operator of di erentiation d, i.e. implicitly C d is proved. Hence, the situation resembles the situation in classical recursion theory, where by many-one reduction K m A of the self-applicability problem K the non-recursiveness of a large class of natural sets A IN can be shown. Of course, the constant function with non-computable value is a simple example of a function for which this method of proving computable non-invariance fails. The main theorem of this paper in Section 3 is concerned with algebraic properties of operators F in metric spaces which are sucient to conclude C F . A special case of our properties is given by the First Main Theorem of Pour-El & Richards (cf. [20]) which deals with closed linear operators on Banach spaces. In Section 4 we prove that the assumptions of this theorem also guarantee C F . Especially, this gives a partial answer to the open problem no. 7 of Pour-El & Richards (cf. [20]). The neccessity to have a theorem that implies C F for a larger class than closed linear operators F in Banach spaces lies in the fact that many interesting operations are not of this 2 The operator C , reducibilities and corresponding hierarchies have been investigated for di erent purposes in [21, 19, 23, 24, 10]. In [3] we have shown that C is complete w.r.t. to a topological variant of in the class of F -measurable functions in Baire's space ININ .
2
type. In Section 5 we will illustrate this with operators on the space K(IR) of non-empty compact subsets of the real line. For instance, it is easy to see that the boundary operator @ : K(IR) ! K(IR); A 7! @A is discontinuous w.r.t. to the usual topology induced by the Hausdor metric on K(IR), i.e. it is also non-computable. Furthermore, with the help of our main theorem one can easily prove that @ is not computably invariant, i.e. there are computably compact sets A IR with a non-computable boundary @A. In Section 5 we prove that several other set-valued operators are not computably invariant.
2 Preliminaries In this section we will de ne some basic notions. First, we need computable metric spaces (cf. [25, 2, 6]).
De nition 2.1 (Computable metric space) (X; d; ) is called computable metric space, if
and only if
(1) d : X X ! IR is a metric on X , (2) : IN ! X is a function such that range( ) is dense in X , (3) d ( ) : IN2 ! IR is computable. Especially, each computable metric space is separable. Sometimes, we will say by abuse of notation that X or (X; d) is a computable metric space if resp. d is xed or out of consideration. One can precisely de ne computable operations in computable metric spaces via Cauchy representations (cf. [25, 26]) or equivalently recursive operations via recursion operators (cf. [4, 6]). We will omit these de nitions and refer the reader to the references. As a further notion we will need computable Banach spaces which additionally have computable vector space operations.
De nition 2.2 (Computable Banach space) (X; jj jj; +; ; ) is called computable Banach space, if and only if
(1) jj jj : X ! IR is a norm on X , (2) (X; d; ) is a computable complete metric space, where d(x; y) := jjx ? yjj for all x; y 2 X , (3) (X; +; ) is a computable vector space over IR, i.e. the vector space operations + : X X ! X and : IR X ! X are computable. For short, we will say that X or (X; jj jj; +; ) is a computable Banach space. Additionally, we need computable groups which will be de ned only for metric spaces.
De nition 2.3 (Computable group) Let X be a computable metric space. Then (X; +) is
called computable group, if and only if (1) (X; +) is a group,
3
(2) the addition + : X X ! X; (x; y) 7! x + y is computable, (3) the inverse mapping ? : X ! X; x 7! ?x is computable. Now we will de ne the central notion of this paper: De nition 2.4 (Computable invariance) Let X; Y be computable metric spaces and let Xc X resp. Yc Y be the subsets of computable elements. Then f : X ! Y is called computably invariant, if and only if f (Xc ) Yc. Now, we can de ne several reducibilities for functions. These reducibilities have been already considered in other investigations (cf. [21, 19, 23, 24, 3]). For the rest of this section let X; Y; U; V be computable metric spaces and let f : X ! Y , g : U ! V be mappings. De nition 2.5 (Reducibility) We de ne for i 2 f1; 2g: (1) f 1 g : () (9 computable A; B )(8x 2 dom(f )) f (x) = BgA(x), (2) f 2 g : () (9 computable A; B )(8x 2 dom(f )) f (x) = B (x; gA(x)), where A : X ! U and B : V ! Y resp. B : X V ! Y denote functions. One could also consider a more general Turing reducibility for functions, de ned by f T g : () f is fgg-recursive; where f is called fgg-recursive, if and only if it can be generated from some basic operations extended by g with the help of nitely many applications of the recursion operators (cf. [4, 6]). But Turing reducibility is out of the scope of this paper. One easy basic observation for this paper is that computable invariance is preserved by reducibility from the right to the left. Lemma 2.6 (Reducibility and computable invariance) For i 2 f1; 2g f i g and g computably invariant =) f computably invariant. Proof. Follows immediately, since computable functions are computably invariant. 2 As a consequence we can express our sucient condition for computable non-invariance with help of the operator C : ININ ! ININ , de ned in the introduction. Therefore, we consider Baire's space ININ as a computable metric space in the usual way (cf. [25]). Lemma 2.7 (Sucient condition for computable non-invariance) For i 2 f1; 2g C i f =) f not computably invariant. Proof. We only have to show that C is not computably invariant. But this follows immediately, since there is a computable p 2 ININ such that K = fn : (9k)p(k) = n + 1g for some recursively enumerable but non-recursive set K IN. Thus, C (p) is the characteristic function of K and hence non-computable. 2 4
3 Computable transformation spaces In this section we will provide an algebraic condition for functions f in computable metric spaces with an additional algebraic structure that is sucient for the reduction C i f . Intuitively, such a reduction means that we can use f to translate an enumeration p of a subset A IN into a characteristic function C (p) of A. The idea is to use for each n 2 IN a point of discontinutiy x1 of f to encode the characteristic value of n, which describes whether n 2 A or not. This distinction will be realized with the help of a sequence (xi )i2IN which converges e ectively to x1 such that the sets ff (xn ) : n 2 INg and ff (x1)g are suciently separated. Now, depending on the position of the value n + 1 in the sequence p we will choose n . If k is the rst position with p(k) = n +1 then n := xk , if no such position k exists, n := x1 . This procedure is repeated for each n with a suitable sequence and a corresponding point of discontinuity. This is the place where our algebraic structure P comes into consideration: we will use it to combine all those values n to one point x := i . The following de nition introduces the structure. Therefore, we will use some technical notations: for each function T : X IN ! Y we will write for short Tn (x) := T (x; n) for all x 2 X; n 2 IN. P If + : X X ! X is an operation and (xn )n2IN a sequence in X then we will write ki=j xi for the iterated operation (:::(((xj + xj +1) + xj +2P ) + xj +3 )::: + xk ) with k j . If there is a neutral P1 element 0 w.r.t.P+n and k k < j then we de ne iQ i=j xi := limn!1 i=j xi . =j xi := 0. We will use the abbreviation Correspondingly, we use as notation for a second iterated operation : X X ! X .
De nition 3.1 (Transformation space) A tuple (X; d; +; T; D) is called computable transformation space, if and only if
(1) (2) (3) (4)
(X; d) is a computable complete metric space, + : X X ! X is a computable operation, T : X IN ! X is a computable operation, sn := Pni=0 Ti (xi ) exists for each n 2 IN and d(sn ; sk ) < 2?k for all n > k and both for each sequence (xn )n2IN in D X .
The next de nition introduces homomorphisms of transformation spaces. If f is a discontinuous function between transformation spaces, then the homomorphism property will guarantee the existence of suciently many points of discontinuity and that f is algebraically well-behaved in a certain sense.
De nition 3.2 (Homomorphism of transformation spaces) Let X = (X; dX ; ; T; DX ), Y = (Y; dY ; +; S; DY ) be computable transformation spaces. Then f : X ! Y is called a Q1 homomorphism of X to Y , if and only if i Ti (xi ) 2 dom(f ) and =0
f
1 Y i=0
!
Ti (xi ) =
1 X i=0
Sif (xi )
for all sequences (xn )n2IN in DX , such that (f (xn ))n2IN is a sequence in DY . 5
The previous de nition can be generalized to k-ary functions f straightforwardly. The following de nition describes a property of transformation spaces P Q that will allow to retrieve the characteristic function C (p) from the value Si f (xi ) = f ( Ti (xi )). We will distinguish two cases. By B (x; ") := fy 2 X : d(x; y) < "g we denote the open balls, by B (x; ") the corresponding closed balls in metric spaces (X; d).
De nition 3.3 (Reversible transformation spaces) Let X = (X; d; +; T; D) be a computable transformation space, let D ; D D with D \ D = ; and let x 2 D, " > 0. (1) X is called reversible w.r.t. fD ; D g, if and only if there is a computable test function : X IN ! IN such that ! ( 1 X D Ti (xi ) = 10 ifif xxn 22 D () n n i k for all sequences (xi )i2 in D [ D and n; k 2 IN with n = k. (2) X is called strongly reversible w.r.t. fD ; D g, if and only if there is a computable test function : X IN ! IN such that condition () holds for all sequences (xi )i2 in D [ D ; n 2 IN and k = 0. (3) X is called (strongly) reversible in (x; "; ), if and only if X is (strongly) reversible w.r.t. fB (x; "); D n B(x; )g. 0
1
0
0
1
1
0
1
=
IN
0
1
0
1
IN
0
1
P In the case of reversibility one can imagine 1 i=k Ti (xi ) as a \stack memory" which allows to nd the iPwith xn 2 Di only for the top element xn with n = k while in the case of strong reversibility 1 i=0 Ti (xi ) behaves like a \random access memory" which allows to nd the i with xn 2 Di for all n. The following theorem shows that a homomorphism of suitable transformation spaces together with a suitable sequence (xn )n2IN which converges e ectively to a point of discontinuity x1 of f implies C i f .
Theorem 3.4 (Computable non-invariance of homomorphisms) Let X = (X ,dX ,,T ,DX ), Y = (Y; dY ; +; S; DY ) be computable transformation spaces with a homomorphism f : X ! Y . Let (xn )n2 be a computable sequence in DX and let D ; D DY such that (a) dX (xn ; xk ) < 2?k for all n > k and x1 := limn!1 xn 2 DX , (b) f (xn ) 2 D for all n 2 IN and f (x1 ) 2 D . IN
0
0
1
1
Then the following holds: (1) If Y is strongly reversible w.r.t. fD0 ; D1 g, then C 1 f . (2) If Y is reversible w.r.t. fD0 ; D1 g, (Y; +) is a computable group, (f (xn ))n2IN is a computable sequence and f (x1 ) a computable point in Y , then C 2 f . In both cases f is not computably invariant. Moreover, there exists a computable x 2 X , depending only on X and (xn )n2IN such that f (x) 2 Y is non-computable.
6
Proof. Since X is complete x1 exists. De ne : IN IN ! X by ( if k = minfm : p(m) = n + 1g exists n(p) := xk IN
x1
else
for all p 2 ININ and n 2 IN. Since (xn )n2IN is a computable sequence and dX (xk ; x1 ) 2?k for all k 2 IN, is computable. Since X is a computable transformation spaceQ and (n (p))n2IN is a sequence in DX for all p 2 ININ , s : ININ IN ! X , de ned by sn (p) := ni=0 Ti i (p) exists for all p 2 ININ , n 2 IN and dX (sn (p); sk (p)) < 2?k for all n > k. Since T; ; are computable, s is computable and hence A : ININ ! X de ned by 1 Y A(p) := nlim s ( p ) = Ti i(p) n !1 i=0
for all p 2 ININ is computable too. By (b) (f (xn ))n2IN is a sequence in DY and f (x1 ) 2 DY , thus (fn(p))n2IN is a sequence in DY for all p 2 ININ too. Since f is a homomorphism A(p) 2 dom(f ) for all p 2 ININ . (1) Let Y be strongly reversible w.r.t. fD0 ; D1 g and let : Y IN ! IN be the corresponding computable test function. De ne B : Y ! ININ by B (y)(n) := (y; n) for all y 2 Y and n 2 IN. Then B is computable and since f is a homomorphism we obtain
BfA(p)(n) = (fA(p); n) = n f 1 X
1 Y
i=0
Ti i (p)
!!
!
Si fi(p) = n i =0 ( 0 if fn(p) 2 D0 = 1 if fn(p) 2 D1 ( 0 if (9k)n (p) = xk = 1 if n (p) = x1 ( 0 if (9k)p(k) = n + 1 = 1 else = C (p)(n)
for all p 2 ININ , n 2 IN, i.e. C 1 f . (2) Now, let Y be reversible w.r.t. fD0 ; D1 g and let : Y IN ! IN be the corresponding computable test function. Furthermore, let (Y; +) be a computable group, and let (f (xn ))n2IN , f (x1 ) be computable. De ne B : ININ Y ! ININ , : ININ Y IN ! Y recursively by ! nX ?1 B (p; y)(n) := n y ? Si i (p; y) i=0
7
and
(
f (xk[p(k)=i+1]) if B (p; y)(i) = 0 f (x1) else for all p 2 ININ , y 2 Y , n; i 2 IN. Since (Y; +) is a computable group, (f (xn))n2IN , f (x1) are computable and ; S are computable, B; are computable too. Inductively, one can prove that B (p; fA(p))(i) = 0 implies the existence of k[p(k) = i +1] for all i 2 IN and hence (p; fA(p)) 2 dom(B ) for all p 2 ININ . Furthermore, by induction one can show i (p; fA(p)) = fi(p) for all p 2 ININ and i 2 IN. Since f is a homomorphism and (Y; +) is a group i (p; y) :=
fA(p) ?
nX ?1 i=0
Si i (p; fA(p)) = f = =
1 X i=0
1 X i=n
1 Y i=0
! n?1 X
Ti i(p) ?
Sifi (p) ?
nX ?1 i=0
i=0
Sifi(p)
Si fi(p)
Si fi(p):
for all p 2 ININ and n 2 IN. Correspondingly to (1) we obtain
B (p; fA(p))(n) = n fA(p) ? 1 X
= n
i=n
nX ?1
Si i(p; fA(p))
!
i=0!
Si fi(p)
= C (p)(n) for all p 2 ININ , n 2 IN, i.e. C 2 f . Now, there is a computable sequence p 2 ININ which enumerates a non-recursive set, i.e. C (p) is non-computable. Since the de nition of A does depend only on X and (xn )n2IN , the same holds for x := A(p). Since A is computable, x is computable too. Since C (p) = BfA(p) resp. C (p) = B (p; fA(p)) are non-computable, and B is computable, f (x) = fA(p) must be noncomputable. 2 From the proof one recognizes that in case (2) the computability of the group (Y;P+) and the computability of the sequence (f (xn ))n2IN is needed to access the \stack memory" Si fi. Obviously, the theorem can be generalized to k-ary functions f .
4 Computable Banach spaces In this section we will show that for each Banach space there is an induced reversible transformation space structure. Each closed and linear operator is a homomorphism of transformation 8
spaces. Last not least, we will prove that the presumptions of the First Main Theorem of PourEl & Richards (cf. [20]) imply the presumptions of our Main Theorem 3.4 and hence C 2 f . Thus, our theorem of Section 3 is a generalization of the non-trivial direction of the First Main Theorem. Lemma 4.1 (Induced transformation space) Let (X; jj jj; +; ) be a computable Banach space. Then X = (X; jjx ? yjj; +; 4?n?1 x; B (0; 1)) is a computable transformation space which is reversible in (0; 61 ; 56 ). Proof. For each sequence (xn)n2IN in B (0; 1) yn := Pni=0 4?i?1 xi exists and for all n > k n X jjyn ? yk jj = jj 4?i?1 xi jj < 4?k?1 < 2?k : i=k+1
Hence X is a computable transformation space. Now, de ne : X IN ! IN by 8 > if jjxjj < 4?n2?1 1 if jjxjj > 4?n2?1 : div else for all x 2 X; n 2 IN. Obviously, is computable. Since 1 1 ?n?1 X X jj 4?i?1 xi jj 4?i?1 = 4 3 ; i=n+1
i=n+1
we obtain for each sequences (xn )n2IN in fx : jjxjj < 61 g [ fx : 56 < jjxjj 1g and n 2 IN ! ( 1 1 X 6 : n 4?i?1 xi = 01 ifif jjjjxxn jjjj < n > 65 i=n
Hence is a computable test function for X and X is reversible w.r.t. fB (0; 61 ); B (0; 1) n B (0; 65 )g, i.e. X is reversible in (0; 16 ; 56 ). 2 The transformation space introduced in this lemma will be called the induced transformation space of the Banach space X . Especially, R := (IR; jx ? yj; 4?n?1 x; [?1; 1]) is a computable transformation space which is reversible in (0; 61 ; 56 ). It is easy to see that each closed linear operator in Banach spaces is a homomorphism of the corresponding induced transformation spaces. Lemma 4.2 (Induced homomorphism) Let X; Y be computable Banach spaces with induced computable transformation spaces X ; Y . If f : X ! Y is a closed linear operator, then f is a homomorphism of X to Y . Proof. Let (xn)n2IN be a sequence with jjxn jj 1 and jjf (xn)jj 1 for all n 2 IN. Since f is 1 linear, f (4?i?1 xi ) = 4?i?P f (xi) for all i 2 IN. Since X ; Y are computable transformation spaces, P 1 4?i?1 xi , as well as 1 4?i?1 f (xi ) exist. Since f is closed, P1 4?i?1 xi 2 dom(f ) and i=0 i=0 i=0 ! 1 1 X ?i?1 X 4 xi = 4?i?1 f (xi ) f i=0
follows. Thus, f is a homomorphism of X to Y . 9
i=0
2
Now we will prove that our Theorem 3.4 is a generalization of the non-trivial direction of the First Main Theorem of Pour-El & Richards (cf. [20]). Especially, the presumptions of the First Main Theorem do even imply C 2 f .
Theorem 4.3 (Computable non-invariance of unbounded linear operators) Let X; Y be computable Banach spaces and let f : X ! Y be a closed linear and unbounded operator. Let (en )n2IN be a computable sequence in dom(f ) whose linear span is dense in X and let (f (en ))n2IN be computable in Y . Then C 2 f . Especially, f is not computably invariant. Proof. Let X ; Y be the computable transformation spaces induced by X; Y . Since f is closed and linear, f is a homomorphism of X to Y . It is easy to see, that (Y; +) is a computable group
since Y is a computable Banach space. To apply Theorem 3.4 it remains to construct a suitable sequence (xn )n2IN . This can be done correspondingly to the proof of the First Main Theorem in [20]: since the linear span of (en )n2IN is dense in X and f : X ! Y is closed and unbounded, f has to be unbounded on the linear span of (en )n2IN . Hence, we can nd a computable sequence (bn )n2IN , consisting of nite rational linear combinations of (en )n2IN such that jjf (bk )jj > 2k+1 jjbk jj for all k 2 IN. Since f is linear and (f (en))n2IN is computable, (f (bn))n2IN is a computable sequence too. Let xn := jjf (bbnn )jj . Then jjxk jj < 2?k?1 and jjf (xk )jj = 1 for all k 2 IN. Especially, (f (xn ))n2IN is a computable sequence. Furthermore, for n > k
jjxn ? xk jj jjxnjj + jjxk jj < 2?k and jjf (xn )jj = 1 while jjf (limn!1 xn )jj = jjf (0)jj = 0. Thus, by Theorem 3.4 C f follows. 2 2
So, in contrast to the general situation of Theorem 3.4, in the special case of closed linear and unbounded operators in Banach spaces there is a canonical sequence (xn )n2IN which converges e ectively to 0 such that ff (xn ) : n 2 INg and f0g are suciently separated. In [20] it has been shown, that several unbounded closed and linear operators f from analysis and physics, like the operator of di erentiation d, ful ll the presumptions of the previous theorem. Hence we can conclude C 2 f for all those operators.
5 Operators on the space of compact subsets In this section we will apply our Main Theorem 3.4 to several set-valued operators. Moreover, we will prove the computable non-invariance of further set-valued operators by reduction. We will start with introducing the space of compact subsets (for this topic cf. [6, 16, 27, 28]). Let
K(X ) := fA X : A non-empty and compactg for each X IR. Let dA : IR ! IR; x 7! inf a2A jx ? aj be the distance function of A K(IR) and let dK : K(IR) K(IR) ! K(IR) be the Hausdor metric on K(IR), de ned by (IR)
(
dK(IR) (A; B ) := max sup dB (a); sup dA (b) a2A
10
b2B
)
for all A; B 2 K(IR). Furthermore, consider the transformation t : IR IN ! IR; (x; n) 7! x + n1+1 and the corresponding induced transformation on sets 4n+1 2 T : K(IR) IN ! K(IR); (A; n) 7! tn(A) = 4nx+1 + 2n1+1 : x 2 A : Let I := [0; 1] be the unit interval and let In := Tn (I ) for all n 2 IN. Now we can de ne a computable transformation space on K(IR).
Lemma 5.1 (Transformation space of compact subsets) K := (K(IR); dK(IR); [; T; K(I )) is a computable transformation space which is strongly reversible in (I; 31 ; 13 ). Proof. (K(IR); dK(IR)) can be considered as a computable complete metric space in the usual way. Then K(IR) K(IR) ! K(IR); (A; B ) 7! A [ B is a computable operation. It is easy to see that T is a computable operation too. Now, let (An )n2IN be a sequence in KS(I ). Then (Tn (An ))n2IN is a locally nite sequence of pairwise disjoint compact sets, i.e. Bn := ni=0 Ti (Ai ) is a compact set too. Furthermore, dK(IR) (Bn; Bk ) 2k1+1 + 4k1+1 < 2?k for all n > k. Thus K is a computable transformation space. I1 I0 0 I2 1 Figure 1: The transformed intervals In = Tn (I ) Now we have to show that there is a computable test function : K(IR) IN ! IN w.r.t. (I; ; 31 ). De ne by 8 > if supx2In dA (x) < 341n+1 1 if sup dA (x) > 341n+1 : div else x2In 1 3
for all A 2 K(IR); n 2 IN. Then is computable and for all sequences (An )n2IN in B (I; 13 ) [ K(I ) n B (I; 13 ) we obtain ! ( 1 [ An ) < 13 n Ti (Ai ) = 01 ifif ddK(IR) ((I; I; A ) > 1 K(IR)
i=0
for all n 2 IN. Thus K is strongly reversible in (I; 13 ; 13 ).
n
3
2
The next proposition shows, that many set-valued operators are homomorphisms of the de ned transformation spaces in a natural way. Therefore, let A denote the closure, A the set of inner points, @A the boundary, and A0 the derived set (of accumulation points) of A. Furthermore, let be the usual Lebesgue measure on the real numbers. 11
Proposition 5.2 (Set-valued homomorphisms) The following operators are homomorphisms of the corresponding computable transformation spaces:
(1) : K(IR) ! K(IR); A 7! A, (2) @ : K(IR) ! K(IR); A 7! @A, (3) d : K(IR) ! K(IR); A 7! A0 , (4) \ : K(IR) K(IR) ! K(IR); (A; B ) 7! A \ B , (5) : K(IR) ! IR; A 7! (A). Furthermore, C 1 F holds for each of these operators F . Especially, none of these operators is computably invariant.
Proof. First, we prove that @ is a homomorphism of K to K. Let (An)n2 be a sequence in K(I ) such that (@An )n2 is a sequence Sin K(I ) too. Since (Tn (An ))n2 is a locally nite sequence of pairwise disjoint compact sets, 1 i Ti (Ai ) 2 dom(@ ) and IN
IN
@
IN
1 [
i=0
!
=0
Ti (Ai ) =
1 [
i=0
@Ti (Ai ) =
1 [
i=0
Ti (@Ai );
i.e. @ is a homomorphism of K to K. The operators ; d can be treated correspondingly. Now, let (An )n2IN ; (Bn )n2IN be sequences in K(I ) such that (An \ Bn )n2IN is a sequence in K(I ) too. Since (Tn (An ))n2IN , (Tn (Bn ))n2IN are locally nite sequence of pairwise disjoint compact sets, S 1 (Ti (Ai ); Ti (Bi )) 2 dom(\) and i=0 ! ! 1 1 1 1 [ \ [ [ [ Ti (Ai ) Ti (Bi ) = (Ti(Ai ) \ Ti (Bi )) = Ti (Ai \ Bi ); i=0
i=0
i=0
i=0
i.e. \ is a homomorphism of K K to K. Now, let (An )n2IN be a sequence in K(I ). Then ((An ))n2IN is a sequence S1 in I . Since (Tn (An ))n2IN is a locally nite sequence of pairwise disjoint compact sets, i=0 Ti (Ai ) 2 dom() and ! 1 1 1 [ X X Ti (Ai ) = Ti (Ai ) = 4?i?1 (Ai); i=0
i=0
i=0
i.e. is a homomorphism of K to R. Now, to apply Theorem 3.4 we have to construct suitable sequences converging to points of discontinuity. Let Jn := [0; 71 ] [f 2nk+1 : k = 0; :::; 2n+1 g and Jn0 := [0; 17 ] [f 3nk+1 : k = 0; :::; 3n+1 g. Then (Jn )n2IN , (Jn0 )n2IN are computable sequences in K(I ) which converge to I . Furthermore, dK(IR) (Jn ; Jk ) 221k+1 < 2?k and dK(IR) (Jn0 ; Jk0 ) 231k+1 < 2?k for all n > k. We obtain (1) (Jn ) = [0; 71 ] for all n 2 IN and (I ) = I , (2) @ (Jn ) = f0; 71 g [ (Jn n [0; 17 ]) for all n 2 IN and @ (I ) = f0; 1g, (3) d(Jn ) = [0; 17 ] for all n 2 IN and d(I ) = I , 12
(4) Jn \ Jn0 = [0; 17 ] [ f1g for all n 2 IN and I \ I = I , (5) (Jn ) = 71 for all n 2 IN and (I ) = 1. and dK(IR) (I; [0; 71 ]) = 67 > 13 , and dK(IR) (I; f0; 71 g[(Jn n[0; 71 ])) 41 < 13 , dK(IR) (I; f0; 1g) = 12 > 13 , dK(IR) (I; [0; 17 ] [ f1g) = 73 > 13 . Thus, by Theorem 3.4 C 1 F for each of these operators F 2 f; @; d; \; g follows. 2 A carefull look at the proof shows, that we have used one transformation space K and one sequence (Jn )n2IN simultaneously for all operators ; @; d; . Hence by Theorem 3.4 we can conclude the following corollary.
Corollary 5.3 There is a computable compact set A [0; 1] such that neither A , nor the 0
boundary @A, nor the derived set A are computable compact sets, nor the measure (A) is a computable real number.
Now we will illustrate how one can use the method of reduction to prove that further setvalued operators are not computably invariant.
Proposition 5.4 (Not computably invariant set-valued operators) The following oper-
ators are not computably invariant: (1) c : K(I ) ! K(I ); A 7! Ac ,
(2) D : K(IR) K(IR) ! K(IR); (A; B ) 7! A n B , (3) : K(IR) K(IR) ! K(IR); (A; B ) 7! (A n B ) [ (B n A).
Proof.
(1) One can deduce from the previous proposition that 0 := jK(I ) also is not computably invariant. Since 0 (A) = A = Ac c = c2 (A) for all A 2 K(I ), it follows 0 = c2 . Hence, c2 and thus c are not computably invariant. (3) Since c(A) = Ac = (I n A) [ (A n I ) = (I; A) for all A 2 K(I ), it follows c 1 . Thus, is not computably invariant. (2) Since (A; B ) = (A n B ) [ (B n A) = A n B [ B n A for all A; B 2 K(IR), it follows 1 (D D). Hence, D D and thus D are not computably invariant.
2 13
Although the operators in the previous proposition are no homomorphisms w.r.t. to the introduced transformation space K, one can directly prove C 1 F for each of these operators F. Finally, we will introduce another transformation space structure on the space C (I ) of continuous functions which is di erent from the induced transformation space structure of C (I ) from Section 4, but which is even strongly reversible. Therefore, consider the usual metric dC(I ) : C (I ) C (I ) ! IR of uniform convergence on C (I ), de ned by dC(I ) (f; g) := sup jf (x) ? g(x)j x2I
for all f; g 2 C (I ). Furthermore, consider the transformation t on the real numbers, de ned at the beginning of this section. Let S : C (I ) IN ! C (I ) be the transformation, de ned by ( 1 ?1 Sn(f )(x) := 02n+1 f tn (x) ifelsex 2 In for all f 2 C0 (I ) := ff 2 C (I ) : f (0) = f (1) = 0; jjf jj < 1g, n 2 IN and x 2 I . Then we obtain
Lemma 5.5 (Transformation space of continuous functions) C := (C (I ); dC(I ) ; +; S; C0 (I )) is a computable transformation space which is strongly reversible in (0; 21 ; 12 ). Proof. (C (I ); dC I ) can be considered as a computable complete metric space in the usual way. ( )
Then + is a computable operation. It is easy to seePthat S is a computable operation too. Now, let (fn)n2IN be a sequence in C0 (I ). Then gn := ni=0 Si (fi ) 2 C0 (I ) for all n 2 IN and since (Sn (fn ))n2IN is a sequence of functions with pairwise disjoint support we obtain jjgn ? gk jj 2k1+1 < 2?k for all n > k. Thus C is a computable transformation space. Now we have to show that there is a computable test function : C (I ) IN ! IN w.r.t. (0; 21 ; 21 ). De ne by 8 >0 if supx2In jf (x)j < 221n+1 < n(f ) := > 1 if supx2In jf (x)j > 221n+1 : div else for all f 2 C (I ); n 2 IN. Then is computable and for all sequences (fn )n2IN in B (0; 21 ) [ C0 (I ) n B (0; 21 ) we obtain ! ( 1 1 X 2 n Si (fi) = 01 ifif jjjjffn jjjj < >1 i=0
for all n 2 IN. Thus C is strongly reversible w.r.t. (0; 12 ; 12 ).
n
2
2
One can use this strong reversible transformation space for a direct proof of C 1 d for the operator of di erentiation d (cf. [21, 19]). We will prove that the support operator and the zero operator are homomorphisms of computable transformation spaces and thus they are not computably invariant. 14
Proposition 5.6 (The support and zero operators) The following operators are homomorphisms of transformation spaces: (1) supp : C (I ) ! K(I ); f 7! supp(f ) := f ?1f0gc , (2) zero : C (I ) ! K(I ); f 7! f ?1 f0g. Furthermore, C 1 supp and C 1 zero. Especially, supp; zero are not computably invariant.
Proof. Let (fn)n2IN be a sequence in C (I ) such that (supp(fn))n2IN is a sequence inPK(I ). Since (Sn (fn ))n2IN is a sequence of functions with pairwise disjoint non-empty supports, 1 i=0 Si (fi ) 2 dom(supp) and ! 1 1 1 X [ [ Si (fi) = supp(Si (fi )) = Ti (supp(fi)); supp i=0
i=0
i=0
i.e. supp is a homomorphism of C to K. For the zero operator we consider a modi cationSof K. De ne the transformation T 0 : K(IR) IN ! IN by Tn0 (A) := Tn (A) [ D where D := I n ( 1 i=0 Ii ). Since D 2 K(I ) is computable, it is 0 easy to see that T is computable. One can prove that the corresponding modi ed transformation space K0 is strongly reversible in (I; 13 ; 13 ), correspondingly to K.PNow, let (fn )n2IN be a sequence in C (I ) such that (zero(fn))n2IN is a sequence in K(I ). Again, 1 i=0 Si(fi ) 2 dom(zero) and ! 1 1 X [ zero Si (fi) = Ti0(zero(fi)); i=0
i=0
i.e. zero is a homomorphism of C to K0 . Now, to apply Theorem 3.4 we have to construct suitable sequences converging to points of discontinuity. Therefore, let (fn )n2IN be a sequence of rational polygons, such that fn is de ned by the vertices (0; 0); ( 41 ; 1); ( 21 ; 21n ); ( 43 ; 21n ); (1; 0):
fn(x) 6 1
1 2n
0
??
H ?? HHHH H ?? r
r
1 2
3 4
r
0
HH HH HH -
r
r
1 4
1 x
Figure 2: The rational polygon fn Then (fn )n2IN is a computable sequence in C0 (I ) which converges to a function f 2 C0 (I ) such that jjfn ? fk jj 21k ? 21n < 21k for all n > k. We obtain (1) supp(fn) = [0; 1] and supp(f ) = [0; 21 ], 15
(2) zero(fn ) = f0; 1g and zero(f ) = f0g [ [ 12 ; 1] and dK(IR) (I; [0; 12 ]) = 12 > 31 , dK(IR) (I; f0; 1g) = 12 > 13 , and dK(IR) (I; f0g[ [ 21 ; 1]) = 41 < 13 . Thus, by Theorem 3.4 C 1 F for F 2 fsupp; zerog follows. 2 Again, a carefull look at the proof shows, that we have used one transformation space C and one sequence (fn) simultaneously for both operators supp and zero. Hence by Theorem 3.4 we can conclude the following corollary.
Corollary 5.7 (Support and Zero) There is a computable function f 2 C (I ) such that neither f ? f0g nor f ? f0gc is a computable compact set in K(I ). 1
0
1
6 Conclusion In this paper we have investigated methods to prove the computable non-invariance of operators F . We have expressed a sucient condition for computable non-invariance by C 2 F . Topologically, the best what one can expect of such an operator is to be F -measurable. It would be an interesting question, how non-computable operators could be classi ed precisely in Borel's hierarchy k . In [3] we have de ned generalizations Ck of C which are complete in the class of k -measurable functions in Baire's space ININ w.r.t. a topological version of 2 . Most natural operators should be computably equivalent to Ck for some k 2 IN. Furthermore, we have investigated an algebraic condition for operators F in metric spaces which implies C 2 F . We have applied the corresponding Theorem 3.4 to several set-valued operators. It would be a further interesting question whether there are more general or more suitable conditions which also imply C 2 F . Moreover, we have shown that our algebraic condition is a generalization of the First Main Theorem of Pour-El & Richards (cf. [20]). Especially, C 2 F holds for all those closed linear and unbounded operators F of analysis and physics which ful ll the additional condition of the theorem (i.e. they have to admit a suitable computable sequence (en )n2IN ). A further topic of interest is the polynomial-time invariance of operators. Ko and others (cf. [7, 11, 12, 13, 14, 1]) have constructed a lot of counterexamples of the type such that there is a polynomial-time computable x and F (x) is not even computable. Since there is no welldeveloped general theory of complexity in metric spaces, it is quite dicult to modify Theorem 3.4 in this direction. But if we assume that (X; d; +; T; D) is a computable transformation space such that there is a reasonable notion of complexity for X and D is compact, then we have uniform complexity bounds for functions on D. If, furthermore, T and + are polynomial-time computable in a certain sense, then it should be possible to derive a polynomial-time version of Theorem 3.4.
References
[1] M. Blaser, Uniform computational complexity of the derivatives of C 1 -functions, in: K.-I Ko & K. Weihrauch, eds., Proc. of the Workshop on Computability and Complexity in Analysis, Informatik Berichte, FernUniversitat Hagen 190 (1995) 99-104
16
[2] J. Blanck, Domain representability of metric spaces, in: K.-I Ko & K. Weihrauch, eds., Proc. of the Workshop on Computability and Complexity in Analysis, Informatik Berichte, FernUniversitat Hagen 190 (1995) 1-10 [3] V. Brattka, Grade der Nichtstetigkeit in der Analysis, Diplomarbeit, FernUniversitat Hagen (1993) [4] V. Brattka, Recursive characterization of computable real-valued functions and relations, Theoretical Computer Science 162 (1996) 45-77 [5] V. Brattka, Computable selection in analysis, in: K.-I Ko & K. Weihrauch, eds., Proc. of the Workshop on Computability and Complexity in Analysis, Informatik Berichte, FernUniversitat Hagen 190 (1995) 125-138 [6] V. Brattka, Order-free recursion on the real numbers, Mathematical Logic Quarterly 43 (1997) [7] A. Chou, K.-I Ko, Computational complexity of two-dimensional regions, SIAM J. on Comput. 24 (1995) 923-947 [8] R. Engelking, General Topology, Heldermann, Berlin (1989) [9] A. Grzegorczyk, On the de nition of computable real continuous functions, Fund. Math. 44 (1957) 61-71 [10] P. Hertling, Topologische Komplexitatsgerade von Funktionen mit endlichem Bild, InformatikBerichte, FernUniversitat Hagen 152 (1993) [11] K.-I Ko, Complexity Theory of Real Functions, Birkhauser, Boston (1991) [12] K.-I Ko, A polynomial-time computable curve whose interior has a nonrecursive measure, Theoretical Computer Science 145 (1995) 241-270 [13] K.-I Ko & K. Weihrauch, On the measure of two-dimensional regions with polynomial-time computable boundaries, in: Porc. of the 11th IEEE Conference on Computational Complexity, to appear (1996) [14] K.-I Ko, On the complexity of fractal dimensions and Julia sets, Theoretical Computer Science, submitted (1996) [15] K. Kuratowski, Topology, Vol. I&II, Academic Press, New York (1966, 1968) [16] Ch. Kreitz & K. Weihrauch, Compactness in constructive analysis revisited, Annals of Pure and Applied Logic 36 (1987) 29-38 [17] D. Lacombe, Extension de la notion de fonction rcursive aux fonctions d'une ou plusieurs variables relles I-III, Comptes Rendus 240/241 (1955) 2478-2480/13-14,151-153,1250-1252 [18] J.R. Myhill, A recursive function, de ned on a compact interval and having a continuous derivative that is not recursive, Michigan Math. J. 18 (1971) 97-98 [19] U. Mylatz, Vergleich unstetiger Funktionen in der Analysis, Diplomarbeit, FernUniversitat Hagen (1992) [20] M.B. Pour-El & J. Richards, Computability in Analysis and Physics, Springer, Berlin (1989) [21] Th. von Stein, Vergleich nicht konstruktiv losbarer Probleme in der Analysis, Diplomarbeit, FernUniversitat Hagen (1989)
17
[22] K. Weihrauch, Computability, Springer, Berlin (1987) [23] K. Weihrauch, The degrees of discontinuity of some Translators between representations of the real numbers, Informatik-Berichte, FernUniversitat Hagen 129 (1992) [24] K. Weihrauch, The TTE-interpretation of three hierarchies of omniscience principles, InformatikBerichte, FernUniversitat Hagen 130 (1992) [25] K. Weihrauch, Computability on computable metric spaces, Theoretical Computer Science 113 (1993) 191-210 [26] K. Weihrauch, A Simple Introduction to Computable Analysis, Informatik Berichte 171, FernUniversitat Hagen (1995) [27] K. Weihrauch & Ch. Kreitz, Representations of the real numbers and of the open subsets of the set of real numbers, Annals of Pure and Applied Logic 35 (1987) 247-260 [28] Q. Zhou, Computable real-valued functions on recursive open and closed subsets of Euclidean space, Mathematical Logic Quarterly 42 (1996) 379-409
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e-mail: [email protected] October 1, 1996
Abstract In Computable Analysis each computable function is continuous and computably invariant, i.e. it maps computable points to computable points. On the other hand, discontinuity is a sucient condition for non-computability, but a discontinuous function might still be computably invariant. We investigate algebraic conditions which guarantee that a discontinuous function is suciently discontinuous and suciently e ective such that it is not computably invariant. Our main theorem generalizes the First Main Theorem ouf Pour-El & Richards (cf. [20]). We apply our theorem to prove that several set-valued operators are not computably invariant.
1 Introduction In the model of computability of Computable Analysis, as introduced by Grzegorczyk and Lacombe (cf. [9, 17]) and further developed by Pour-El & Richards, Friedman & Ko, Kreitz & Weihrauch and others (cf. [20, 11, 26]) each computable function is continuous. Consequently, a lot of operations fail to be computable, simply because they are discontinuous. For instance, the operator of di erentiation d : C [0; 1] ! C [0; 1], f 7! f 0 is discontinuous w.r.t. the usual topology of uniform convergence on C [0; 1] and hence non-computable, i.e. there is no uniform algorithm which, given a program of a continuously di erentiable function f as input, computes a program of f 0 . In this case it makes sense to ask, whether at least f 0 is computable for each computable and continuously di erentiable f . A negative answer to this question has been given by Myhill (cf. [18]). Let us formalize this situation: we will call an operator1 F : X ! Y computably invariant, if and only if F maps the subset Xc X of computable points to the subset Yc Y of computable points, i.e. F (Xc ) Yc. We can immediately conclude the following implications:
F continuous (= F computable =) F computably invariant. 1 We use the notation f : X ! Y for partial functions with domain dom(f ) X .
1
Now, the question arises, how continuity and computable invariance is related. On the one hand, a constant function f : IR ! IR with a non-computable value shows that continuity does not imply computable invariance. On the other hand, the characteristic function f : IR ! f0; 1g of a non-trivial subset A IR shows that computable invariance does not imply continuity. Nevertheless, in almost all natural cases it appears that non-computable operators are also discontinuous and not computably invariant. Typically, it is very easy to prove that an operator is discontinuous and hence non-computable, but it is more complicated to prove that it is not computably invariant. The classical example of the operator of di erentiation illustrates this situation. In this paper we will show that an operator which is suciently discontinuous and suciently e ective in a certain sense is not computably invariant. In Section 2 we will express this sucient condition for computable non-invariance in the following way:
C F =) F is not computably invariant; where C is a certain discontinuous operator and is a computable reducibility for functions. More precisely, C : ININ ! ININ is de ned by ( C (p)(n) := 01 ifelse(9k)p(k) = n + 1 for all p 2 ININ ; n 2 IN, i.e. C is the operator which translates an enumeration p of a set A IN into a characteristic function of A.2 So, on the one hand the topological part of the reduction C F implies that F is at least as discontinuous as C and on the other hand the computable part of the reduction C F implies that F is suciently e ective such that it can be used to translate enumerations into characteristic functions. We claim, that in almost all non-trivial natural situations, where it can be shown that an operator F is not computably invariant, the proof implicitly contains a proof of C F . For instance, this is the case with Myhill's proof of the computable non-invariance of the operator of di erentiation d, i.e. implicitly C d is proved. Hence, the situation resembles the situation in classical recursion theory, where by many-one reduction K m A of the self-applicability problem K the non-recursiveness of a large class of natural sets A IN can be shown. Of course, the constant function with non-computable value is a simple example of a function for which this method of proving computable non-invariance fails. The main theorem of this paper in Section 3 is concerned with algebraic properties of operators F in metric spaces which are sucient to conclude C F . A special case of our properties is given by the First Main Theorem of Pour-El & Richards (cf. [20]) which deals with closed linear operators on Banach spaces. In Section 4 we prove that the assumptions of this theorem also guarantee C F . Especially, this gives a partial answer to the open problem no. 7 of Pour-El & Richards (cf. [20]). The neccessity to have a theorem that implies C F for a larger class than closed linear operators F in Banach spaces lies in the fact that many interesting operations are not of this 2 The operator C , reducibilities and corresponding hierarchies have been investigated for di erent purposes in [21, 19, 23, 24, 10]. In [3] we have shown that C is complete w.r.t. to a topological variant of in the class of F -measurable functions in Baire's space ININ .
2
type. In Section 5 we will illustrate this with operators on the space K(IR) of non-empty compact subsets of the real line. For instance, it is easy to see that the boundary operator @ : K(IR) ! K(IR); A 7! @A is discontinuous w.r.t. to the usual topology induced by the Hausdor metric on K(IR), i.e. it is also non-computable. Furthermore, with the help of our main theorem one can easily prove that @ is not computably invariant, i.e. there are computably compact sets A IR with a non-computable boundary @A. In Section 5 we prove that several other set-valued operators are not computably invariant.
2 Preliminaries In this section we will de ne some basic notions. First, we need computable metric spaces (cf. [25, 2, 6]).
De nition 2.1 (Computable metric space) (X; d; ) is called computable metric space, if
and only if
(1) d : X X ! IR is a metric on X , (2) : IN ! X is a function such that range( ) is dense in X , (3) d ( ) : IN2 ! IR is computable. Especially, each computable metric space is separable. Sometimes, we will say by abuse of notation that X or (X; d) is a computable metric space if resp. d is xed or out of consideration. One can precisely de ne computable operations in computable metric spaces via Cauchy representations (cf. [25, 26]) or equivalently recursive operations via recursion operators (cf. [4, 6]). We will omit these de nitions and refer the reader to the references. As a further notion we will need computable Banach spaces which additionally have computable vector space operations.
De nition 2.2 (Computable Banach space) (X; jj jj; +; ; ) is called computable Banach space, if and only if
(1) jj jj : X ! IR is a norm on X , (2) (X; d; ) is a computable complete metric space, where d(x; y) := jjx ? yjj for all x; y 2 X , (3) (X; +; ) is a computable vector space over IR, i.e. the vector space operations + : X X ! X and : IR X ! X are computable. For short, we will say that X or (X; jj jj; +; ) is a computable Banach space. Additionally, we need computable groups which will be de ned only for metric spaces.
De nition 2.3 (Computable group) Let X be a computable metric space. Then (X; +) is
called computable group, if and only if (1) (X; +) is a group,
3
(2) the addition + : X X ! X; (x; y) 7! x + y is computable, (3) the inverse mapping ? : X ! X; x 7! ?x is computable. Now we will de ne the central notion of this paper: De nition 2.4 (Computable invariance) Let X; Y be computable metric spaces and let Xc X resp. Yc Y be the subsets of computable elements. Then f : X ! Y is called computably invariant, if and only if f (Xc ) Yc. Now, we can de ne several reducibilities for functions. These reducibilities have been already considered in other investigations (cf. [21, 19, 23, 24, 3]). For the rest of this section let X; Y; U; V be computable metric spaces and let f : X ! Y , g : U ! V be mappings. De nition 2.5 (Reducibility) We de ne for i 2 f1; 2g: (1) f 1 g : () (9 computable A; B )(8x 2 dom(f )) f (x) = BgA(x), (2) f 2 g : () (9 computable A; B )(8x 2 dom(f )) f (x) = B (x; gA(x)), where A : X ! U and B : V ! Y resp. B : X V ! Y denote functions. One could also consider a more general Turing reducibility for functions, de ned by f T g : () f is fgg-recursive; where f is called fgg-recursive, if and only if it can be generated from some basic operations extended by g with the help of nitely many applications of the recursion operators (cf. [4, 6]). But Turing reducibility is out of the scope of this paper. One easy basic observation for this paper is that computable invariance is preserved by reducibility from the right to the left. Lemma 2.6 (Reducibility and computable invariance) For i 2 f1; 2g f i g and g computably invariant =) f computably invariant. Proof. Follows immediately, since computable functions are computably invariant. 2 As a consequence we can express our sucient condition for computable non-invariance with help of the operator C : ININ ! ININ , de ned in the introduction. Therefore, we consider Baire's space ININ as a computable metric space in the usual way (cf. [25]). Lemma 2.7 (Sucient condition for computable non-invariance) For i 2 f1; 2g C i f =) f not computably invariant. Proof. We only have to show that C is not computably invariant. But this follows immediately, since there is a computable p 2 ININ such that K = fn : (9k)p(k) = n + 1g for some recursively enumerable but non-recursive set K IN. Thus, C (p) is the characteristic function of K and hence non-computable. 2 4
3 Computable transformation spaces In this section we will provide an algebraic condition for functions f in computable metric spaces with an additional algebraic structure that is sucient for the reduction C i f . Intuitively, such a reduction means that we can use f to translate an enumeration p of a subset A IN into a characteristic function C (p) of A. The idea is to use for each n 2 IN a point of discontinutiy x1 of f to encode the characteristic value of n, which describes whether n 2 A or not. This distinction will be realized with the help of a sequence (xi )i2IN which converges e ectively to x1 such that the sets ff (xn ) : n 2 INg and ff (x1)g are suciently separated. Now, depending on the position of the value n + 1 in the sequence p we will choose n . If k is the rst position with p(k) = n +1 then n := xk , if no such position k exists, n := x1 . This procedure is repeated for each n with a suitable sequence and a corresponding point of discontinuity. This is the place where our algebraic structure P comes into consideration: we will use it to combine all those values n to one point x := i . The following de nition introduces the structure. Therefore, we will use some technical notations: for each function T : X IN ! Y we will write for short Tn (x) := T (x; n) for all x 2 X; n 2 IN. P If + : X X ! X is an operation and (xn )n2IN a sequence in X then we will write ki=j xi for the iterated operation (:::(((xj + xj +1) + xj +2P ) + xj +3 )::: + xk ) with k j . If there is a neutral P1 element 0 w.r.t.P+n and k k < j then we de ne iQ i=j xi := limn!1 i=j xi . =j xi := 0. We will use the abbreviation Correspondingly, we use as notation for a second iterated operation : X X ! X .
De nition 3.1 (Transformation space) A tuple (X; d; +; T; D) is called computable transformation space, if and only if
(1) (2) (3) (4)
(X; d) is a computable complete metric space, + : X X ! X is a computable operation, T : X IN ! X is a computable operation, sn := Pni=0 Ti (xi ) exists for each n 2 IN and d(sn ; sk ) < 2?k for all n > k and both for each sequence (xn )n2IN in D X .
The next de nition introduces homomorphisms of transformation spaces. If f is a discontinuous function between transformation spaces, then the homomorphism property will guarantee the existence of suciently many points of discontinuity and that f is algebraically well-behaved in a certain sense.
De nition 3.2 (Homomorphism of transformation spaces) Let X = (X; dX ; ; T; DX ), Y = (Y; dY ; +; S; DY ) be computable transformation spaces. Then f : X ! Y is called a Q1 homomorphism of X to Y , if and only if i Ti (xi ) 2 dom(f ) and =0
f
1 Y i=0
!
Ti (xi ) =
1 X i=0
Sif (xi )
for all sequences (xn )n2IN in DX , such that (f (xn ))n2IN is a sequence in DY . 5
The previous de nition can be generalized to k-ary functions f straightforwardly. The following de nition describes a property of transformation spaces P Q that will allow to retrieve the characteristic function C (p) from the value Si f (xi ) = f ( Ti (xi )). We will distinguish two cases. By B (x; ") := fy 2 X : d(x; y) < "g we denote the open balls, by B (x; ") the corresponding closed balls in metric spaces (X; d).
De nition 3.3 (Reversible transformation spaces) Let X = (X; d; +; T; D) be a computable transformation space, let D ; D D with D \ D = ; and let x 2 D, " > 0. (1) X is called reversible w.r.t. fD ; D g, if and only if there is a computable test function : X IN ! IN such that ! ( 1 X D Ti (xi ) = 10 ifif xxn 22 D () n n i k for all sequences (xi )i2 in D [ D and n; k 2 IN with n = k. (2) X is called strongly reversible w.r.t. fD ; D g, if and only if there is a computable test function : X IN ! IN such that condition () holds for all sequences (xi )i2 in D [ D ; n 2 IN and k = 0. (3) X is called (strongly) reversible in (x; "; ), if and only if X is (strongly) reversible w.r.t. fB (x; "); D n B(x; )g. 0
1
0
0
1
1
0
1
=
IN
0
1
0
1
IN
0
1
P In the case of reversibility one can imagine 1 i=k Ti (xi ) as a \stack memory" which allows to nd the iPwith xn 2 Di only for the top element xn with n = k while in the case of strong reversibility 1 i=0 Ti (xi ) behaves like a \random access memory" which allows to nd the i with xn 2 Di for all n. The following theorem shows that a homomorphism of suitable transformation spaces together with a suitable sequence (xn )n2IN which converges e ectively to a point of discontinuity x1 of f implies C i f .
Theorem 3.4 (Computable non-invariance of homomorphisms) Let X = (X ,dX ,,T ,DX ), Y = (Y; dY ; +; S; DY ) be computable transformation spaces with a homomorphism f : X ! Y . Let (xn )n2 be a computable sequence in DX and let D ; D DY such that (a) dX (xn ; xk ) < 2?k for all n > k and x1 := limn!1 xn 2 DX , (b) f (xn ) 2 D for all n 2 IN and f (x1 ) 2 D . IN
0
0
1
1
Then the following holds: (1) If Y is strongly reversible w.r.t. fD0 ; D1 g, then C 1 f . (2) If Y is reversible w.r.t. fD0 ; D1 g, (Y; +) is a computable group, (f (xn ))n2IN is a computable sequence and f (x1 ) a computable point in Y , then C 2 f . In both cases f is not computably invariant. Moreover, there exists a computable x 2 X , depending only on X and (xn )n2IN such that f (x) 2 Y is non-computable.
6
Proof. Since X is complete x1 exists. De ne : IN IN ! X by ( if k = minfm : p(m) = n + 1g exists n(p) := xk IN
x1
else
for all p 2 ININ and n 2 IN. Since (xn )n2IN is a computable sequence and dX (xk ; x1 ) 2?k for all k 2 IN, is computable. Since X is a computable transformation spaceQ and (n (p))n2IN is a sequence in DX for all p 2 ININ , s : ININ IN ! X , de ned by sn (p) := ni=0 Ti i (p) exists for all p 2 ININ , n 2 IN and dX (sn (p); sk (p)) < 2?k for all n > k. Since T; ; are computable, s is computable and hence A : ININ ! X de ned by 1 Y A(p) := nlim s ( p ) = Ti i(p) n !1 i=0
for all p 2 ININ is computable too. By (b) (f (xn ))n2IN is a sequence in DY and f (x1 ) 2 DY , thus (fn(p))n2IN is a sequence in DY for all p 2 ININ too. Since f is a homomorphism A(p) 2 dom(f ) for all p 2 ININ . (1) Let Y be strongly reversible w.r.t. fD0 ; D1 g and let : Y IN ! IN be the corresponding computable test function. De ne B : Y ! ININ by B (y)(n) := (y; n) for all y 2 Y and n 2 IN. Then B is computable and since f is a homomorphism we obtain
BfA(p)(n) = (fA(p); n) = n f 1 X
1 Y
i=0
Ti i (p)
!!
!
Si fi(p) = n i =0 ( 0 if fn(p) 2 D0 = 1 if fn(p) 2 D1 ( 0 if (9k)n (p) = xk = 1 if n (p) = x1 ( 0 if (9k)p(k) = n + 1 = 1 else = C (p)(n)
for all p 2 ININ , n 2 IN, i.e. C 1 f . (2) Now, let Y be reversible w.r.t. fD0 ; D1 g and let : Y IN ! IN be the corresponding computable test function. Furthermore, let (Y; +) be a computable group, and let (f (xn ))n2IN , f (x1 ) be computable. De ne B : ININ Y ! ININ , : ININ Y IN ! Y recursively by ! nX ?1 B (p; y)(n) := n y ? Si i (p; y) i=0
7
and
(
f (xk[p(k)=i+1]) if B (p; y)(i) = 0 f (x1) else for all p 2 ININ , y 2 Y , n; i 2 IN. Since (Y; +) is a computable group, (f (xn))n2IN , f (x1) are computable and ; S are computable, B; are computable too. Inductively, one can prove that B (p; fA(p))(i) = 0 implies the existence of k[p(k) = i +1] for all i 2 IN and hence (p; fA(p)) 2 dom(B ) for all p 2 ININ . Furthermore, by induction one can show i (p; fA(p)) = fi(p) for all p 2 ININ and i 2 IN. Since f is a homomorphism and (Y; +) is a group i (p; y) :=
fA(p) ?
nX ?1 i=0
Si i (p; fA(p)) = f = =
1 X i=0
1 X i=n
1 Y i=0
! n?1 X
Ti i(p) ?
Sifi (p) ?
nX ?1 i=0
i=0
Sifi(p)
Si fi(p)
Si fi(p):
for all p 2 ININ and n 2 IN. Correspondingly to (1) we obtain
B (p; fA(p))(n) = n fA(p) ? 1 X
= n
i=n
nX ?1
Si i(p; fA(p))
!
i=0!
Si fi(p)
= C (p)(n) for all p 2 ININ , n 2 IN, i.e. C 2 f . Now, there is a computable sequence p 2 ININ which enumerates a non-recursive set, i.e. C (p) is non-computable. Since the de nition of A does depend only on X and (xn )n2IN , the same holds for x := A(p). Since A is computable, x is computable too. Since C (p) = BfA(p) resp. C (p) = B (p; fA(p)) are non-computable, and B is computable, f (x) = fA(p) must be noncomputable. 2 From the proof one recognizes that in case (2) the computability of the group (Y;P+) and the computability of the sequence (f (xn ))n2IN is needed to access the \stack memory" Si fi. Obviously, the theorem can be generalized to k-ary functions f .
4 Computable Banach spaces In this section we will show that for each Banach space there is an induced reversible transformation space structure. Each closed and linear operator is a homomorphism of transformation 8
spaces. Last not least, we will prove that the presumptions of the First Main Theorem of PourEl & Richards (cf. [20]) imply the presumptions of our Main Theorem 3.4 and hence C 2 f . Thus, our theorem of Section 3 is a generalization of the non-trivial direction of the First Main Theorem. Lemma 4.1 (Induced transformation space) Let (X; jj jj; +; ) be a computable Banach space. Then X = (X; jjx ? yjj; +; 4?n?1 x; B (0; 1)) is a computable transformation space which is reversible in (0; 61 ; 56 ). Proof. For each sequence (xn)n2IN in B (0; 1) yn := Pni=0 4?i?1 xi exists and for all n > k n X jjyn ? yk jj = jj 4?i?1 xi jj < 4?k?1 < 2?k : i=k+1
Hence X is a computable transformation space. Now, de ne : X IN ! IN by 8 > if jjxjj < 4?n2?1 1 if jjxjj > 4?n2?1 : div else for all x 2 X; n 2 IN. Obviously, is computable. Since 1 1 ?n?1 X X jj 4?i?1 xi jj 4?i?1 = 4 3 ; i=n+1
i=n+1
we obtain for each sequences (xn )n2IN in fx : jjxjj < 61 g [ fx : 56 < jjxjj 1g and n 2 IN ! ( 1 1 X 6 : n 4?i?1 xi = 01 ifif jjjjxxn jjjj < n > 65 i=n
Hence is a computable test function for X and X is reversible w.r.t. fB (0; 61 ); B (0; 1) n B (0; 65 )g, i.e. X is reversible in (0; 16 ; 56 ). 2 The transformation space introduced in this lemma will be called the induced transformation space of the Banach space X . Especially, R := (IR; jx ? yj; 4?n?1 x; [?1; 1]) is a computable transformation space which is reversible in (0; 61 ; 56 ). It is easy to see that each closed linear operator in Banach spaces is a homomorphism of the corresponding induced transformation spaces. Lemma 4.2 (Induced homomorphism) Let X; Y be computable Banach spaces with induced computable transformation spaces X ; Y . If f : X ! Y is a closed linear operator, then f is a homomorphism of X to Y . Proof. Let (xn)n2IN be a sequence with jjxn jj 1 and jjf (xn)jj 1 for all n 2 IN. Since f is 1 linear, f (4?i?1 xi ) = 4?i?P f (xi) for all i 2 IN. Since X ; Y are computable transformation spaces, P 1 4?i?1 xi , as well as 1 4?i?1 f (xi ) exist. Since f is closed, P1 4?i?1 xi 2 dom(f ) and i=0 i=0 i=0 ! 1 1 X ?i?1 X 4 xi = 4?i?1 f (xi ) f i=0
follows. Thus, f is a homomorphism of X to Y . 9
i=0
2
Now we will prove that our Theorem 3.4 is a generalization of the non-trivial direction of the First Main Theorem of Pour-El & Richards (cf. [20]). Especially, the presumptions of the First Main Theorem do even imply C 2 f .
Theorem 4.3 (Computable non-invariance of unbounded linear operators) Let X; Y be computable Banach spaces and let f : X ! Y be a closed linear and unbounded operator. Let (en )n2IN be a computable sequence in dom(f ) whose linear span is dense in X and let (f (en ))n2IN be computable in Y . Then C 2 f . Especially, f is not computably invariant. Proof. Let X ; Y be the computable transformation spaces induced by X; Y . Since f is closed and linear, f is a homomorphism of X to Y . It is easy to see, that (Y; +) is a computable group
since Y is a computable Banach space. To apply Theorem 3.4 it remains to construct a suitable sequence (xn )n2IN . This can be done correspondingly to the proof of the First Main Theorem in [20]: since the linear span of (en )n2IN is dense in X and f : X ! Y is closed and unbounded, f has to be unbounded on the linear span of (en )n2IN . Hence, we can nd a computable sequence (bn )n2IN , consisting of nite rational linear combinations of (en )n2IN such that jjf (bk )jj > 2k+1 jjbk jj for all k 2 IN. Since f is linear and (f (en))n2IN is computable, (f (bn))n2IN is a computable sequence too. Let xn := jjf (bbnn )jj . Then jjxk jj < 2?k?1 and jjf (xk )jj = 1 for all k 2 IN. Especially, (f (xn ))n2IN is a computable sequence. Furthermore, for n > k
jjxn ? xk jj jjxnjj + jjxk jj < 2?k and jjf (xn )jj = 1 while jjf (limn!1 xn )jj = jjf (0)jj = 0. Thus, by Theorem 3.4 C f follows. 2 2
So, in contrast to the general situation of Theorem 3.4, in the special case of closed linear and unbounded operators in Banach spaces there is a canonical sequence (xn )n2IN which converges e ectively to 0 such that ff (xn ) : n 2 INg and f0g are suciently separated. In [20] it has been shown, that several unbounded closed and linear operators f from analysis and physics, like the operator of di erentiation d, ful ll the presumptions of the previous theorem. Hence we can conclude C 2 f for all those operators.
5 Operators on the space of compact subsets In this section we will apply our Main Theorem 3.4 to several set-valued operators. Moreover, we will prove the computable non-invariance of further set-valued operators by reduction. We will start with introducing the space of compact subsets (for this topic cf. [6, 16, 27, 28]). Let
K(X ) := fA X : A non-empty and compactg for each X IR. Let dA : IR ! IR; x 7! inf a2A jx ? aj be the distance function of A K(IR) and let dK : K(IR) K(IR) ! K(IR) be the Hausdor metric on K(IR), de ned by (IR)
(
dK(IR) (A; B ) := max sup dB (a); sup dA (b) a2A
10
b2B
)
for all A; B 2 K(IR). Furthermore, consider the transformation t : IR IN ! IR; (x; n) 7! x + n1+1 and the corresponding induced transformation on sets 4n+1 2 T : K(IR) IN ! K(IR); (A; n) 7! tn(A) = 4nx+1 + 2n1+1 : x 2 A : Let I := [0; 1] be the unit interval and let In := Tn (I ) for all n 2 IN. Now we can de ne a computable transformation space on K(IR).
Lemma 5.1 (Transformation space of compact subsets) K := (K(IR); dK(IR); [; T; K(I )) is a computable transformation space which is strongly reversible in (I; 31 ; 13 ). Proof. (K(IR); dK(IR)) can be considered as a computable complete metric space in the usual way. Then K(IR) K(IR) ! K(IR); (A; B ) 7! A [ B is a computable operation. It is easy to see that T is a computable operation too. Now, let (An )n2IN be a sequence in KS(I ). Then (Tn (An ))n2IN is a locally nite sequence of pairwise disjoint compact sets, i.e. Bn := ni=0 Ti (Ai ) is a compact set too. Furthermore, dK(IR) (Bn; Bk ) 2k1+1 + 4k1+1 < 2?k for all n > k. Thus K is a computable transformation space. I1 I0 0 I2 1 Figure 1: The transformed intervals In = Tn (I ) Now we have to show that there is a computable test function : K(IR) IN ! IN w.r.t. (I; ; 31 ). De ne by 8 > if supx2In dA (x) < 341n+1 1 if sup dA (x) > 341n+1 : div else x2In 1 3
for all A 2 K(IR); n 2 IN. Then is computable and for all sequences (An )n2IN in B (I; 13 ) [ K(I ) n B (I; 13 ) we obtain ! ( 1 [ An ) < 13 n Ti (Ai ) = 01 ifif ddK(IR) ((I; I; A ) > 1 K(IR)
i=0
for all n 2 IN. Thus K is strongly reversible in (I; 13 ; 13 ).
n
3
2
The next proposition shows, that many set-valued operators are homomorphisms of the de ned transformation spaces in a natural way. Therefore, let A denote the closure, A the set of inner points, @A the boundary, and A0 the derived set (of accumulation points) of A. Furthermore, let be the usual Lebesgue measure on the real numbers. 11
Proposition 5.2 (Set-valued homomorphisms) The following operators are homomorphisms of the corresponding computable transformation spaces:
(1) : K(IR) ! K(IR); A 7! A, (2) @ : K(IR) ! K(IR); A 7! @A, (3) d : K(IR) ! K(IR); A 7! A0 , (4) \ : K(IR) K(IR) ! K(IR); (A; B ) 7! A \ B , (5) : K(IR) ! IR; A 7! (A). Furthermore, C 1 F holds for each of these operators F . Especially, none of these operators is computably invariant.
Proof. First, we prove that @ is a homomorphism of K to K. Let (An)n2 be a sequence in K(I ) such that (@An )n2 is a sequence Sin K(I ) too. Since (Tn (An ))n2 is a locally nite sequence of pairwise disjoint compact sets, 1 i Ti (Ai ) 2 dom(@ ) and IN
IN
@
IN
1 [
i=0
!
=0
Ti (Ai ) =
1 [
i=0
@Ti (Ai ) =
1 [
i=0
Ti (@Ai );
i.e. @ is a homomorphism of K to K. The operators ; d can be treated correspondingly. Now, let (An )n2IN ; (Bn )n2IN be sequences in K(I ) such that (An \ Bn )n2IN is a sequence in K(I ) too. Since (Tn (An ))n2IN , (Tn (Bn ))n2IN are locally nite sequence of pairwise disjoint compact sets, S 1 (Ti (Ai ); Ti (Bi )) 2 dom(\) and i=0 ! ! 1 1 1 1 [ \ [ [ [ Ti (Ai ) Ti (Bi ) = (Ti(Ai ) \ Ti (Bi )) = Ti (Ai \ Bi ); i=0
i=0
i=0
i=0
i.e. \ is a homomorphism of K K to K. Now, let (An )n2IN be a sequence in K(I ). Then ((An ))n2IN is a sequence S1 in I . Since (Tn (An ))n2IN is a locally nite sequence of pairwise disjoint compact sets, i=0 Ti (Ai ) 2 dom() and ! 1 1 1 [ X X Ti (Ai ) = Ti (Ai ) = 4?i?1 (Ai); i=0
i=0
i=0
i.e. is a homomorphism of K to R. Now, to apply Theorem 3.4 we have to construct suitable sequences converging to points of discontinuity. Let Jn := [0; 71 ] [f 2nk+1 : k = 0; :::; 2n+1 g and Jn0 := [0; 17 ] [f 3nk+1 : k = 0; :::; 3n+1 g. Then (Jn )n2IN , (Jn0 )n2IN are computable sequences in K(I ) which converge to I . Furthermore, dK(IR) (Jn ; Jk ) 221k+1 < 2?k and dK(IR) (Jn0 ; Jk0 ) 231k+1 < 2?k for all n > k. We obtain (1) (Jn ) = [0; 71 ] for all n 2 IN and (I ) = I , (2) @ (Jn ) = f0; 71 g [ (Jn n [0; 17 ]) for all n 2 IN and @ (I ) = f0; 1g, (3) d(Jn ) = [0; 17 ] for all n 2 IN and d(I ) = I , 12
(4) Jn \ Jn0 = [0; 17 ] [ f1g for all n 2 IN and I \ I = I , (5) (Jn ) = 71 for all n 2 IN and (I ) = 1. and dK(IR) (I; [0; 71 ]) = 67 > 13 , and dK(IR) (I; f0; 71 g[(Jn n[0; 71 ])) 41 < 13 , dK(IR) (I; f0; 1g) = 12 > 13 , dK(IR) (I; [0; 17 ] [ f1g) = 73 > 13 . Thus, by Theorem 3.4 C 1 F for each of these operators F 2 f; @; d; \; g follows. 2 A carefull look at the proof shows, that we have used one transformation space K and one sequence (Jn )n2IN simultaneously for all operators ; @; d; . Hence by Theorem 3.4 we can conclude the following corollary.
Corollary 5.3 There is a computable compact set A [0; 1] such that neither A , nor the 0
boundary @A, nor the derived set A are computable compact sets, nor the measure (A) is a computable real number.
Now we will illustrate how one can use the method of reduction to prove that further setvalued operators are not computably invariant.
Proposition 5.4 (Not computably invariant set-valued operators) The following oper-
ators are not computably invariant: (1) c : K(I ) ! K(I ); A 7! Ac ,
(2) D : K(IR) K(IR) ! K(IR); (A; B ) 7! A n B , (3) : K(IR) K(IR) ! K(IR); (A; B ) 7! (A n B ) [ (B n A).
Proof.
(1) One can deduce from the previous proposition that 0 := jK(I ) also is not computably invariant. Since 0 (A) = A = Ac c = c2 (A) for all A 2 K(I ), it follows 0 = c2 . Hence, c2 and thus c are not computably invariant. (3) Since c(A) = Ac = (I n A) [ (A n I ) = (I; A) for all A 2 K(I ), it follows c 1 . Thus, is not computably invariant. (2) Since (A; B ) = (A n B ) [ (B n A) = A n B [ B n A for all A; B 2 K(IR), it follows 1 (D D). Hence, D D and thus D are not computably invariant.
2 13
Although the operators in the previous proposition are no homomorphisms w.r.t. to the introduced transformation space K, one can directly prove C 1 F for each of these operators F. Finally, we will introduce another transformation space structure on the space C (I ) of continuous functions which is di erent from the induced transformation space structure of C (I ) from Section 4, but which is even strongly reversible. Therefore, consider the usual metric dC(I ) : C (I ) C (I ) ! IR of uniform convergence on C (I ), de ned by dC(I ) (f; g) := sup jf (x) ? g(x)j x2I
for all f; g 2 C (I ). Furthermore, consider the transformation t on the real numbers, de ned at the beginning of this section. Let S : C (I ) IN ! C (I ) be the transformation, de ned by ( 1 ?1 Sn(f )(x) := 02n+1 f tn (x) ifelsex 2 In for all f 2 C0 (I ) := ff 2 C (I ) : f (0) = f (1) = 0; jjf jj < 1g, n 2 IN and x 2 I . Then we obtain
Lemma 5.5 (Transformation space of continuous functions) C := (C (I ); dC(I ) ; +; S; C0 (I )) is a computable transformation space which is strongly reversible in (0; 21 ; 12 ). Proof. (C (I ); dC I ) can be considered as a computable complete metric space in the usual way. ( )
Then + is a computable operation. It is easy to seePthat S is a computable operation too. Now, let (fn)n2IN be a sequence in C0 (I ). Then gn := ni=0 Si (fi ) 2 C0 (I ) for all n 2 IN and since (Sn (fn ))n2IN is a sequence of functions with pairwise disjoint support we obtain jjgn ? gk jj 2k1+1 < 2?k for all n > k. Thus C is a computable transformation space. Now we have to show that there is a computable test function : C (I ) IN ! IN w.r.t. (0; 21 ; 21 ). De ne by 8 >0 if supx2In jf (x)j < 221n+1 < n(f ) := > 1 if supx2In jf (x)j > 221n+1 : div else for all f 2 C (I ); n 2 IN. Then is computable and for all sequences (fn )n2IN in B (0; 21 ) [ C0 (I ) n B (0; 21 ) we obtain ! ( 1 1 X 2 n Si (fi) = 01 ifif jjjjffn jjjj < >1 i=0
for all n 2 IN. Thus C is strongly reversible w.r.t. (0; 12 ; 12 ).
n
2
2
One can use this strong reversible transformation space for a direct proof of C 1 d for the operator of di erentiation d (cf. [21, 19]). We will prove that the support operator and the zero operator are homomorphisms of computable transformation spaces and thus they are not computably invariant. 14
Proposition 5.6 (The support and zero operators) The following operators are homomorphisms of transformation spaces: (1) supp : C (I ) ! K(I ); f 7! supp(f ) := f ?1f0gc , (2) zero : C (I ) ! K(I ); f 7! f ?1 f0g. Furthermore, C 1 supp and C 1 zero. Especially, supp; zero are not computably invariant.
Proof. Let (fn)n2IN be a sequence in C (I ) such that (supp(fn))n2IN is a sequence inPK(I ). Since (Sn (fn ))n2IN is a sequence of functions with pairwise disjoint non-empty supports, 1 i=0 Si (fi ) 2 dom(supp) and ! 1 1 1 X [ [ Si (fi) = supp(Si (fi )) = Ti (supp(fi)); supp i=0
i=0
i=0
i.e. supp is a homomorphism of C to K. For the zero operator we consider a modi cationSof K. De ne the transformation T 0 : K(IR) IN ! IN by Tn0 (A) := Tn (A) [ D where D := I n ( 1 i=0 Ii ). Since D 2 K(I ) is computable, it is 0 easy to see that T is computable. One can prove that the corresponding modi ed transformation space K0 is strongly reversible in (I; 13 ; 13 ), correspondingly to K.PNow, let (fn )n2IN be a sequence in C (I ) such that (zero(fn))n2IN is a sequence in K(I ). Again, 1 i=0 Si(fi ) 2 dom(zero) and ! 1 1 X [ zero Si (fi) = Ti0(zero(fi)); i=0
i=0
i.e. zero is a homomorphism of C to K0 . Now, to apply Theorem 3.4 we have to construct suitable sequences converging to points of discontinuity. Therefore, let (fn )n2IN be a sequence of rational polygons, such that fn is de ned by the vertices (0; 0); ( 41 ; 1); ( 21 ; 21n ); ( 43 ; 21n ); (1; 0):
fn(x) 6 1
1 2n
0
??
H ?? HHHH H ?? r
r
1 2
3 4
r
0
HH HH HH -
r
r
1 4
1 x
Figure 2: The rational polygon fn Then (fn )n2IN is a computable sequence in C0 (I ) which converges to a function f 2 C0 (I ) such that jjfn ? fk jj 21k ? 21n < 21k for all n > k. We obtain (1) supp(fn) = [0; 1] and supp(f ) = [0; 21 ], 15
(2) zero(fn ) = f0; 1g and zero(f ) = f0g [ [ 12 ; 1] and dK(IR) (I; [0; 12 ]) = 12 > 31 , dK(IR) (I; f0; 1g) = 12 > 13 , and dK(IR) (I; f0g[ [ 21 ; 1]) = 41 < 13 . Thus, by Theorem 3.4 C 1 F for F 2 fsupp; zerog follows. 2 Again, a carefull look at the proof shows, that we have used one transformation space C and one sequence (fn) simultaneously for both operators supp and zero. Hence by Theorem 3.4 we can conclude the following corollary.
Corollary 5.7 (Support and Zero) There is a computable function f 2 C (I ) such that neither f ? f0g nor f ? f0gc is a computable compact set in K(I ). 1
0
1
6 Conclusion In this paper we have investigated methods to prove the computable non-invariance of operators F . We have expressed a sucient condition for computable non-invariance by C 2 F . Topologically, the best what one can expect of such an operator is to be F -measurable. It would be an interesting question, how non-computable operators could be classi ed precisely in Borel's hierarchy k . In [3] we have de ned generalizations Ck of C which are complete in the class of k -measurable functions in Baire's space ININ w.r.t. a topological version of 2 . Most natural operators should be computably equivalent to Ck for some k 2 IN. Furthermore, we have investigated an algebraic condition for operators F in metric spaces which implies C 2 F . We have applied the corresponding Theorem 3.4 to several set-valued operators. It would be a further interesting question whether there are more general or more suitable conditions which also imply C 2 F . Moreover, we have shown that our algebraic condition is a generalization of the First Main Theorem of Pour-El & Richards (cf. [20]). Especially, C 2 F holds for all those closed linear and unbounded operators F of analysis and physics which ful ll the additional condition of the theorem (i.e. they have to admit a suitable computable sequence (en )n2IN ). A further topic of interest is the polynomial-time invariance of operators. Ko and others (cf. [7, 11, 12, 13, 14, 1]) have constructed a lot of counterexamples of the type such that there is a polynomial-time computable x and F (x) is not even computable. Since there is no welldeveloped general theory of complexity in metric spaces, it is quite dicult to modify Theorem 3.4 in this direction. But if we assume that (X; d; +; T; D) is a computable transformation space such that there is a reasonable notion of complexity for X and D is compact, then we have uniform complexity bounds for functions on D. If, furthermore, T and + are polynomial-time computable in a certain sense, then it should be possible to derive a polynomial-time version of Theorem 3.4.
References
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