Computable Real Functions: Type 1 Computability Versus ... - CiteSeerX
Computable Real Functions: Type 1 Computability Versus Type 2 Computability Peter Hertling Theoretische Informatik I, FernUniversitat, D-58084 Hagen, Germany August, 1996
Abstract
Based on the Turing machine model there are essentially two di erent notions of computable functions over the real numbers. The e ective functions are de ned only on computable real numbers and are Type 1 computable with respect to a numbering of the computable real numbers. The e ectively continuous functions may be de ned on arbitrary real nunbers. They are exactly those functions which are Type 2 computable with respect to an appropriate representation of the real numbers. We characterize the Type 2 computable functions on computable real numbers as exactly those Type 1 computable functions which satisfy a certain additional condition concerning their domain of de nition. Our result is a sharp strengthening of the well-known continuity result of Tseitin and Moschovakis for e ective functions. The result is presented for arbitrary computable metric spaces.
1 Introduction In this paper we compare two approaches for de ning computability of functions between computable metric spaces. The rst approach can be described as e ectivity or computability with respect to numberings, the second as e ective continuity or computability with respect to representations. We shall give special attention to the space of real numbers. In the rst approach one considers countable metric spaces M , M 0 with numberings. One of the most important examples is the space IRc of computable real numbers. A numbering of this space is derived from a numbering of the algorithms or machines that compute such a number. A function f : M ! M 0 1 is called Type 1 computable or e ective if there is partial computable function g 2 P (1) which transforms any index of an element x 2 dom f into an index of the result f (x) 2 M 0 . This approach, with some variations, has been considered in the Russian school of constructive analysis, represented by Markov [10], Shanin [15], Tseitin [21], Kushner [9], and others. It has also been considered by Moschovakis [11], Aberth [1] and by many others. Closely related is the constructive approach to analyis presented by Bishop and Bridges [3]. In the second approach one considers metric spaces M , M 0 with a numbering of a dense subset, respectively. Consider for example the space IR of real numbers with a standard numbering of the rational numbers. A representation of M (respectively of M 0 ) is de ned via the Email: [email protected] 1
We use the notation f : X ! Y for a (partial) function f with domain dom f X .
1
numbering of the dense subset. One can represent an element x by sequences p in the Baire space IB = ININ = fp : IN ! INg which describe normed Cauchy sequences consisting of elements in the enumerated dense subset with limit x. A function f : M ! M 0 is Type 2 computable if there is a partial recursive operator in the sense of Rogers [14] on the Baire space IB that transforms any name p 2 IB for an element x 2 dom f into a name for f (x). This is equivalent to e ective continuity in the following sense: there is an algorithm which, given a desired output precision 2?n , needs only a nite approximation of an element x 2 dom f in order to compute an approximation of the result f (x) up to precision 2?n , and which is able to compute a suf cient input precision. For real functions this leads to the notion of computability considered by Grzegorczyk [4], Klaua [6], Pour-El and Richards [13], Weihrauch [22], Ko [7], and by many others. How are the two computability notions related? From the normed Cauchy representation of a metric space M (with a numbering of a dense subset) one can derive a numbering of the subspace Mc M of the computable elements: a number i is an index for a computable element x if it is an index of a name p 2 R(1) = fp : IN ! IN j p recursiveg for a recursive normed Cauchy sequence with limit x. In the following we consider Type 2 computability with respect to normed Cauchy representations and Type 1 computability with respect to the derived numberings of the computable elements. We assume that the metric spaces M and M 0 are computable and recursively separable; see Section 3.1. It is easy to see that the restriction to Mc of a Type 2 computable function f : M ! M 0 is Type 1 computable. Note that this implies f (Mc) M 0 c . Hence, for a function f : Mc ! M 0 Type 1 computability is a necessary condition for Type 2 computability. A simple example of Myhill (see Remark 2 in [8]) shows that it is not a sucient condition. But Weihrauch's [22] version of the well-known Tseitin/Moschovakis continuity result shows that a Type 1 computable function with a recursively separable domain must be Type 2 computable. Yet, in general, recursive separability of the domain is a too strong additional condition: there is a Type 2 computable function f : Mc ! M 0 which cannot be extended to a Type 1 computable function with a recursively separable domain. It is the aim of this paper to give a sharp condition. We show that a function f : Mc ! M 0 is Type 2 computable if and only if it is Type 1 computable and satis es a certain additional condition which in general is weaker than recursive separability of the domain (to be precise: it is weaker than the possibility to extend the function to a Type 1 computable function with a recursively separable domain). This result is a sharp strengthening of the continuity result of Tseitin [20, 21] and Moschovakis [11]. It seems to be new even for functions between the space R(1) of total computable functions or the space IRc of computable real numbers. Note that a special version of the continuity result had been obtained by Kreisel, Lacombe, and Shoen eld [8]. There is also an analogue for a complete partial order, c.f. Myhill and Sheperdson [12]. A combined generalization has been obtained by Spreen and Young [19] and by Spreen [17]. Another generalization for complete partial orders, which also implies the result of Tseitin and Moschovakis for metric spaces, has been obtained by Berger [2]. But in all these papers recursive separability of the domain of a function has been assumed in order to deduce the (global) e ective continuity (compare Remark 4.3). In the papers [19, 17, 18] Spreen also investigates the relation between the two computability notions. He considers more general spaces and gives a condition which in combination with e ectivity and recursive separability of the domain implies e ective continuity. But in the general case the problem to give a sharp additional condition, i.e. to give a characterization of e ective continuity as above, seems to be still open. For a discussion of the two approaches for 2
de ning computability the reader is referred to Kushner [9], pp. 304 { 306, and to Weihrauch [22], Ch. 3.6. First we formulate a corollary of the main result for the special case of operators on the space of total recursive functions, which explains already the kind of characterization. In the following section the notions concerning computable metric spaces, Type 1 computable functions (e ective functions), and Type 2 computable functions (e ectively continuous functions) are introduced. Then we discuss the classical results more precisely. In the last section the new condition is introduced and the main results are stated. Actually we give two conditions, which are equivalent in all interesting cases. Due to lack of space the proofs are omitted.
Notations
IN = f0; 1; 2; : : : g is the set of nonnegative integers, Ql is the space of rational numbers, IR is the space of real numbers, IB = fp : IN ! INg is the Baire space (see Example 3.8), h:; :i : IN2 ! IN is a standard pairing function, and the maps i : IN ! IN for 1 i 2 are the corresponding inverse maps with j hn1 ; n2 i = nj . By P (1) := fp : IN ! IN j p computableg we denote the set of all (partial) computable number functions, by R(1) := fp : IN ! IN j p computableg the space of total recursive functions. A numbering of a denumerable set S is a (partial) surjective function : IN ! S . For i 2 dom we denote the corresponding element in S either by i or by (i). A numbering is reducible to a numbering 0 (written 0 ) if there is a g 2 P (1) such that (i) = 0 g(i), for all i 2 dom . If 0 and 0 then the numberings , 0 are equivalent (written 0 ). The map Ql : IN ! Ql is the standard total numbering of Ql de ned by Ql hi; j; ki = (i ? j )=(k + 1). ' : IN ! P (1) is a total standard numbering of P (1) and Wi := dom 'i for all i 2 IN. Hence, W is a total numbering of all recursively enumerable (r.e.) sets.
2 Operators on the Space of Total Recursive Functions In order to give an impression of our main result we formulate a corollary for the important special case of an operator mapping a subset of the total recursive functions R(1) to R(1) . The restriction '0 := 'j'? (R ) of ' is a partial numbering of R(1) . A function f : R(1) ! R(1) is called ('0 ; '0 ){computable or e ective if there is a function g 2 P (1) such that for all i 2 '?1 (dom f ) we have i 2 dom g, 'g(i) 2 R(1) , and f ('i ) = 'g(i) . The function f is called e ectively continuous if there is a (partial) recursive operator F in the sense of Rogers [14] with F jdom f = f ; compare Section 3.3. Kreisel, Lacombe and Shoen eld [8] proved that an e ective function f : R(1) ! R(1) with a recursively separable domain is e ectively continuous (see Theorem 1 in [8] and the remark preceding Theorem 3 in [8]). The domain dom f is recursively separable if there is an r.e. set E IN with 1
(1)
(1) E '?1 (dom f ); (2) dom f '(E ) : If g 2 P (1) is a function which \computes" f , i.e. f ('i ) = 'g(i) for all i 2 dom f , then dom f and hence also '(E ) must be contained in the set
X (g) := fp 2 R(1) j (8i; j 2 '?1 fpg) g(i) and g(j ) are de ned and 'g(i) = 'g(j) 2 R(1) g 3
of elements p 2 R(1) at which g \de nes a value in R(1) ". Now the main point is that it is not necessary that g really \de nes a value in R(1) " at the elements in '(E ). Instead it is sucient to demand that for each n 2 IN one can nd an r.e. set of elements whose closure contains dom f and at which g \de nes a value up to precision n". Therefore we de ne the set Xn(g) := fp 2 R(1) j (8i; j 2 '?1 fpg) g(i) and g(j ) are de ned and (8k n) 'g(i) (k) = 'g(j ) (k) are de nedg of elements p at which g de nes a value up to precision n. We call a function f : R(1) ! R(1) D'-e ective if there is a function g 2 P (1) with f ('i ) = 'g(i) for all i 2 '?1 (dom f ) and if additionally there is a function e 2 R(1) such that for all n 2 IN (1) We(n) '?1 (Xn (g)); (2) dom f '(We(n) ) : Obviously, if a function f : R(1) ! R(1) is e ective and has a recursively separable domain, then it is D'-e ective.
Theorem 2.1 For a function f : R(1) ! R(1) the following two conditions are equivalent: 1. f is D'-e ective, 2. f is e ectively continuous.
3 Notions In this section the basic notions are introduced and simple statements connecting them are deduced. First we consider computable metric spaces. Then e ective functions are de ned, i.e. functions that are (Type 1) computable with respect to numberings. Finally we introduce e ectively continuous functions and functions that are (Type 2) computable with respect to normed Cauchy representations.
3.1 Computable Metric Spaces
The various approaches for introducing e ectivity on a metric space M can be summarized by saying that one imposes e ectivity conditions on the metric with respect to a numbering of a dense subset. Hence e ectivity for metric spaces has been introduced only for separable metric spaces. Under a metric space we always understand a triple (M; d; ) where M is a nonempty set, d : M M ! IR is a metric, and : IN ! M is a numbering of a dense subset of M . Sometimes we write only (M; ), leaving aside d.
De nition 3.1 A metric space (M; d; ) (or just (M; )) is called semicomputable if there is an r.e. set D< IN such that for all i; j 2 dom and all n 2 IN hi; j; ni 2 D< () d( i ; j ) < Ql (n) : It is called a computable metric space if furthermore there is an r.e. set D> IN such that for all i; j 2 dom and all n 2 IN hi; j; ni 2 D> () d( i ; j ) > Ql (n) : 4
Computationally interesting spaces are usually not just separable but recursively separable.
De nition 3.2 A subset X of a metric space (M; ) is called -recursively separable if there is an r.e. set E ?1 (X ) such that X (E ). Often we simply write \recursively separable" instead of \ -recursively separable" if it is clear which numbering is meant. Note that in general a subspace of a recursively separable metric space is of course separable, but it does not have to be recursively separable; compare Slisenko [16]. The e ectivity properties of a metric space depend only on the equivalence class of the numbering of a dense subset, as the following easily proved lemma shows.
Lemma 3.3 Let (M; ) be a metric space, and let 0 be a numbering of a dense subset of M with 0 . If (M; ) is semicomputable (computable), then (M; 0 ) is semicomputable (computable). If (M; 0 ) is recursively separable, then (M; ) is recursively separable.
We remark that, if a metric space (M; ) is recursively separable, then one can easily construct a total numbering 0 of a dense subset of M with 0 . We wish to consider computations over the real numbers that are based on approximations. The idea is to describe the real numbers by Cauchy sequences of rational numbers that converge recursively. Among these it is sucient to consider the Cauchy sequences with a normed rate of convergence. This can be done also for arbitrary metric spaces with a numbering of a dense subset. A normed Cauchy sequence is a sequence of points (xi )i2IN in a metric space M with d(xi ; xj ) < 2? minfi;jg . Using normed Cauchy sequences and a numbering of a dense subset one can de ne a natural representation of the metric space, de ne its computable elements and derive a numbering of its computable elements.
De nition 3.4 Let (M; ) be a metric space. The Cauchy representation : IB ! M of (M; ) is de ned by
(p) = x i ( p(n) )n2IN is a normed Cauchy sequence with limit x 2 M : By (i) := 'i for i 2 dom := fi 2 IN j 'i 2 dom g we de ne a partial numbering of the set Mc := range of the computable elements of M . The normed Cauchy representations are special cases of the admissible representations of
T0 -spaces with countable bases considered in the Type 2 Theory of E ectivity; see Weihrauch
[22]. In Section 3.3 we will de ne the computability of functions with respect to representations. Here we are interested in the numbering .
Lemma 3.5 If (M; ) is a semicomputable (computable, recursively separable) metric space
then also (M; ) and (Mc ; ) are semicomputable (computable, recursively separable) metric spaces.
Furthermore one easily checks for any metric space (M; ). A metric space (M; ) is called weakly complete if the converse relation is true also, i.e. given an index of a recursive normed Cauchy sequence converging to an enumerated point, one can compute an index of the point:
j ? (range ) : 1
5
The most important special property of the derived numbering is that the spaces (M; ) and (Mc ; ) are weakly complete. The expression \weakly complete" is taken from Kushner [9]. For computable metric spaces with a surjective numbering the condition above is equivalent to the Condition (A) introduced by Moschovakis [11]. Example 3.6 One of the simplest examples of a computable, recursively separable, and weakly complete metric space is the discrete space IN with the trivial total numbering idIN (and the distance d(n; m) = 1 for n 6= m). Example 3.7 The space (IR; Ql ) is a computable recursively separable metric space, but it is not weakly complete. The space (IRc ; ) is the standard space of the computable real numbers. It is computable, recursively separable, and weakly complete. Example 3.8 Finally let : IN ! IN be a total bijective standard numbering of the nite words over IN, With (i) := (i)0! the space (IB; ) is a computable recursively separable metric space. We use the metric d : IB IB ! IR de ned by ( ? minfi j p(i)6=q(i)g if p 6= q d(p; q) := 02 else for any p = p(0)p(1)p(2) : : : 2 IB, q = q(0)q(1)q(2) : : : 2 IB. Because of range = R(1) , the computable elements of (IB; ) are just the total recursive functions. The restriction '0 := 'j'? (R ) is also a partial numbering of R(1) . One easily checks '0 . The spaces (R(1) ; ) and (R(1) ; '0 ) are computable, recursively separable, and weakly complete metric spaces. Q l
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3.2 E ective Functions
In this section (M; d; ) and (M 0 ; d0 ; ) are two xed metric spaces. We introduce e ective or Type 1 computable functions between M and M 0 . De nition 3.9 A function f : M ! M 0 is called ( ; ){computable if dom f range and there is a g 2 P (1) with (8i 2 ?1 (dom f )) f ( i ) = g(i) : The rst property to be mentioned is that the computability of a function depends only on the equivalence class of the numbering. This follows immediately from the following obvious lemma. Lemma 3.10 Let ~ : IN ! M and ~ : IN ! M 0 be additional numberings of dense subsets with ~ and ~. If f : M ! M 0 is ( ; ){computable, then f is (~ ; ~){computable. We are mainly interested in ( ; ){computable functions. If it is clear which numberings and are meant then we will call the ( ; ){computable functions e ective functions. Remark 3.11 The above notion of computability with respect to numberings is the notion which is used e.g. in the early paper [10] by Markov, by Helm [5], by Weihrauch [22] and by many other authors. In constructive analysis (see e.g. Kushner [9]) a more restrictive de nition of an e ective function f : M ! M 0 between enumerated sets M and M 0 with numberings : IN ! M and : IN ! M 0 (i.e. and are surjective) is often used. One demands that there is a partial recursive function g 2 P (1) with f ( i ) = g(i) for all i 2 ?1 (dom f ) and additionally with ?1 (dom f ) = dom \ dom g. If a function f can be computed in this way, Weihrauch [22] calls it strongly ( ; ){computable. For example Tseitin [21] and Moschovakis [11] have formulated their continuity results for e ective operators in this stronger sense. 6
3.3 Computable and E ectively Continuous Functions
Let (M; ) and (M 0 ; ) be metric spaces. We de ne Type 2 computable functions and two forms of e ective continuity for functions f : M ! M 0 . In order to de ne e ective continuity we use a numbering B of a base of (M; ) de ned by
Bi := B ( (i) ; 2? (i) ) = fx 2 M j d( (i) ; x) < 2? (i) g 1
2
2
1
for i 2 dom B := fi j 1 (i) 2 dom g. The corresponding numbering of a base of (M 0 ; ) will be denoted by B 0 in this section. The rst of the following three de nitions of global e ective continuity expresses that the preimage of a basic open set is open in an e ective way. The second is based on "{ continuity. The third expresses the computability used in Type 2 Theory of E ectivity, c.f. Weihrauch [22].
De nition 3.12 A function f : M ! M 0 is called 1. weakly e ectively continuous if there is a function g 2 P (1) with
[
(8j 2 dom B 0 )(Wg(j ) dom B and f ?1(Bj0 ) = dom f \ fBi j i 2 Wg(j ) g); 2. e ectively continuous if there is an r.e. set V IN such that (8hi; j i 2 V )(i 2 dom B ^ j 2 dom B 0 ^ f (Bi ) Bj0 ) and (8x 2 dom f )(8" > 0)(9hi; j i 2 V )(x 2 Bi ^ 2? (j ) < ") ; 2
3. ( ; )-computable if there is an e ectively continuous function F : IB ! IB such that for all p 2 ? 1 (dom f ) f (p) = F (p) :
X of M is called a Lacombe set if there is an r.e. set A dom B such that X = SfBA jsubset i 2 A g . Hence a function is weakly e ectively continuous if the preimage of any ball Bj0 is i a Lacombe set and given an index j of such a ball one can compute an index of this Lacombe set. The rst condition in Part (2) (e ective continuity) of the de nition states that V enumerates only pairs of balls Bi , Bj0 such that Bi is mapped to Bj0 . By the second condition suciently many such pairs are enumerated. Finally note that for a function F : IB ! IB the following conditions are equivalent: (1) F is weakly e ectively continuous, (2) F is e ectively continuous, (3) F can be extended to a partial recursive operator G : fp : IN ! INg ! fp : IN ! INg in the sense of Rogers [14], x9.8, (4) F can be extended to a total recursive operator G in the
same sense. The metric spaces considered in Type 2 Theory of E ectivity (Weihrauch [22, 23]) are usually semicomputable metric spaces with a total numbering of a dense subset, hence semicomputable, recursively separable metric spaces. A more e ective version of the next proposition has been given by Weihrauch [23].
Proposition 3.13 If (M; ) and (M 0 ; ) are semicomputable and recursively separable, then for a function f : M ! M 0 the conditions (1), (2), and (3) in De nition 3.12 are equivalent.
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4 On the Classical Results We discuss shortly the known results on the relation between Type 1 computability and Type 2 computability for functions between computable metric spaces. The following proposition is well-known and easy to prove.
Proposition 4.1 Let (M; ) be a semicomputable metric space and (M 0 ; ) be a metric space. If f : M ! M 0 is e ectively continuous then f jMc is ( ; ){computable. On the other hand Remark 2 in [8] contains a simple example, due to Myhill, of an e ective function f : R(1) ! IN which is discontinuous and, hence, cannot be Type 2 computable. Thus, e ectivity alone does not imply continuity. But the famous result of Tseitin [20, 21] implies that a strongly ( ; )-computable function f : Mc ! Mc0 is e ectively continuous if (M; ) is computable and recursively separable and (M; ) is computable. In this case the domain of f must itself be recursively separable; compare Kushner [9], Ch. 9. x2, Lemma 5. In fact one can prove the following result; see Weihrauch [22].
Proposition 4.2 Let (M; ) be a semicomputable metric space and (M 0 ; ) be a computable, recursively separable metric space. If a function f : Mc ! M 0 is ( ; ){computable and has
a -recursively separable domain, then it is ( ; )-computable.
Actually Tseitin formulated his result not for the derived numberings , , but more generally for weakly complete numberings. Moschovakis gave a similar but slightly weaker formulation ([11], Theorem 12)
Remark 4.3 Moschovakis also gave another continuity result in [11], Theorem 3. Using slightly weaker assumptions he derived e ective \pointwise" continuity of e ective functions. Helm [5] has given an example of an e ective and e ectively pointwise continuous function which is not e ectively continuous.
The question arises whether the last proposition is sharp, i.e. whether it covers all e ectively continuous functions. The answer is Yes if the image space is discrete.
Theorem 4.4 Let (M; ) be a computable, recursively separable metric space and (M 0 ; ) be a discrete, computable, recursively separable metric space. Then for a function f : Mc ! M 0 the following three conditions are equivalent. 1. f can be extended to a strongly ( ; )-computable function. 2. f can be extended to a ( ; )-computable function with a -recursively separable domain. 3. f is e ectively continuous.
But in the general case the answer is No. There is even a counterexample with a discrete domain.
Theorem 4.5 There is an e ectively continuous function f : IN ! R(1) which cannot be extended to an e ective function g : IN ! R(1) with a recursively separable domain. This poses the problem to characterize the relation between e ectivity and e ective continuity precisely, which is the aim of the paper. 8
5 Characterization of Type 2 Computability by Type 1 Computability and an Additional Condition
In this section (M; d; ) and (M 0 ; d0 ; ) are two xed metric spaces. We have seen that an e ective function with a recursively separable domain is e ectively continuous. We shall show that a weaker condition than recursive separability suces. In fact we give two conditions, which turn out to be equivalent under suitable assumptions. Let us assume that a function f : M ! M 0 is ( ; ){computable via a function g 2 P (1) , i.e. dom f range and f ( (i)) = (g(i)) for all i 2 ?1 (dom f ). Then dom f is recursively separable if and only if there is an r.e. set E IN such that (1) E ?1 (dom f ); (2) dom f (E ): The domain dom f must be contained in the set
CX (g) := fx 2 M j (8i; j 2 ?1 fxg) ( ('g(i) (n)))n2IN and ( ('g(j) (n)))n2IN are normed Cauchy sequences with the same limitg
of points in which g \de nes a value" with respect to and . For any n 2 IN we de ne the set CXn (g) := fx 2 M j (8i; j 2 ?1 fxg) 'g(i) (n) 2 dom ; 'g(j) (n) 2 dom ; d( ('g(i) (n)); ('g(j) (n))) < 3 2?n g of points in which g de nes a value up to precision n. A slightly stronger version is given by the sequence of sets
SCXn(g) := fx 2 M j (8i; j 2 ?1 fxg) (8k n) 'g(i) (k) 2 dom ; 'g(j) (k) 2 dom ; d( ('g(i) (k)); ('g(j) (k))) < 3 2?k g of points in which g strongly de nes a value up to precision n. Obviously CX (g) SCXn (g) CXn (g) for all n 2 IN. De nition 5.1 1. A function f : M ! M 0 is called D-( ; ){computable or D-e ective if it is ( ; ){computable via a function g 2 P (1) and additionally there is a function e 2 R(1) such that for all n 2 IN we have (1) We(n) ?1 (CXn (g)); (2) dom f (We(n) ): 2. A function f : M ! M 0 is called SD-( ; ){computable or strongly D-e ective if it is ( ; ){computable via a function g 2 P (1) and additionally there is a function e 2 R(1) such that for all n 2 IN we have (1) We(n) ?1 (SCXn (g)); (2) dom f (We(n) ): Obviously a strongly D-e ective function is D-e ective. We shall see that under suitable conditions on the spaces M and M 0 a D-e ective function is e ectively continuous and, vice versa, an e ectively continuous function is even strongly D-e ective. 9
Proposition 5.2 Let f : M ! M 0 be ( ; ){computable via a function g 2 P (1) and let dom f be recursively separable. Then there is a function e 2 R(1) such that f is SD-( ; ){ computable via g and e.
Proof. Let E IN be an r.e. set such that E ?1(dom f ) and dom f (E ). The statement is proved by a function e 2 R(1) with We(n) = E for all n 2 IN. 2 By the last proposition and by Theorem 4.5 the following result is a strengthening of the Tseitin/Moschovakis result (for a recursively separable image space).
Theorem 5.3 Let (M; ) be a semicomputable metric space, and let (M 0; ) be a computable recursively separable metric space. If f : Mc ! M 0 is D-( ; ){computable then it is ( ; ){ computable.
We have copied from Weihrauch [22] the observation that the domain does not need to be computable but only semicomputable. The theorem is a sharp result since the converse is also true.
Theorem 5.4 Let (M; ) be a computable recursively separable metric space and let (M 0 ; ) be a semicomputable metric space. If f : M ! M 0 is e ectively continuous then f jMc is SD-( ; ){computable.
Note that here we do not need that the image space (M 0 ; ) is computable. Instead semicomputability suces. Summarizing Theorem 5.3 and Theorem 5.4 we obtain our main result.
Theorem 5.5 Let (M; ) and (M 0 ; ) be computable recursively separable metric spaces. Then for a function f : Mc ! M 0 the following conditions are equivalent. 1. f is D-( ; ){computable. 2. f is SD-( ; ){computable. 3. f is weakly e ectively continuous. 4. f is e ectively continuous. 5. f is ( ; ){computable.
Proof. The equivalences (3) () (4) () (5) have been stated already in Proposition 3.13. (4) ) (2) is the content of Theorem 5.4, (2) ) (1) is trivial, and (1) ) (5) is the content of
2
Theorem 5.3.
Theorem 5.5 describes precisely the di erence between the Type 1 computability notion for real functions based on the numbering of the computable real numbers and the Type 2 computability notion based on representations like the Cauchy representation ; see the introduction. Q l
Q l
10
References [1] O. Aberth. Computable analysis. McGraw{Hill, New York, 1980. [2] U. Berger. Total sets and objects in domain theory. Annals of Pure and Applied Logic, 60:91{117, 1993. [3] E. Bishop and D. S. Bridges. Constructive Analysis. Springer-Verlag, Berlin, Heidelberg, 1985. [4] A. Grzegorczyk. On the de nitions of computable real functions. Fund. Math., 44:61{71, 1957. [5] J. P. Helm. On e ectively computable operators. Zeitschrift f. math. Logik und Grundlagen d. Math., 17:231{244, 1971. [6] D. Klaua. Konstruktive Analysis. VEB Deutscher Verlag der Wiss., Berlin, 1961. [7] K.-I Ko. Complexity Theory of Real Functions. Birkhauser, Boston, 1991. [8] G. Kreisel, D. Lacombe, and J. R. Shoen eld. Partial recursive functionals and e ective operations. In A. Heyting, editor, Constructivity in Mathematics, Proceedings of the Colloquium held at Amsterdam 1957, pages 290{297, Amsterdam, 1959. North{Holland Publishing Company. [9] B. A. Kushner. Lectures on Constructive Mathematical Analysis, volume 60 of Translations of Mathematical Monographs. American Math. Soc., Providence, Rhode Island, 1984. [10] A. A. Markov. On the continuity of constructive functions. Uspekhi Mat. Nauk, 9(3 (61)):226{230, 1954. (Russian). [11] Y. N. Moschovakis. Recursive metric spaces. Fund. Math., 15:215{238, 1964. [12] J. Myhill and J. C. Shepherdson. E ective operations on partial recursive functions. Zeitschrift f. math. Logik und Grundlagen d. Math., 1:310{317, 1955. [13] M. B. Pour-El and J. I. Richards. Computability in Analysis and Physics. Springer{Verlag, Berlin, Heidelberg, 1989. [14] H. Rogers, Jr. Theory of Recursive Functions and E ective Computability. McGraw{Hill Book Company, New York, 1967. [15] N. A. Shanin. Constructive real numbers and constructive function spaces, volume 21 of Translations of Math. Monographs. Amer. Math. Soc., Providence, R. I., 1968. [16] A. O. Slisenko. On constructive nonseparable spaces. Amer. Math. Soc Transl., 2(100):195{ 199, 1972. (Trudy Mat. Inst. Steklov. 72 (1964), 533{536, Russian). [17] D. Spreen. On e ective topological spaces. Technical Report 94{01, Universitat Siegen, 1994. 41 pages. [18] D. Spreen. Representations versus numberings: On two computability notions. Technical Report 96{04, Universitat Siegen, 1996. 11
[19] D. Spreen and P. Young. E ective operators in a topological setting. In M. M. Richter et al., editors, Computation and Proof Theory, Proc. Logic Colloquium Aachen 1983, Part II, number 1104 in Lect. Notes Math., pages 437{451, Berlin, 1984. Springer{Verlag. [20] G. S. Tseitin. Algorithmic operators in constructive complete separable metric spaces. Dokl. Akad. Nauk. SSSR, 128:49{52, 1959. (Russian). [21] G. S. Tseitin. Algorithmic operators in constructive metric spaces. Amer. Math. Soc. Transl., 2(64):1{80, 1967. (Trudy Mat. Inst. Steklov. 67 (1962), 259{361, Russian). [22] K. Weihrauch. Computability, Springer{Verlag, Berlin, 1987. [23] K. Weihrauch. Computability on computable metric spaces. Theoretical Computer Science, 113:191{210, 1993.
12
Abstract
Based on the Turing machine model there are essentially two di erent notions of computable functions over the real numbers. The e ective functions are de ned only on computable real numbers and are Type 1 computable with respect to a numbering of the computable real numbers. The e ectively continuous functions may be de ned on arbitrary real nunbers. They are exactly those functions which are Type 2 computable with respect to an appropriate representation of the real numbers. We characterize the Type 2 computable functions on computable real numbers as exactly those Type 1 computable functions which satisfy a certain additional condition concerning their domain of de nition. Our result is a sharp strengthening of the well-known continuity result of Tseitin and Moschovakis for e ective functions. The result is presented for arbitrary computable metric spaces.
1 Introduction In this paper we compare two approaches for de ning computability of functions between computable metric spaces. The rst approach can be described as e ectivity or computability with respect to numberings, the second as e ective continuity or computability with respect to representations. We shall give special attention to the space of real numbers. In the rst approach one considers countable metric spaces M , M 0 with numberings. One of the most important examples is the space IRc of computable real numbers. A numbering of this space is derived from a numbering of the algorithms or machines that compute such a number. A function f : M ! M 0 1 is called Type 1 computable or e ective if there is partial computable function g 2 P (1) which transforms any index of an element x 2 dom f into an index of the result f (x) 2 M 0 . This approach, with some variations, has been considered in the Russian school of constructive analysis, represented by Markov [10], Shanin [15], Tseitin [21], Kushner [9], and others. It has also been considered by Moschovakis [11], Aberth [1] and by many others. Closely related is the constructive approach to analyis presented by Bishop and Bridges [3]. In the second approach one considers metric spaces M , M 0 with a numbering of a dense subset, respectively. Consider for example the space IR of real numbers with a standard numbering of the rational numbers. A representation of M (respectively of M 0 ) is de ned via the Email: [email protected] 1
We use the notation f : X ! Y for a (partial) function f with domain dom f X .
1
numbering of the dense subset. One can represent an element x by sequences p in the Baire space IB = ININ = fp : IN ! INg which describe normed Cauchy sequences consisting of elements in the enumerated dense subset with limit x. A function f : M ! M 0 is Type 2 computable if there is a partial recursive operator in the sense of Rogers [14] on the Baire space IB that transforms any name p 2 IB for an element x 2 dom f into a name for f (x). This is equivalent to e ective continuity in the following sense: there is an algorithm which, given a desired output precision 2?n , needs only a nite approximation of an element x 2 dom f in order to compute an approximation of the result f (x) up to precision 2?n , and which is able to compute a suf cient input precision. For real functions this leads to the notion of computability considered by Grzegorczyk [4], Klaua [6], Pour-El and Richards [13], Weihrauch [22], Ko [7], and by many others. How are the two computability notions related? From the normed Cauchy representation of a metric space M (with a numbering of a dense subset) one can derive a numbering of the subspace Mc M of the computable elements: a number i is an index for a computable element x if it is an index of a name p 2 R(1) = fp : IN ! IN j p recursiveg for a recursive normed Cauchy sequence with limit x. In the following we consider Type 2 computability with respect to normed Cauchy representations and Type 1 computability with respect to the derived numberings of the computable elements. We assume that the metric spaces M and M 0 are computable and recursively separable; see Section 3.1. It is easy to see that the restriction to Mc of a Type 2 computable function f : M ! M 0 is Type 1 computable. Note that this implies f (Mc) M 0 c . Hence, for a function f : Mc ! M 0 Type 1 computability is a necessary condition for Type 2 computability. A simple example of Myhill (see Remark 2 in [8]) shows that it is not a sucient condition. But Weihrauch's [22] version of the well-known Tseitin/Moschovakis continuity result shows that a Type 1 computable function with a recursively separable domain must be Type 2 computable. Yet, in general, recursive separability of the domain is a too strong additional condition: there is a Type 2 computable function f : Mc ! M 0 which cannot be extended to a Type 1 computable function with a recursively separable domain. It is the aim of this paper to give a sharp condition. We show that a function f : Mc ! M 0 is Type 2 computable if and only if it is Type 1 computable and satis es a certain additional condition which in general is weaker than recursive separability of the domain (to be precise: it is weaker than the possibility to extend the function to a Type 1 computable function with a recursively separable domain). This result is a sharp strengthening of the continuity result of Tseitin [20, 21] and Moschovakis [11]. It seems to be new even for functions between the space R(1) of total computable functions or the space IRc of computable real numbers. Note that a special version of the continuity result had been obtained by Kreisel, Lacombe, and Shoen eld [8]. There is also an analogue for a complete partial order, c.f. Myhill and Sheperdson [12]. A combined generalization has been obtained by Spreen and Young [19] and by Spreen [17]. Another generalization for complete partial orders, which also implies the result of Tseitin and Moschovakis for metric spaces, has been obtained by Berger [2]. But in all these papers recursive separability of the domain of a function has been assumed in order to deduce the (global) e ective continuity (compare Remark 4.3). In the papers [19, 17, 18] Spreen also investigates the relation between the two computability notions. He considers more general spaces and gives a condition which in combination with e ectivity and recursive separability of the domain implies e ective continuity. But in the general case the problem to give a sharp additional condition, i.e. to give a characterization of e ective continuity as above, seems to be still open. For a discussion of the two approaches for 2
de ning computability the reader is referred to Kushner [9], pp. 304 { 306, and to Weihrauch [22], Ch. 3.6. First we formulate a corollary of the main result for the special case of operators on the space of total recursive functions, which explains already the kind of characterization. In the following section the notions concerning computable metric spaces, Type 1 computable functions (e ective functions), and Type 2 computable functions (e ectively continuous functions) are introduced. Then we discuss the classical results more precisely. In the last section the new condition is introduced and the main results are stated. Actually we give two conditions, which are equivalent in all interesting cases. Due to lack of space the proofs are omitted.
Notations
IN = f0; 1; 2; : : : g is the set of nonnegative integers, Ql is the space of rational numbers, IR is the space of real numbers, IB = fp : IN ! INg is the Baire space (see Example 3.8), h:; :i : IN2 ! IN is a standard pairing function, and the maps i : IN ! IN for 1 i 2 are the corresponding inverse maps with j hn1 ; n2 i = nj . By P (1) := fp : IN ! IN j p computableg we denote the set of all (partial) computable number functions, by R(1) := fp : IN ! IN j p computableg the space of total recursive functions. A numbering of a denumerable set S is a (partial) surjective function : IN ! S . For i 2 dom we denote the corresponding element in S either by i or by (i). A numbering is reducible to a numbering 0 (written 0 ) if there is a g 2 P (1) such that (i) = 0 g(i), for all i 2 dom . If 0 and 0 then the numberings , 0 are equivalent (written 0 ). The map Ql : IN ! Ql is the standard total numbering of Ql de ned by Ql hi; j; ki = (i ? j )=(k + 1). ' : IN ! P (1) is a total standard numbering of P (1) and Wi := dom 'i for all i 2 IN. Hence, W is a total numbering of all recursively enumerable (r.e.) sets.
2 Operators on the Space of Total Recursive Functions In order to give an impression of our main result we formulate a corollary for the important special case of an operator mapping a subset of the total recursive functions R(1) to R(1) . The restriction '0 := 'j'? (R ) of ' is a partial numbering of R(1) . A function f : R(1) ! R(1) is called ('0 ; '0 ){computable or e ective if there is a function g 2 P (1) such that for all i 2 '?1 (dom f ) we have i 2 dom g, 'g(i) 2 R(1) , and f ('i ) = 'g(i) . The function f is called e ectively continuous if there is a (partial) recursive operator F in the sense of Rogers [14] with F jdom f = f ; compare Section 3.3. Kreisel, Lacombe and Shoen eld [8] proved that an e ective function f : R(1) ! R(1) with a recursively separable domain is e ectively continuous (see Theorem 1 in [8] and the remark preceding Theorem 3 in [8]). The domain dom f is recursively separable if there is an r.e. set E IN with 1
(1)
(1) E '?1 (dom f ); (2) dom f '(E ) : If g 2 P (1) is a function which \computes" f , i.e. f ('i ) = 'g(i) for all i 2 dom f , then dom f and hence also '(E ) must be contained in the set
X (g) := fp 2 R(1) j (8i; j 2 '?1 fpg) g(i) and g(j ) are de ned and 'g(i) = 'g(j) 2 R(1) g 3
of elements p 2 R(1) at which g \de nes a value in R(1) ". Now the main point is that it is not necessary that g really \de nes a value in R(1) " at the elements in '(E ). Instead it is sucient to demand that for each n 2 IN one can nd an r.e. set of elements whose closure contains dom f and at which g \de nes a value up to precision n". Therefore we de ne the set Xn(g) := fp 2 R(1) j (8i; j 2 '?1 fpg) g(i) and g(j ) are de ned and (8k n) 'g(i) (k) = 'g(j ) (k) are de nedg of elements p at which g de nes a value up to precision n. We call a function f : R(1) ! R(1) D'-e ective if there is a function g 2 P (1) with f ('i ) = 'g(i) for all i 2 '?1 (dom f ) and if additionally there is a function e 2 R(1) such that for all n 2 IN (1) We(n) '?1 (Xn (g)); (2) dom f '(We(n) ) : Obviously, if a function f : R(1) ! R(1) is e ective and has a recursively separable domain, then it is D'-e ective.
Theorem 2.1 For a function f : R(1) ! R(1) the following two conditions are equivalent: 1. f is D'-e ective, 2. f is e ectively continuous.
3 Notions In this section the basic notions are introduced and simple statements connecting them are deduced. First we consider computable metric spaces. Then e ective functions are de ned, i.e. functions that are (Type 1) computable with respect to numberings. Finally we introduce e ectively continuous functions and functions that are (Type 2) computable with respect to normed Cauchy representations.
3.1 Computable Metric Spaces
The various approaches for introducing e ectivity on a metric space M can be summarized by saying that one imposes e ectivity conditions on the metric with respect to a numbering of a dense subset. Hence e ectivity for metric spaces has been introduced only for separable metric spaces. Under a metric space we always understand a triple (M; d; ) where M is a nonempty set, d : M M ! IR is a metric, and : IN ! M is a numbering of a dense subset of M . Sometimes we write only (M; ), leaving aside d.
De nition 3.1 A metric space (M; d; ) (or just (M; )) is called semicomputable if there is an r.e. set D< IN such that for all i; j 2 dom and all n 2 IN hi; j; ni 2 D< () d( i ; j ) < Ql (n) : It is called a computable metric space if furthermore there is an r.e. set D> IN such that for all i; j 2 dom and all n 2 IN hi; j; ni 2 D> () d( i ; j ) > Ql (n) : 4
Computationally interesting spaces are usually not just separable but recursively separable.
De nition 3.2 A subset X of a metric space (M; ) is called -recursively separable if there is an r.e. set E ?1 (X ) such that X (E ). Often we simply write \recursively separable" instead of \ -recursively separable" if it is clear which numbering is meant. Note that in general a subspace of a recursively separable metric space is of course separable, but it does not have to be recursively separable; compare Slisenko [16]. The e ectivity properties of a metric space depend only on the equivalence class of the numbering of a dense subset, as the following easily proved lemma shows.
Lemma 3.3 Let (M; ) be a metric space, and let 0 be a numbering of a dense subset of M with 0 . If (M; ) is semicomputable (computable), then (M; 0 ) is semicomputable (computable). If (M; 0 ) is recursively separable, then (M; ) is recursively separable.
We remark that, if a metric space (M; ) is recursively separable, then one can easily construct a total numbering 0 of a dense subset of M with 0 . We wish to consider computations over the real numbers that are based on approximations. The idea is to describe the real numbers by Cauchy sequences of rational numbers that converge recursively. Among these it is sucient to consider the Cauchy sequences with a normed rate of convergence. This can be done also for arbitrary metric spaces with a numbering of a dense subset. A normed Cauchy sequence is a sequence of points (xi )i2IN in a metric space M with d(xi ; xj ) < 2? minfi;jg . Using normed Cauchy sequences and a numbering of a dense subset one can de ne a natural representation of the metric space, de ne its computable elements and derive a numbering of its computable elements.
De nition 3.4 Let (M; ) be a metric space. The Cauchy representation : IB ! M of (M; ) is de ned by
(p) = x i ( p(n) )n2IN is a normed Cauchy sequence with limit x 2 M : By (i) := 'i for i 2 dom := fi 2 IN j 'i 2 dom g we de ne a partial numbering of the set Mc := range of the computable elements of M . The normed Cauchy representations are special cases of the admissible representations of
T0 -spaces with countable bases considered in the Type 2 Theory of E ectivity; see Weihrauch
[22]. In Section 3.3 we will de ne the computability of functions with respect to representations. Here we are interested in the numbering .
Lemma 3.5 If (M; ) is a semicomputable (computable, recursively separable) metric space
then also (M; ) and (Mc ; ) are semicomputable (computable, recursively separable) metric spaces.
Furthermore one easily checks for any metric space (M; ). A metric space (M; ) is called weakly complete if the converse relation is true also, i.e. given an index of a recursive normed Cauchy sequence converging to an enumerated point, one can compute an index of the point:
j ? (range ) : 1
5
The most important special property of the derived numbering is that the spaces (M; ) and (Mc ; ) are weakly complete. The expression \weakly complete" is taken from Kushner [9]. For computable metric spaces with a surjective numbering the condition above is equivalent to the Condition (A) introduced by Moschovakis [11]. Example 3.6 One of the simplest examples of a computable, recursively separable, and weakly complete metric space is the discrete space IN with the trivial total numbering idIN (and the distance d(n; m) = 1 for n 6= m). Example 3.7 The space (IR; Ql ) is a computable recursively separable metric space, but it is not weakly complete. The space (IRc ; ) is the standard space of the computable real numbers. It is computable, recursively separable, and weakly complete. Example 3.8 Finally let : IN ! IN be a total bijective standard numbering of the nite words over IN, With (i) := (i)0! the space (IB; ) is a computable recursively separable metric space. We use the metric d : IB IB ! IR de ned by ( ? minfi j p(i)6=q(i)g if p 6= q d(p; q) := 02 else for any p = p(0)p(1)p(2) : : : 2 IB, q = q(0)q(1)q(2) : : : 2 IB. Because of range = R(1) , the computable elements of (IB; ) are just the total recursive functions. The restriction '0 := 'j'? (R ) is also a partial numbering of R(1) . One easily checks '0 . The spaces (R(1) ; ) and (R(1) ; '0 ) are computable, recursively separable, and weakly complete metric spaces. Q l
1
(1)
3.2 E ective Functions
In this section (M; d; ) and (M 0 ; d0 ; ) are two xed metric spaces. We introduce e ective or Type 1 computable functions between M and M 0 . De nition 3.9 A function f : M ! M 0 is called ( ; ){computable if dom f range and there is a g 2 P (1) with (8i 2 ?1 (dom f )) f ( i ) = g(i) : The rst property to be mentioned is that the computability of a function depends only on the equivalence class of the numbering. This follows immediately from the following obvious lemma. Lemma 3.10 Let ~ : IN ! M and ~ : IN ! M 0 be additional numberings of dense subsets with ~ and ~. If f : M ! M 0 is ( ; ){computable, then f is (~ ; ~){computable. We are mainly interested in ( ; ){computable functions. If it is clear which numberings and are meant then we will call the ( ; ){computable functions e ective functions. Remark 3.11 The above notion of computability with respect to numberings is the notion which is used e.g. in the early paper [10] by Markov, by Helm [5], by Weihrauch [22] and by many other authors. In constructive analysis (see e.g. Kushner [9]) a more restrictive de nition of an e ective function f : M ! M 0 between enumerated sets M and M 0 with numberings : IN ! M and : IN ! M 0 (i.e. and are surjective) is often used. One demands that there is a partial recursive function g 2 P (1) with f ( i ) = g(i) for all i 2 ?1 (dom f ) and additionally with ?1 (dom f ) = dom \ dom g. If a function f can be computed in this way, Weihrauch [22] calls it strongly ( ; ){computable. For example Tseitin [21] and Moschovakis [11] have formulated their continuity results for e ective operators in this stronger sense. 6
3.3 Computable and E ectively Continuous Functions
Let (M; ) and (M 0 ; ) be metric spaces. We de ne Type 2 computable functions and two forms of e ective continuity for functions f : M ! M 0 . In order to de ne e ective continuity we use a numbering B of a base of (M; ) de ned by
Bi := B ( (i) ; 2? (i) ) = fx 2 M j d( (i) ; x) < 2? (i) g 1
2
2
1
for i 2 dom B := fi j 1 (i) 2 dom g. The corresponding numbering of a base of (M 0 ; ) will be denoted by B 0 in this section. The rst of the following three de nitions of global e ective continuity expresses that the preimage of a basic open set is open in an e ective way. The second is based on "{ continuity. The third expresses the computability used in Type 2 Theory of E ectivity, c.f. Weihrauch [22].
De nition 3.12 A function f : M ! M 0 is called 1. weakly e ectively continuous if there is a function g 2 P (1) with
[
(8j 2 dom B 0 )(Wg(j ) dom B and f ?1(Bj0 ) = dom f \ fBi j i 2 Wg(j ) g); 2. e ectively continuous if there is an r.e. set V IN such that (8hi; j i 2 V )(i 2 dom B ^ j 2 dom B 0 ^ f (Bi ) Bj0 ) and (8x 2 dom f )(8" > 0)(9hi; j i 2 V )(x 2 Bi ^ 2? (j ) < ") ; 2
3. ( ; )-computable if there is an e ectively continuous function F : IB ! IB such that for all p 2 ? 1 (dom f ) f (p) = F (p) :
X of M is called a Lacombe set if there is an r.e. set A dom B such that X = SfBA jsubset i 2 A g . Hence a function is weakly e ectively continuous if the preimage of any ball Bj0 is i a Lacombe set and given an index j of such a ball one can compute an index of this Lacombe set. The rst condition in Part (2) (e ective continuity) of the de nition states that V enumerates only pairs of balls Bi , Bj0 such that Bi is mapped to Bj0 . By the second condition suciently many such pairs are enumerated. Finally note that for a function F : IB ! IB the following conditions are equivalent: (1) F is weakly e ectively continuous, (2) F is e ectively continuous, (3) F can be extended to a partial recursive operator G : fp : IN ! INg ! fp : IN ! INg in the sense of Rogers [14], x9.8, (4) F can be extended to a total recursive operator G in the
same sense. The metric spaces considered in Type 2 Theory of E ectivity (Weihrauch [22, 23]) are usually semicomputable metric spaces with a total numbering of a dense subset, hence semicomputable, recursively separable metric spaces. A more e ective version of the next proposition has been given by Weihrauch [23].
Proposition 3.13 If (M; ) and (M 0 ; ) are semicomputable and recursively separable, then for a function f : M ! M 0 the conditions (1), (2), and (3) in De nition 3.12 are equivalent.
7
4 On the Classical Results We discuss shortly the known results on the relation between Type 1 computability and Type 2 computability for functions between computable metric spaces. The following proposition is well-known and easy to prove.
Proposition 4.1 Let (M; ) be a semicomputable metric space and (M 0 ; ) be a metric space. If f : M ! M 0 is e ectively continuous then f jMc is ( ; ){computable. On the other hand Remark 2 in [8] contains a simple example, due to Myhill, of an e ective function f : R(1) ! IN which is discontinuous and, hence, cannot be Type 2 computable. Thus, e ectivity alone does not imply continuity. But the famous result of Tseitin [20, 21] implies that a strongly ( ; )-computable function f : Mc ! Mc0 is e ectively continuous if (M; ) is computable and recursively separable and (M; ) is computable. In this case the domain of f must itself be recursively separable; compare Kushner [9], Ch. 9. x2, Lemma 5. In fact one can prove the following result; see Weihrauch [22].
Proposition 4.2 Let (M; ) be a semicomputable metric space and (M 0 ; ) be a computable, recursively separable metric space. If a function f : Mc ! M 0 is ( ; ){computable and has
a -recursively separable domain, then it is ( ; )-computable.
Actually Tseitin formulated his result not for the derived numberings , , but more generally for weakly complete numberings. Moschovakis gave a similar but slightly weaker formulation ([11], Theorem 12)
Remark 4.3 Moschovakis also gave another continuity result in [11], Theorem 3. Using slightly weaker assumptions he derived e ective \pointwise" continuity of e ective functions. Helm [5] has given an example of an e ective and e ectively pointwise continuous function which is not e ectively continuous.
The question arises whether the last proposition is sharp, i.e. whether it covers all e ectively continuous functions. The answer is Yes if the image space is discrete.
Theorem 4.4 Let (M; ) be a computable, recursively separable metric space and (M 0 ; ) be a discrete, computable, recursively separable metric space. Then for a function f : Mc ! M 0 the following three conditions are equivalent. 1. f can be extended to a strongly ( ; )-computable function. 2. f can be extended to a ( ; )-computable function with a -recursively separable domain. 3. f is e ectively continuous.
But in the general case the answer is No. There is even a counterexample with a discrete domain.
Theorem 4.5 There is an e ectively continuous function f : IN ! R(1) which cannot be extended to an e ective function g : IN ! R(1) with a recursively separable domain. This poses the problem to characterize the relation between e ectivity and e ective continuity precisely, which is the aim of the paper. 8
5 Characterization of Type 2 Computability by Type 1 Computability and an Additional Condition
In this section (M; d; ) and (M 0 ; d0 ; ) are two xed metric spaces. We have seen that an e ective function with a recursively separable domain is e ectively continuous. We shall show that a weaker condition than recursive separability suces. In fact we give two conditions, which turn out to be equivalent under suitable assumptions. Let us assume that a function f : M ! M 0 is ( ; ){computable via a function g 2 P (1) , i.e. dom f range and f ( (i)) = (g(i)) for all i 2 ?1 (dom f ). Then dom f is recursively separable if and only if there is an r.e. set E IN such that (1) E ?1 (dom f ); (2) dom f (E ): The domain dom f must be contained in the set
CX (g) := fx 2 M j (8i; j 2 ?1 fxg) ( ('g(i) (n)))n2IN and ( ('g(j) (n)))n2IN are normed Cauchy sequences with the same limitg
of points in which g \de nes a value" with respect to and . For any n 2 IN we de ne the set CXn (g) := fx 2 M j (8i; j 2 ?1 fxg) 'g(i) (n) 2 dom ; 'g(j) (n) 2 dom ; d( ('g(i) (n)); ('g(j) (n))) < 3 2?n g of points in which g de nes a value up to precision n. A slightly stronger version is given by the sequence of sets
SCXn(g) := fx 2 M j (8i; j 2 ?1 fxg) (8k n) 'g(i) (k) 2 dom ; 'g(j) (k) 2 dom ; d( ('g(i) (k)); ('g(j) (k))) < 3 2?k g of points in which g strongly de nes a value up to precision n. Obviously CX (g) SCXn (g) CXn (g) for all n 2 IN. De nition 5.1 1. A function f : M ! M 0 is called D-( ; ){computable or D-e ective if it is ( ; ){computable via a function g 2 P (1) and additionally there is a function e 2 R(1) such that for all n 2 IN we have (1) We(n) ?1 (CXn (g)); (2) dom f (We(n) ): 2. A function f : M ! M 0 is called SD-( ; ){computable or strongly D-e ective if it is ( ; ){computable via a function g 2 P (1) and additionally there is a function e 2 R(1) such that for all n 2 IN we have (1) We(n) ?1 (SCXn (g)); (2) dom f (We(n) ): Obviously a strongly D-e ective function is D-e ective. We shall see that under suitable conditions on the spaces M and M 0 a D-e ective function is e ectively continuous and, vice versa, an e ectively continuous function is even strongly D-e ective. 9
Proposition 5.2 Let f : M ! M 0 be ( ; ){computable via a function g 2 P (1) and let dom f be recursively separable. Then there is a function e 2 R(1) such that f is SD-( ; ){ computable via g and e.
Proof. Let E IN be an r.e. set such that E ?1(dom f ) and dom f (E ). The statement is proved by a function e 2 R(1) with We(n) = E for all n 2 IN. 2 By the last proposition and by Theorem 4.5 the following result is a strengthening of the Tseitin/Moschovakis result (for a recursively separable image space).
Theorem 5.3 Let (M; ) be a semicomputable metric space, and let (M 0; ) be a computable recursively separable metric space. If f : Mc ! M 0 is D-( ; ){computable then it is ( ; ){ computable.
We have copied from Weihrauch [22] the observation that the domain does not need to be computable but only semicomputable. The theorem is a sharp result since the converse is also true.
Theorem 5.4 Let (M; ) be a computable recursively separable metric space and let (M 0 ; ) be a semicomputable metric space. If f : M ! M 0 is e ectively continuous then f jMc is SD-( ; ){computable.
Note that here we do not need that the image space (M 0 ; ) is computable. Instead semicomputability suces. Summarizing Theorem 5.3 and Theorem 5.4 we obtain our main result.
Theorem 5.5 Let (M; ) and (M 0 ; ) be computable recursively separable metric spaces. Then for a function f : Mc ! M 0 the following conditions are equivalent. 1. f is D-( ; ){computable. 2. f is SD-( ; ){computable. 3. f is weakly e ectively continuous. 4. f is e ectively continuous. 5. f is ( ; ){computable.
Proof. The equivalences (3) () (4) () (5) have been stated already in Proposition 3.13. (4) ) (2) is the content of Theorem 5.4, (2) ) (1) is trivial, and (1) ) (5) is the content of
2
Theorem 5.3.
Theorem 5.5 describes precisely the di erence between the Type 1 computability notion for real functions based on the numbering of the computable real numbers and the Type 2 computability notion based on representations like the Cauchy representation ; see the introduction. Q l
Q l
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References [1] O. Aberth. Computable analysis. McGraw{Hill, New York, 1980. [2] U. Berger. Total sets and objects in domain theory. Annals of Pure and Applied Logic, 60:91{117, 1993. [3] E. Bishop and D. S. Bridges. Constructive Analysis. Springer-Verlag, Berlin, Heidelberg, 1985. [4] A. Grzegorczyk. On the de nitions of computable real functions. Fund. Math., 44:61{71, 1957. [5] J. P. Helm. On e ectively computable operators. Zeitschrift f. math. Logik und Grundlagen d. Math., 17:231{244, 1971. [6] D. Klaua. Konstruktive Analysis. VEB Deutscher Verlag der Wiss., Berlin, 1961. [7] K.-I Ko. Complexity Theory of Real Functions. Birkhauser, Boston, 1991. [8] G. Kreisel, D. Lacombe, and J. R. Shoen eld. Partial recursive functionals and e ective operations. In A. Heyting, editor, Constructivity in Mathematics, Proceedings of the Colloquium held at Amsterdam 1957, pages 290{297, Amsterdam, 1959. North{Holland Publishing Company. [9] B. A. Kushner. Lectures on Constructive Mathematical Analysis, volume 60 of Translations of Mathematical Monographs. American Math. Soc., Providence, Rhode Island, 1984. [10] A. A. Markov. On the continuity of constructive functions. Uspekhi Mat. Nauk, 9(3 (61)):226{230, 1954. (Russian). [11] Y. N. Moschovakis. Recursive metric spaces. Fund. Math., 15:215{238, 1964. [12] J. Myhill and J. C. Shepherdson. E ective operations on partial recursive functions. Zeitschrift f. math. Logik und Grundlagen d. Math., 1:310{317, 1955. [13] M. B. Pour-El and J. I. Richards. Computability in Analysis and Physics. Springer{Verlag, Berlin, Heidelberg, 1989. [14] H. Rogers, Jr. Theory of Recursive Functions and E ective Computability. McGraw{Hill Book Company, New York, 1967. [15] N. A. Shanin. Constructive real numbers and constructive function spaces, volume 21 of Translations of Math. Monographs. Amer. Math. Soc., Providence, R. I., 1968. [16] A. O. Slisenko. On constructive nonseparable spaces. Amer. Math. Soc Transl., 2(100):195{ 199, 1972. (Trudy Mat. Inst. Steklov. 72 (1964), 533{536, Russian). [17] D. Spreen. On e ective topological spaces. Technical Report 94{01, Universitat Siegen, 1994. 41 pages. [18] D. Spreen. Representations versus numberings: On two computability notions. Technical Report 96{04, Universitat Siegen, 1996. 11
[19] D. Spreen and P. Young. E ective operators in a topological setting. In M. M. Richter et al., editors, Computation and Proof Theory, Proc. Logic Colloquium Aachen 1983, Part II, number 1104 in Lect. Notes Math., pages 437{451, Berlin, 1984. Springer{Verlag. [20] G. S. Tseitin. Algorithmic operators in constructive complete separable metric spaces. Dokl. Akad. Nauk. SSSR, 128:49{52, 1959. (Russian). [21] G. S. Tseitin. Algorithmic operators in constructive metric spaces. Amer. Math. Soc. Transl., 2(64):1{80, 1967. (Trudy Mat. Inst. Steklov. 67 (1962), 259{361, Russian). [22] K. Weihrauch. Computability, Springer{Verlag, Berlin, 1987. [23] K. Weihrauch. Computability on computable metric spaces. Theoretical Computer Science, 113:191{210, 1993.
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