Computability on the Probability Measures on the ... - Semantic Scholar
Computability on the Probability Measures on the Borel Sets of the Unit Interval Klaus Weihrauch FernUniversitat Hagen D { 58084 Hagen e{mail: [email protected] September 11, 1996
Abstract
Scientists apply digital computers to perform computations on natural numbers, nite strings, real numbers and more general objects like sets, functions and measures. While computability theory on many countable sets is well established and for computability on the real numbers several (unfortunately mutually non-equivalent) de nitions are being studied, in particular for measures no computability concept at all has been available until now. In this contribution we introduce a natural computability concept on the set M of probability measures on the Borel subsets of the unit interval [0; 1]. As background we consider TTE, Type 2 Theory of E ectivity, [KW84, KW85], where computability is de ned on nite and in nite sequences of symbols explicitly by Turing machines and on other sets by means of notations and representations. A standard representation m : ! ?! M is introduced via some natural information structure [Wei95a] (M; ; ), where is a subbase of some T0 -topology m and : ?! is a standard notation of . While several modi cations of m suggesting itself at rst glance have undesirable properties, m itself has several very natural properties and hence should induce an important computability theory. Many interesting functions on measures turn out to be computable, in particular linear combination, integration of continuous functions and any transformation de ned by a computable iterated function system with probabilities. Some other representations of M are introduced, among them a Cauchy representation associated with the Hutchinson metric, and proved to be equivalent to m . In particular, the nal topology m of m is the well known weak topology on M.
1 Introduction Measure and integration is a central branch of mathematics pervading almost all parts of abstract analysis. Several authors have already considered questions of e ectivity, constructivity, computability or computational complexity in measure or integration theory. Kushner [Kus85] studies computability and Ko [Ko91] computational complexity of integration. Bishop and Bridges [BB85] present constructive measure theory extensively. Although they do not consider computability, certainly many of their concepts and results have computational counterparts. Edalat gives a domain theoretic approach 1
to e ective integration [Eda95]. He also does not consider computability, but it should be possible to extend his topological approach by computability concepts. Traub et al. [TWW88] investigate the computational complexity of numerical algorithms for integration in the real number model of computation. However, a systematic study of computability in integration and measure theory does not yet exist. In this paper we extend TTE, Type 2 Theory of E ectivity, to measure theory. TTE has been introduced by Kreitz and Weihrauch [KW84, KW85] as a general framework for studying e ectivity, i.e. continuity, computability and computational complexity, in Analysis. For details the reader is referred to the introduction [Wei95b] and a recent short survey [Wei95a] containing most of the notations we shall use in this paper. More details can be found in [KW85, Wei87]. Since this paper is a rst attempt, we consider only the space of probability measures on the Borel subsets of the real unit interval. By f : A ?! B we denote a partial function, i.e. a function from a subset of A to B . Throughout this paper let be a suciently large nite alphabet. Let be the set of nite and ! = fp j p : ! ?! g the set of !{words over . On we consider the discrete topology and on ! the cantor topology de ned by the basis fw! j w 2 g. For Y0 ; Y1 ; : : : ; Yk 2 f ; ! g, a function f : Y1 : : : Yk ?! Y0 is called computable, i it is computed by a Turing machine with a one{way output tape. Every computable function is continuous. The basic idea of TTE is to use nite or in nite sequences as names of \abstract" objects. As naming systems we consider notations, i.e. surjections : ?! S , and representations, i.e. surjections : ! ?! M . Continuity and computability concepts are transferred from and ! via notations and representations, respectively, to the named sets straightforwardly, see [KW85, Wei87, Wei95b, Wei95a]. Mainly notations or representations which are compatible with some relevant structure on the set under consideration are of practical interest. We do not discuss this for notations (see [RW80, Wei87] and Appendix C in [Wei95b]), but we will introduce \e ective" notations explicitly whenever necessary. In particular, for the rational numbers let Q : ?! be the standard representation via fractions of integers in binary notation. We shall abbreviate Q(w) by w. Standard notations of the natural numbers, pairs of rational numbers etc. will be used without further de nitions. Q I
For uncountable sets M we shall consider mainly representations derived from \information structures" (M; ; ), where is a countable subset of 2M of \atomic properties" which identi es points, and is a notation of [Wei95a]. It is assumed that a computer (Turing machine) manipulates {names of atomic properties. As a name of an object x 2 M we consider any in nite list of all properties A 2 which hold for x. Concretely, the standard representation : ! ?! M is de ned by (p) = x () x = w0 ]w1 ] : : : and fwi j i 2 !g = fw j x 2 (w)g: Every nite pre x of a {name p of x contains nitely many atomic properties of x which \approximate" x. Mathematically, this kind of approximation is described by the topology on M , which has as a subbase. Computability on and via on M are xed by the notation which expresses how atomic properties can be handled concretely. Thus, for any information structure (M; ; ), characterizes approximation and computability on M . The topology and the standard representation are closely related: X 2 () ?1 X is open in dom( ) (for all X M ), i.e. is the nal topology of . By the \main theorem for admissible representations" [KW85] a function is continuous relative to standard representations, i it is continuous w.r.t. the associated nal topologies in the usual sense. For more details see [KW85, Wei87, Wei95b, Wei95a].
2
For the real numbers, we need three representations < ; > ; : ! ?! , derived from information structures. They can be de ned explicitly as follows [Wei87, Wei95a]: (p) = x : () p = w0 ]w1 ] : : : with fwi j i 2 !g = fw j w > xg; (p) = x : () p = v0 ]w0 ]v1 ]w1 : : : with f(vi ; wi ) j i 2 !g = f(v; w) j v < x < wg: The nal topologies are < = f(y; 1) j y 2 g [ f g, > = f(?1; y) j y 2 g [ f g and the set IR of ordinary open subsets of , respectively. IR
IR
IR
IR
IR
IR
Notice that induces the standard computability theory on the real line. The translatability or reducibility properties [Wei87, Wei95a] < , > , < 6t , > 6t , < 6t > , > 6t < can be proved easily. In Section 2 we introduce a standard representation m of the set M of probability measures on the Borel sets of the interval [0; 1] by a very natural information structure. We prove a stability theorem for this de nition. We discuss some further modi cations of the de nition and show that that they have undesirable properties. The results indicate that the computability theory on M induced by the representation m is indeed very natural. In Section 3 we prove computability of several interesting functions on measures, in particular linear combination and integration of continuous functions. Also the measure transformation induced by a computable iterated function system with probabilities [Bar93] is computable. Finally in Section 4, we introduce representations based on other natural information structures and a Cauchy representation for the Hutchinson metric [Hut81, Bar93] . We prove that all these representations are equivalent and that their nal topology is the well known weak topology [Bau74].
2 The standard representation of measures In this section we introduce the standard representation m of the probability measures and show that it induces a very natural computability theory. Let Int := f(a; b); [0; a); (b; 1]; [0; 1] j a; b 2 ; 0 < a < b < 1g be the set of open subintervals of [0; 1] with rational boundaries, and let I : ?! Int be some standard notation of Int with dom(I ) ( n fcj ; ]g) . We write Iw for I (w). By B we denote the set of Borel subsets of [0; 1], i.e. the smallest {Algebra containing Int. By M we denote the set of probability measures : B ?! on the space ([0; 1]; B). By a basic theorem of measure theory [Bau74], every measure 2 M is de ned uniquely by its values on the generating set Int. We introduce a standard representation of M via an information structure. The informations available from some standard name of a measure shall be all (r; J ) with r 2 and J 2 Int such that r < (J ). Q I
IR
Q I
De nition 2.1
De ne an information structure (M; ; ) by := range ( ) , where
2 (ucj v) : () u < (Iv ) for all u 2 dom(Q ), v 2 dom(I ) and 2 M. Let m be the topology on M with subbase and let m be the standard representation of M derived form .
3
It remains to show that identi es the points of M. Consider measures ; 0 2 M such that r < (J ) () r < 0 (J ) for all r 2 and J 2 Int. Then obviously, (J ) = 0 (J ) for all J 2 Int, i.e. = 0 . Q I
The de nition of the representation m looks somewhat arbitrary. By the next stability lemma, we obtain an equivalent representation, if we replace Q and I by adequate other notations. For any X let cls(X ) be the closure of X . IR
Theorem 2.2 (stability of m ) Let S : ?! S be a notation of a set S which is dense in such that f(u; v) j S (u) < Q(v)g and f(u; v) j Q(u) < S (v)g are r.e. . Let D be a countable dense subset of [0; 1] and let I 0 be a notation of Int0 := f(a; b); [0; a)(a; 1]; [0; 1] j a; b 2 D; 0 < a < b < 1g such that f(u; v) j cls(Iu0 ) Iv g and f(u; v) j cls(Iu ) Iv0 g are r.e. De ne m0 and m0 by IR
substituting S for Q and I 0 for I in De nition 1. Then m0 = m and m0 m .
Proof:
We show that there is a machine which maps any p 2 ! into some q 2 ! such that the following holds: If p is a list of all (u; v) such that Q (u) < Iv for some 2 M, then q is a list of all (x; y) such that S (x) < Iy0 . We have S (x) < Iy0 , i there are words u; v such that S (x) < Q (u) , Q (u) < Iv and cls(Iv ) Iy0 . Since the rst and the third property are r.e. and the input p lists the second one by assumption, there is indeed a machine with the desired property. This shows m m0 . By symmetry we have also m0 m0 , hence m m0 . Since m and m0 are the nal topologies of m and m0 [Wei95a], respectively, and since equivalent representations have the same nal topology, we obtain m = m0 .
2
If we replace, for example, rational numbers by nite binary fractions or by nite decimal fractions in the de nition of the set Int and in De nition 1, we obtain an equivalent representation with the same nal topology. If we replace the relation \ < " in De nition 1 by \ ", \ > " or \ ", we obtain rerpesentations which violate Lemma 3. Remember that by de nition, the topology m has the subbase = fUr;J j r 2 and J 2 Int g where Ur;J = f 2 M j r < (J )g . We prepare the proof of the theorem by two Lemmas. First, we consider the cases \r (J )", \r (J )". Q I
Lemma 2.3 For Q let (Q) be the topology on M generated by the subbase (Q) := fUr;J j r 2 Q; J 2 Intg, where Ur;J = f 2 M j r (J )g. Then (P ) 6 (Q), if t 2 P n Q for some t 2 (0; 1) (for all P; Q ). The statement holds accordingly, if \ " is replaced by \ ". Proof: The set U := f 2 M j t [0; 1)g is in (P ). For x 2 (0; 1) de ne x 2 M by x f0g := x; x f1g := 1 ? x . Then t 2 U . Assume U 2 (Q). Then there are V1 ; : : : ; Vk 2 (Q) with t 2 V1 \ : : : \ Vk U . For i = 1; : : : ; k there are ri 2 Q and Ji 2 Int with Vi = f j ri (Ji )g. If 1 2= Ji , from t 2 Vi we obtain ri t (Ji ), hence ri < t (since t 2= Q). Let r := max(fri j 1 2= Ji g [ f0g). Then r < t. If 0; 1 2= Ji , then ri t (Ji ) = 0 = r (Ji ), if 0 2 Ji and 1 2= Ji then ri r r (Ji ), if 0 2= Ji and 1 2 Ji then ri t (Ji ) = 1 ? t < 1 ? r = r (Ji ), and if Ji = [0; 1] then ri t (Ji ) = 1 = r (Ji ). We obtain r 2 V1 \ : : : \ Vk , but t > r = r [0; 1), hence r 2= U . This is a contradiction. IR
IR
4
The case \ " can be proved accordingly.
2
The next lemma considers the case \r > (J )".
Lemma 2.4 For D (0; 1) let Int(D) := f(a; b); [0; a); (a; 1]; [0; 1] j a; b 2 D; 0 < a < b < 1g. Let (D) be the topology on M generated by the subbase (D) := fUr;J j r 2 ; J 2 Int (D)g where Ur;J = f 2 M j r > (J )g. Then Q I
D E () (D) (E ) (for all D; E (0; 1) ):
Proof: \ =) ": (obvious) \ (= ": Assume D 6 E . Then there is some d 2 D n E . The set U := f 2 M j 1 > [0; d)g is an element of (D). For any x , 0 x 1, de ne a measure x 2 M by x (A) := (1 if x 2 A; 0 otherwise). In particular, we have 1 > 0 = d [0; d), hence d 2 U . We assume U 2 (E ). Then there are V1 ; : : : ; Vk 2 (E ) with d 2 V1 \ : : : \ Vk U . For i = 1; : : : ; k there are ri > 0 and Ji 2 Int (E ) with Vi = f j ri > (Ji )g. De ne e0 := max(fsup Ji j Ji < dg [ finf Ji j d 2 Ji g). Then e0 d, hence e0 < d since d 2= E . De ne e := (e0 + d)=2. For i = 1; : : : ; k we have d 2 Ji () e 2 Ji , hence d(Ji ) = e(Ji ) and d 2 Vi () e 2 Vi , consequently e 2 V1 \ : : : \ Vk . But e[0; d) = 1 since e < d, hence e 2= U . This is a contradiction. 2
Theorem 2.5 If in De nition 2.1 the relation \u < (Iv )" is replaced by \u (Iv )"; \u (Iv )" or \u < (Iv )", the resulting representations m violate the stability theorem (Theorem 2).
Proof:
By Lemma 3, in the cases \ " and \ " a change of the set Q of bounds changes the topology m , by lemma 4, in the case \ > " a change of the set D of interval boundaries changes the topology m . Since the equivalent representations have the same nal topology, each of the modi cations produces a non{equivalent representation. 2 By De nition 1 and Lemmata 3 and 4, many di erent more or less natural representations and hence computability theories for the set M of probability measures on [0; 1]; B) can be introduced. The \user" has to decide, which of them is adequate for his application. The stable representation m from De nition 1 is certainly the most important one, since its computability theory will occur most frequently. We shall study it in the following exclusively. As a simple consequence of De nition 1, all rational lower bounds of (J ) can be obtained from any m {name of and any I {name of J . This property characterizes the representation m except for equivalence: The representation m is {complete in the set of all representations of M, for which (; J ) 7! (J ) is (; I; < ){computable.
Theorem 2.6 For any representation of M: m () (; J ) 7! (J ) is (; I; < ){computable.
Proof:
Consider m . By de nition there is some computable function f : ! ?! ! with (p) = m f (p) 5
for all p 2 dom(). For p 2 dom(), f (p) is a list of all (u; v) with u < m f (p)(Iv ). There is a Turing machine, which for any inputs p 2 dom() and v 2 dom(I ) determines internally the sequence f (p) and simultaneously writes all words u such that (u; v) is listed by f (p). Therefore, (; J ) 7! (J ) is (; I; < ){computable. Consider, on the other hand, that (; J ) 7! (J ) is (; I; < ){computable. Then there is a machine M1 which for any p 2 dom() and v 2 Int produces a list of all u 2 with u < (p)(Iv ) From this machine another machine M2 can be constructed, which for any p 2 dom() constructs a list of all (u; v) with u < (p)(Iw ). This machine translates into m .
2
Notice, that in particular (; J ) 7! (J ) is (m ; I; < ){computable. Computing only lower rational bounds does not seem to be satisfactory. We would like to compute also arbitrarily close upper bounds of (Iv ). We prove a negative and a positive answer. For any x 2 [0; 1] de ne x 2 M by x(A) := (1 if x 2 A; 0 otherwise). For any good and useful representation of M it should be possible to determine a {name of the measure x e ectively from a name of x. Let M 0 := fx j x 2 [0; 1]g.
Theorem 2.7 For any representation of M, for which x 7! x is (; ){continuous on (0; 1), 7! [0; 1=2) is not (; > ){continuous on M 0 . m is such a representation.
Proof:
Assume, 7! [0; 1=2) is (; > ){continuous on M 0 . Then f : x 7! x [0; 1=2) is (; > ){continuous, hence ( IR ; > ){continuous by the main theorem for admissible representations. Since (?1; 1=2) 2 >, f ?1 (?1; 1=2) 2 IR , but f ?1(?1; 1=2) = [1=2; 1) 2= IR . We prove now that g : x 7! x is (; m ){ continuous, i.e. ( IR ; m ){continuous by the main theorem for admissible representations. Let Ur;J := f 2 M j r < (J )g, r 2 , J 2 Int, be some arbitrary subbase element of m . We have g(x) 2 Ur;J , i r < x (J ). For r 1, g?1 Ur;J = ;, for r < 0, g?1 Ur;J = (0; 1) and for 0 r < 1, g?1 Ur;J = J . Therefore, ?1 g Ur;J is IR {open in dom(g) = (0; 1). This shows that g is ( IR ; m ){continuous. Q I
2
Therefore, for reasonable representations of M, in particular for our standard representation m , arbitrarily close rational upper bounds of measures of open intervals cannot be computed. Although this contradicts intuition at rst glance, it has to be accepted as a matter of fact. Notice, that for proving Lemma 2.3, Lemma 2.4 and Theorem 2.7 we have used measures 2 M with fxg > 0 for some x 2 . Since the arguments have been purely topological without reference to computability, we have also shown that the nal topology m of the representation m , which formalizes a concept of \approximation" on the set M of measures, is quite natural. If we exclude measures with fxg > 0 for some x 2 [0; 1], (; J ) 7! (J ) becomes (m ; I; ){computable. Let M0 := f 2 M j 8x 2 [0; 1]:fxg = 0g IR
Theorem 2.8 The function (; J ) 7! (J ) is (m ; I; ){computable for J 2 Int and 2 M0 .
Proof:
By Theorem 2.6, (; J ) 7! (J ) is (m ; I; < ){computable. It suces to show, that this function is (m ; I; > ){computable for 2 M0 . The algorithm branches into four cases depending on the type of the interval J 2 Int. We discuss only the branch for J = (a; b) with 0 < a < b < 1. For 2 M0 we have 1 = [0; a) + (a; b) + (b; 1]. If we can approximate [0; a) and (b; 1] from below, we can approximate (J ) from above. Consider m (p) = 2 M0 and Iw = (a; b) 2 Int, 0 < a < b < 1. 6
Then for any r 2 we have: r > (a; b), i there are rational numbers sl ; sr with sl < [0; a) and sr < (b; 1] such that r > 1 ? sl ? sr . There is a machine, which for input p and w writes the rational number r to the output , i it has found numbers sl and sr with the above properties by reading the input p. 2 Q I
3 Computable Functions on Measures In this section we prove computability of some interesting functions on probability measures. By the next theorem, the linear combination of measures is computable in all variables.
Theorem 3.1
The function (a; ; 0 ) 7! a + (1 ? a)0 is (; m ; m ; m ){computable for 0 a 1
Proof:
Consider m (p) = , m (p0 ) = 0 and (pa ) = a. Then p is a list of all (r; J ) with r < (J ), and p0 is a list of all (r; J ) with r < 0 (J ) (r 2 ; J 2 Int). For r 2 and J 2 Int we have r < a(J )+(1 ? a)0 (J ), i there are r; r0 ; s; s0 2 with r < sr + (1 ? s0 )r0 , r < (J ) and r0 < 0 (J ), and s < a < s0 . Therefore, there is a Turing machine, which for any input (p; p0 ; pa ), (p; p0 2 dom(m ); 0 (pa ) 1) produces some list q 2 dom(m ) of all (r; J ) 2 Int for which there are r; r0 ; s; s0 2 such that the above properties hold or are listed by p, p0 or pa , respectively. Q I
Q I
Q I
Q I
Q I
2
By Theorem 2.6, (; J ) 7! (J ) is (m ; I; < ){computable on M Int. We extend this result to I0R = fU \ [0; 1] j U 2 IRg, the set of all open subsets of [0; 1]. First we need a representation of this topology. For the set IR of open subsets of , the following information structure ( IR ; ; ) and its derived representation o and topology o are natural (see Example 4.2 and Section 6 in [Wei95a]): For any U 2 IR and u; v 2 let U 2 (ucj v) i [u; v ] U . Consequently, o (p) = U i p is a list of all closed intervals with rational boundaries contained in U . We de ne our standard representation of I0R accordingly: o0 (p) = U : () p is a list of all w 2 with cls(Iw ) U (p 2 ! ; U 2 IR0 ). Let L 2 M be the Lebesgue measure on ([0; 1]; B). IR
Theorem 3.2
(1) (; U ) 7! (U ) for 2 M and U 2 IR0 is (m ; o0 ; < ){computable. (2) (; U ) 7! (U ) for = L and U 2 IR0 is not (m ; o0 ; > ){continuous.
Proof:
Consider m (p) = and o0 (q) = U . Then p lists all (r; J ) 2 Int with r < (J ), and q lists all K 2 Int such that cls(K ) U . For s 2 we have s < (U ), i there are intervals J1 ; : : : ; Jk 2 Int with cls(Ji ) \ cls(Jj ) = ; for 1 i < j k and rational numbers r1 ; : : : ; rk such that: s < r1 + : : : + rk , ri < (Ji ) and cls(Ji ) U for i = 1; : : : ; k. Therefore, there is some machine, which from any p 2 dom(m ) and q 2 dom(o0 ) computes some q0 2 ! with ){continuous, hence ( 0 ; > ){continuous, where 0 is the nal topology of o0 de ned by the subbase = fVJ j J 2 Intg, where U 2 VJ i cls(J ) U . Since F [0; 1=2) 2 (?1; 3=4) 2 > , by continuity of F there are intervals J1 ; : : : ; Jk 2 Int such that [0; 1=2) 2 VJ1 : : : VJ F ?1 (?1; 3=4). We obtain [0; 1] 2 VJ for i = 1; : : : ; k, but F [0; 1] = 1 2= (?1; 3=4), a contradiction. Q I
Q I
k
2
7
i
For uniform formulations in the next theorems we need a standard representation ! of the set C [0; 1] of continuous functions f : [0; 1] ?! . We de ne ! and the corresponding nal topology ! by the following information structure (C [0; 1]; ; ): f 2 (ucj vcj w) : () u < f (cls Iv ) < w for all f 2 C [0; 1] and u; v; w 2 . Properties of ! are discussed in [Wei87, Wei95b, Wei95a]. In particular, ! is the compact{open topology on C [0; 1], which is also generated by the metric d(f; g) := maxfjf (x) ? g(x)jj0 x 1g on C [0; 1]. For any measure 2 M and any continuous function f : [0; 1] ?! [0; 1] de ne the measure Tf () by Tf ()(A) := f ?1 (A) for every Borel set A [0; 1] (see [Bau74], page 42). IR
Theorem 3.3
The function (f; ) 7! Tf () for continuous f : [0; 1] ! [0; 1] and 2 M is (! ; m ; m ){ computable.
Proof:
An easy consideration shows that (J; f ) 7! f ?1 (J ) is (I; ! ; o0 ){computable. By Theorem 3.2, (; U ) ?! (U ) is (m ; o0 ; < ){computable. Therefore, (; f; J ) ?! Tf ()(J ) is (m ; ! ; I; < ){ computable, i.e. there is a computable function h such that m (p)(! (q))?1 Iv = < h(p; q; v). There is a machine which for any input (p; q; v) produces a list of all u such that u < < h(p; q; w). Therefore there is a machine, which for inputs p and q with m (p) = and ! (q) = f produces a list of all (u; v) such that u < f ?1 (Iv )) = Tf ()(Iv ). This means that (f; ) 7! Tf () is (! ; m ; m ){computable.
2
We apply this theorem to iterated function systems with probabilities [Bar93]. An interated function system (IFS) on [0; 1] with probabilities is a tuple S = ([0; 1]; f1 ; : : : ; fk ; p1 ; : : : ; pk ) where f1 ; : : : ; fk : [0; 1] ?! [0; 1] are continuous functions and p1 ; : : : ; pk are positive real numbers with p1 + : : : + pk = 1. With S one associates the function TS : M ?! M de ned by
TS () :=
k X i=1
piTf () i
Corollary 3.4 Let S = ([0; 1]; f1 ; : : : ; fk ; p1 ; : : : ; pk ) be an IFS such that f1 ; : : : ; fk are ! {computable and p1 ; : : : ; pk are {computable. Then TS : M ?! M is (m ; m ){computable. Proof:
Since computable functions transform computable elements to computable elements, by Theorem 3.3 Tf : M ?! M is is computable for i = 1; : : : ; k. Since the composition of computable functions is computable, by Theorem 3.1 the operator TS is computable. i
2
Therefore, for any computable iterated function system with S with probabilities, the associated measure transformation TS : M ?! M is is a (m ; m ){computable function. We shall show that integration of continuous functions is computable in both arguments. The integral of a continuous function can be de ned via summations over nite partitions. Consider 2 M and f 2 C [0; 1]. Let Part be the set of all nite partitions Z of [0; 1] into intervals with rational boundaries. For Z 2 Part de ne s+(Z ) := P (J ) sup f (x); J 2Z
x2J
f (x): s?(Z ) := P (J ) xinf 2J J 2Z
8
R
Since f is continuous, we have sup s? (Z ) = Z 2inf s (Z ) =: fd. Part + Z 2Part The following lemma is the key to the next proof.
Lemma 3.5
For any ; > 0 there are a nite set T Int of (pairwise disjoint) open intervals and a nite set L of closed intervals such that T [ L 2 Part, length(J ) < for every J 2 T and (S L) < . (L can be chosen, such that each J 2 L has length 0.)
Proof:
S
De ne Xn := fx j fxg > 2?n g (n 2 !) and X := Xn . Since (X ) is nite and (X ) = sup (Xn ), there is some n such that (X n Xn ) < . Since Xn is nite, there are a nite set T S Int of pairwise disjointS neighboured intervals with 8J 2 T:length(J ) < , L := f[a; a]ja 2 [0; 1] n T g is nite and Xn T . For each y 2 S L we have y 2 X n Xn or fyg = 0, hence (S L) < . 2
Theorem 3.6
R
The function (f; ) 7! fd for f 2 C [0; 1] and 2 M is (! ; m ; ){computable.
Proof:
P
For any T Int let s? (T ) := f(J ) inf f (J ) j J 2 T g. Consider f 2 C [0; 1] and " > 0. By uniform continuity of f there is some > 0 such that jx ? yj < =) jfx ? fyj < "=4. Let M := maxfjf (x)jj0 x 1g, choose := "=(4(1 3.5 there is some set T Int of S + M )). By8JLemma pairwise disjoint intervals such that 1 ? < T 1 and 2 T:length (J ) < . Furthermore, there P are zJ 2 such that zJ < (J ) for J 2 T and 1 ? ){continuous on M. IR
Proof: R fd = [0; 1=2), apply Theorem 2.7. 2
4 Further representations on measures In De nition 2.1 we have used atomic J 2 Int for identifying R properties r < (J ) with r 2for and measures. By Theorem 3.6, (f; ) 7! fd is (! ; m ; ){computable continuous functions. In the R R following we indentify measures by atomic properties r < td or r < td < s, where r; s 2 and t is from a set of simple continuous \test functions". Q I
Q I
De nition 4.1 For n 2 ! and 0 m 2n de ne the triangle function tnm 2 C [0; 1] by 8 >< x ? (m ? 1)2?n if (m ? 1)2?n x m 2?n tnm (x) := > (m + 1)2?n ? x if m 2?n < x (m + 1) 2?n :0 otherwise. Let m0 and m00 be the standard representartion of M induced by the information structures (M; 0 ; 0 ) and (M; 00 ; 00 ), respectively, de ned as follows:
Z
2 0(0n cj 0m cj u) : () u < tnm d; 2 00 (0n cj 0m cj ucj v) : ()
Z
u < tnm d < v
for all 2 M, n 2 !, 0 m < 2n and u; v 2 dom(Q ). We have not yet shown, that the systems 0 and 00 from De nition 4.1 identify points, i.e. m0 and m00 may still be representations of partitions of M which are coarser than ffg j 2 M g.
Theorem 4.2
m0 and m00 are representations of M such that m m0 m00 : 10
Proof:
For any interval J 2 Int let cJ : [0; 1] ?! be the characteristic function of J . Let T be the set of linear combinations with positive integer coecients of functions tnm (n 2 !; 0 m 2n ). An easy consideration shows, that there is a sequence u0 ; u1 ; : : : of functions from T such that 8x; k:uRk (x) R uk+1 (x) and 8x:cJ (x) = sup uk (x). By a basic property of integrals [Bau74], cJ d = sup uk d, k Rk R i.e. r < (J ) () r < cJ d () R 9k:r < uk d. Therefore r < (J ), i there is some function u 2 T with 8x:u(xR) cJ (x) and r < ud. In particular, for each J 2 Int, (J ) is de ned uniquely by the set of all tnm d, therefore 0 and 00 from De nition 4.1 identify points and m0 and m00 are representations of M. There is a machine which fransforms anyR p 2 ! to some q 2 ! with the following property. If p is a list of all (n; m; u) such that u < tnm d for some 2 M, then q is a list of all (u; v) such that vR< (Iv ). This proves m0 m . On the other hand, we know from Theorem 3.6, that (f; ) 7! fd is (! ; m ; ){computable. Furthermore, (n; m) 7! tnm is (Q ; Q ; ! ){computable on ! !. Therefore there is a machine which transforms any p 2 ! to some q 2 ! with the following property. RIf p is a list of all (u; v) with u < (Iv ) for some 2 M, then q is a list of all (n; m; u; v) with u < tnm d < v. This proves m m00 . An easy consideration shows m00 m0 IR
2
By de nition, the weak topology w on the R set M of probability measures on ([0; 1]; B) is the coarsest, i.e. smallest, topology , such that 7! fd is ( ; IR ){continuous for every f 2 C [0; 1] [Bau74]. As a corollary of Theorem 4.2 we obtain:
Corollary 4.3 The weak topology w is the nal topology m of the representation m .
Proof:
R
By Theorem 3.6, 7! fd is (m ; ){continuous, i.e. ( m ; IR ){continuous for all f 2R C [0; 1]. We obtain w m byR de nition of the weak topology. Consider the functions Fnm : 7! tnm d. Let ?1 (r; 1), hence Vnmr 2 w . Vnmr = f j r < tnm dg be a subbase element of m0 . Then Vnmr = Fnm This shows m0 w . From Theorem 4.2 we know m = m0 , hence m w .
2
The weak topology w on ([0; 1]; B) can be generated by a metric [Bau74].
De nition 4.4 (Hutchinson metric) Let Lip := ff 2 C [0; 1] j f (x) = 0 and 8x; y:jf (x) ? f (y)j jx ? yjg. De ne dH : M M ?! by Z Z dH (; 0 ) := supfj fd ? fd0 jjf 2 Lipg: IR
The metric dH is called the Hutchinson metric [Hut81, Bar93].
Lemma 4.5
dH is a metric on M.
Proof:
R
R
For any f 2 Lip; 8x:jf (x)j 1, hence fd 1d = [0; 1] = 1. Therefore, dH (; 0 ) 2 is We have dH (; ) = 0 by de nition. Consider dH (; 0 ) = 0. Then by tnm 2 Lip, Rwell-de ned. R fnm d = tnm d0 for all n and 0 m 2n , hence = 0 since the set 0 from De nition 4.1 11
IR
R
R
R
R
R
R
identi es points. For any f 2 Lip we have j fd ? fd0 j j fd ? fd00 j + j fd00 ? fdj. Taking sups rst on the right hand side and then on the left hand side, we obtain the triangle inequality for dH .
2
Theorem 4.6 dH : M M ?! is (m ; m ; ){computable. Proof: IR
We introduce a simple dense subset of Lip. Let Sn be the set of all polygon functions f 2 C [0; 1] with break points m2?n , 0 m 2n , such that f (0) = 0 and f (m2?n ) ? f ((m ? 1)2?n ) 2 f?2?n ; 0; 2?n g. For all g 2 Lip Sand all n 2 ! there is some f 2 Sn with d(f; g) 2?n . Let be some standard notation of S := Sn. Then ! . It suces, to describe a method for determining dH (; 0 ) with n error < 2?n from p 2 m?1 (), pR0 2 m?1 (R0 ) and n 2 !. From p; p0 and n determine for all f 2 Sn+2 some rf 2 such that jrf ? j fd ? fd0 jj < 2?n?2 . Let r := maxfrf j f 2RSn+2g. RWe show jr ? dH (; 0 )j < 2?n . Let f 2 Sn+2 such that r = rf . FirstR of all, rR ? 2?n?2 < j fd ? fd0j dH (; 0 ). There is some g 2 Lip with dH (; 0) ? 2?n?2 < j gd ? gd0 j. There is some h 2 Sn+2 with d(g; h) 2?n?2 . We obtain: R R dH (; 0 ) < j R gd ? R gd0 j + 2R?n?2 R j gd ? R hdj + Rj hd ? hd0 j + j R hd0 ? R gd0 j + 2?n?2 2?n?2 + j hd ? hd0 j + 2?n?2 + 2?n?2 3 2?n?2 + rh + 2?n?2 < 2?n + r: Q I
R
Therefore, jr ? dH (; 0 )j < 2?n . Since ! and (f; ) 7! fd is (! ; m ; ){computable, there is a machine which determines some r with the above properties form p; p0 and n.
2
By Lemma 2.1 from [Wei93], the metric space (M; dH ) has a countable dense subset. By Corollary 45.4 from [Bau74], the discrete measures are dense. We shall use the discrete measures determined by rational numbers as a dense subset. Let Md be the set of all probability measuresP 2 M such that there arePa nite set K and rational numbers rk ; sk 2 [0; 1] for all k 2 K such that fsk j k 2 K g = 1 and = sk r , where x (A) = (1 if x 2 A; 0 otherwise). Let d be a standard notation of Md . A computable metric space is a quadruple (M; d; A; ) such that (M; d) is a metric space, A is a dense countable subset and is a notation : ?! A of A such that the set f(u; v; w; x) j u < d( (v); (w)) < xg is r.e. [Wei93]. For a computable metric space (M; d; A; ), the Cauchy rerpesentation C [Wei87, Wei95a] is de ned as follows (we assume w.l.o.g. dom( ) ( n f]g) ) : (u ): C (p) = x : () p = u0 ]u1 ] : : : such that 8i > k d( (ui ); (uk )) < 2?k and x = ilim !1 i k
Theorem 4.7 1. d m 2. (M; dH ; Md ; d ) is a computable metric space.
Proof:
3. The Cauchy representation mC for this space is equivalent to m .
1. This can be proved easily. 2. From (1) and Theorem 4.6 we conclude, that u < dH (d (v); d (w)) < x is r.e. It remains to show, that M is dense in M. Consider 2 M and n 2 !. By Lemma 3.5, there are a nite set T and 12
a nite set of closed intervals L such that T 0 := T [ L is a Spartition of [0; 1], length (J ) < 2?n?3 S for all J 2 T and L < P 2?n?3 . Since 1 ?P2?n?3 < T 1, there are rational numbers ? n ?3 tJ < (J ) with 1 ? 2 < ftJ j J 2 T g < f(J ) j J 2 T g 1. De ne S := PftJ j J 2 T g. De ne 0 2 Md by K := T , rJ := the center of J , sJ := tJ =S for all J 2 T . For any f 2 Lip we obtain:
j R fd ? s?(T 0 )j jPs+(T 0) ? s?(T 0)j fj sup f (J ) ? inf f (J )j(J )jJ 2 T 0g 2P?n?3 js?(T 0) ? s?(T )j = Sfj inf f (J )j(J )jJ 2 Lg L ?n?3 2P R js?(T ) ? fd0j = jP finf f (J )(J ) ? f (rJ )sJ jJ 2 T gj fj inf f P (J )(J ) ? f (rJ )(J )j + jf (rJ )(J ) ? f (rJ )sJ jjJ 2 T g 2?n?3 + Pfj(J ) ? sJ jjJ 2 T g P 2?n?3 + fjP(J ) ? tJ jjJ 2 T g + fjtJ ? tJ =S jjJ 2 T g 2 2?n?3 + fjtJ j J 2 T g (1=S ? 1) < 4 2?n?3 (since 7=8 < S 1): R R Combining the three inequalities, we obtain j fd ? fd0 j < 2?n . Therefore M is dense in M. Pfr j k 2 K g, 3. For each function t from De nition 4.1 and each 2 M with = nm s R t d = P r t (s ) 2 . Obviously, (n; m; ) 7! R t d is ( ; ; ; k){computable nm nm Q Q d Q k nm k R t dfor? nR 2 !, 0 m 2n and 2 M . Let mC (p) = with p =R u0 ]u1 ]u2 ] : : :. We have j R t dnm(u )? v < t d () 9 k:v < tnmdd (uk )j dH (; (uk )) 2?k for all k, therefore nm nm d k R 2?k . By the above observation, v < tnm dd (w) < 2?k is decidable in v; w; n; m; k . Therefore, R from p a list of all (n; m; v) can be computed such that v < tnm d. This shows mC m0 . We prove m mC . It sucies to show, that there is a machine, which for any p 2 dom(m ) and n 2 ! determines some v 2 such that dH (m (p); d (v)) < 2?n . The method is already k
Q I
outlined in (2) above. By de nition, p is a list of all (u; J ) with u < (J ) ( := m (p)). Compute v as follows:
By exhaustive search determine a nite set T Int of pairwise disjoint intervals and rational numbers P tJ (J 2 T ) such that length (J ) < 2?n?3 and tJ < (J ) for all J 2 T and 1 ? 2?n?3 < ft j J 2 T g. J
DeterminePv such that d(v) is the measure determined by the numbers rJ ; sJ (J 2 T ) with sJ = tJ = ftJ j J 2 J g. The existence of T and the numbers tJ (J 2 T ) has been shown in (2). Also the property dH (; d (v)) < 2?n has been proved in (2). Therefore, m mC . 2 Since m m0 m00 mC , these four representations of the probability measures M on the space ([0; 1]; B) induce the same computability theory. In measure theory not only probability measures but arbitrary measures : B ?! [f1g are studied. Let Mb be the set of all measures : B ?! , i.e. all bounded measures on ([0; 1]; B). Let < be the representation of M b obtained from De nition 2.1, where M is replaced by Mb . While m (p)[0; 1] = 1, < (p)[0; 1] may be any non{negative real number. An easy proof shows that 7! [0; 1] is only (< ; < ){computable and not (< ; ){continuous. This means, that informations about upper bounds of < (p)[0; 1] are not available from pre xes of p. As a consequence, Theorem 3.6 on integration fails for < . Only the following weak version can be proved: IR
IR
13
R
(f; ) 7! fd for non{negative f 2 C [0; 1] and 2 Mb is (! ; < ; < ){computable. We can, however, include informations about upper bounds of [0; 1] in the names. Let b be the representation of Mb de ned by the following notation of atomic informations: 2 (ucj vcj w) () u < (Iv ) and [0; 1] < w. Then the theorems we have proved for m hold accordingly for b , in particular Theorem 3.6 on integration. The connection to m is given by the following lemma.
Lemma 4.8
The function 7! [0; 1] on Mb is (b ; ){computable, and the function 7! =[0; 1] is (b ; m ){computable for 2 M b ; [0; 1] 6= 0.
References [Bar93] M.F. Barnsley. Fractals everywhere. Academic Press, Boston, San Diego, 1993. [Bau74] H. Bauer. Wahrscheinlichkeitstheorie und Grundzuge der Masstheorie. Walter de Gruyter,Berlin, New York, 1974. [BB85] E. Bishop und D.S. Bridges. Constructive Analysis. Springer-Verlag, Berlin, Heidelberg, 1985. [Eda95] A. Edalat. Domain theory and integration. Theoretical Computer Sciene, 151:163{193, 1995. [Hut81] J. Hutchinson. Fractals and self-similarity. Indiana University Journal of Mathematics, 30:713{747, 1981. [Ko91] K. Ko. Complexity Theory of Real Functions. Birkhauser, Boston, 1991. [Kus85] B.A. Kushner. Lectures on constructive mathematical analysis. American Mathematical Society, Providence, 1985. English translation from Russian original, 1973. [KW84] Ch. Kreitz und K. Weihrauch. A uni ed approach to constructive and recursive analysis. In: Computation and proof theory, (M.M. Richter et al., eds.). Springer{Verlag, Berlin, Heidelberg, 1984. [KW85] Ch. Kreitz und K. Weihrauch. Theory of representations. Theoretical Computer Sciene, 38:35 { 53, 1985. [RW80] A. Reiser and K. Weihrauch. Natural numberings and generalized computability, Elektronische Informationsverarbeitung und Kybernetik, 16 1-3: 11{20, 1980 [TWW88] J.F. Traub, G.W. Wasilkowski und H. Wozniakowski. Information{based Complexity. Academec press, New York, 1988. [Wei87] K. Weihrauch. Computability. Springer{Verlag, Berlin, Heidelberg, 1987. [Wei93] K. Weihrauch. Computabilitiy on computable metric spaces. Theoretical Computer Sciene, 113:191 { 210, 1993. [Wei95a] K. Weihrauch. A foundation of computable analysis; in: Ker-I Ko and Klaus Weihrauch (eds.), Computability and complexity in analysis, Informatik-Bericht 190. Fernuniversitaet, Hagen, 1995. [Wei95b] K. Weihrauch. A simple introduction to computable analysis. In Informatik{Berichte, Band 171, Fernuniversitat Hagen, July 1995. 2nd ed. 14
Abstract
Scientists apply digital computers to perform computations on natural numbers, nite strings, real numbers and more general objects like sets, functions and measures. While computability theory on many countable sets is well established and for computability on the real numbers several (unfortunately mutually non-equivalent) de nitions are being studied, in particular for measures no computability concept at all has been available until now. In this contribution we introduce a natural computability concept on the set M of probability measures on the Borel subsets of the unit interval [0; 1]. As background we consider TTE, Type 2 Theory of E ectivity, [KW84, KW85], where computability is de ned on nite and in nite sequences of symbols explicitly by Turing machines and on other sets by means of notations and representations. A standard representation m : ! ?! M is introduced via some natural information structure [Wei95a] (M; ; ), where is a subbase of some T0 -topology m and : ?! is a standard notation of . While several modi cations of m suggesting itself at rst glance have undesirable properties, m itself has several very natural properties and hence should induce an important computability theory. Many interesting functions on measures turn out to be computable, in particular linear combination, integration of continuous functions and any transformation de ned by a computable iterated function system with probabilities. Some other representations of M are introduced, among them a Cauchy representation associated with the Hutchinson metric, and proved to be equivalent to m . In particular, the nal topology m of m is the well known weak topology on M.
1 Introduction Measure and integration is a central branch of mathematics pervading almost all parts of abstract analysis. Several authors have already considered questions of e ectivity, constructivity, computability or computational complexity in measure or integration theory. Kushner [Kus85] studies computability and Ko [Ko91] computational complexity of integration. Bishop and Bridges [BB85] present constructive measure theory extensively. Although they do not consider computability, certainly many of their concepts and results have computational counterparts. Edalat gives a domain theoretic approach 1
to e ective integration [Eda95]. He also does not consider computability, but it should be possible to extend his topological approach by computability concepts. Traub et al. [TWW88] investigate the computational complexity of numerical algorithms for integration in the real number model of computation. However, a systematic study of computability in integration and measure theory does not yet exist. In this paper we extend TTE, Type 2 Theory of E ectivity, to measure theory. TTE has been introduced by Kreitz and Weihrauch [KW84, KW85] as a general framework for studying e ectivity, i.e. continuity, computability and computational complexity, in Analysis. For details the reader is referred to the introduction [Wei95b] and a recent short survey [Wei95a] containing most of the notations we shall use in this paper. More details can be found in [KW85, Wei87]. Since this paper is a rst attempt, we consider only the space of probability measures on the Borel subsets of the real unit interval. By f : A ?! B we denote a partial function, i.e. a function from a subset of A to B . Throughout this paper let be a suciently large nite alphabet. Let be the set of nite and ! = fp j p : ! ?! g the set of !{words over . On we consider the discrete topology and on ! the cantor topology de ned by the basis fw! j w 2 g. For Y0 ; Y1 ; : : : ; Yk 2 f ; ! g, a function f : Y1 : : : Yk ?! Y0 is called computable, i it is computed by a Turing machine with a one{way output tape. Every computable function is continuous. The basic idea of TTE is to use nite or in nite sequences as names of \abstract" objects. As naming systems we consider notations, i.e. surjections : ?! S , and representations, i.e. surjections : ! ?! M . Continuity and computability concepts are transferred from and ! via notations and representations, respectively, to the named sets straightforwardly, see [KW85, Wei87, Wei95b, Wei95a]. Mainly notations or representations which are compatible with some relevant structure on the set under consideration are of practical interest. We do not discuss this for notations (see [RW80, Wei87] and Appendix C in [Wei95b]), but we will introduce \e ective" notations explicitly whenever necessary. In particular, for the rational numbers let Q : ?! be the standard representation via fractions of integers in binary notation. We shall abbreviate Q(w) by w. Standard notations of the natural numbers, pairs of rational numbers etc. will be used without further de nitions. Q I
For uncountable sets M we shall consider mainly representations derived from \information structures" (M; ; ), where is a countable subset of 2M of \atomic properties" which identi es points, and is a notation of [Wei95a]. It is assumed that a computer (Turing machine) manipulates {names of atomic properties. As a name of an object x 2 M we consider any in nite list of all properties A 2 which hold for x. Concretely, the standard representation : ! ?! M is de ned by (p) = x () x = w0 ]w1 ] : : : and fwi j i 2 !g = fw j x 2 (w)g: Every nite pre x of a {name p of x contains nitely many atomic properties of x which \approximate" x. Mathematically, this kind of approximation is described by the topology on M , which has as a subbase. Computability on and via on M are xed by the notation which expresses how atomic properties can be handled concretely. Thus, for any information structure (M; ; ), characterizes approximation and computability on M . The topology and the standard representation are closely related: X 2 () ?1 X is open in dom( ) (for all X M ), i.e. is the nal topology of . By the \main theorem for admissible representations" [KW85] a function is continuous relative to standard representations, i it is continuous w.r.t. the associated nal topologies in the usual sense. For more details see [KW85, Wei87, Wei95b, Wei95a].
2
For the real numbers, we need three representations < ; > ; : ! ?! , derived from information structures. They can be de ned explicitly as follows [Wei87, Wei95a]: (p) = x : () p = w0 ]w1 ] : : : with fwi j i 2 !g = fw j w > xg; (p) = x : () p = v0 ]w0 ]v1 ]w1 : : : with f(vi ; wi ) j i 2 !g = f(v; w) j v < x < wg: The nal topologies are < = f(y; 1) j y 2 g [ f g, > = f(?1; y) j y 2 g [ f g and the set IR of ordinary open subsets of , respectively. IR
IR
IR
IR
IR
IR
Notice that induces the standard computability theory on the real line. The translatability or reducibility properties [Wei87, Wei95a] < , > , < 6t , > 6t , < 6t > , > 6t < can be proved easily. In Section 2 we introduce a standard representation m of the set M of probability measures on the Borel sets of the interval [0; 1] by a very natural information structure. We prove a stability theorem for this de nition. We discuss some further modi cations of the de nition and show that that they have undesirable properties. The results indicate that the computability theory on M induced by the representation m is indeed very natural. In Section 3 we prove computability of several interesting functions on measures, in particular linear combination and integration of continuous functions. Also the measure transformation induced by a computable iterated function system with probabilities [Bar93] is computable. Finally in Section 4, we introduce representations based on other natural information structures and a Cauchy representation for the Hutchinson metric [Hut81, Bar93] . We prove that all these representations are equivalent and that their nal topology is the well known weak topology [Bau74].
2 The standard representation of measures In this section we introduce the standard representation m of the probability measures and show that it induces a very natural computability theory. Let Int := f(a; b); [0; a); (b; 1]; [0; 1] j a; b 2 ; 0 < a < b < 1g be the set of open subintervals of [0; 1] with rational boundaries, and let I : ?! Int be some standard notation of Int with dom(I ) ( n fcj ; ]g) . We write Iw for I (w). By B we denote the set of Borel subsets of [0; 1], i.e. the smallest {Algebra containing Int. By M we denote the set of probability measures : B ?! on the space ([0; 1]; B). By a basic theorem of measure theory [Bau74], every measure 2 M is de ned uniquely by its values on the generating set Int. We introduce a standard representation of M via an information structure. The informations available from some standard name of a measure shall be all (r; J ) with r 2 and J 2 Int such that r < (J ). Q I
IR
Q I
De nition 2.1
De ne an information structure (M; ; ) by := range ( ) , where
2 (ucj v) : () u < (Iv ) for all u 2 dom(Q ), v 2 dom(I ) and 2 M. Let m be the topology on M with subbase and let m be the standard representation of M derived form .
3
It remains to show that identi es the points of M. Consider measures ; 0 2 M such that r < (J ) () r < 0 (J ) for all r 2 and J 2 Int. Then obviously, (J ) = 0 (J ) for all J 2 Int, i.e. = 0 . Q I
The de nition of the representation m looks somewhat arbitrary. By the next stability lemma, we obtain an equivalent representation, if we replace Q and I by adequate other notations. For any X let cls(X ) be the closure of X . IR
Theorem 2.2 (stability of m ) Let S : ?! S be a notation of a set S which is dense in such that f(u; v) j S (u) < Q(v)g and f(u; v) j Q(u) < S (v)g are r.e. . Let D be a countable dense subset of [0; 1] and let I 0 be a notation of Int0 := f(a; b); [0; a)(a; 1]; [0; 1] j a; b 2 D; 0 < a < b < 1g such that f(u; v) j cls(Iu0 ) Iv g and f(u; v) j cls(Iu ) Iv0 g are r.e. De ne m0 and m0 by IR
substituting S for Q and I 0 for I in De nition 1. Then m0 = m and m0 m .
Proof:
We show that there is a machine which maps any p 2 ! into some q 2 ! such that the following holds: If p is a list of all (u; v) such that Q (u) < Iv for some 2 M, then q is a list of all (x; y) such that S (x) < Iy0 . We have S (x) < Iy0 , i there are words u; v such that S (x) < Q (u) , Q (u) < Iv and cls(Iv ) Iy0 . Since the rst and the third property are r.e. and the input p lists the second one by assumption, there is indeed a machine with the desired property. This shows m m0 . By symmetry we have also m0 m0 , hence m m0 . Since m and m0 are the nal topologies of m and m0 [Wei95a], respectively, and since equivalent representations have the same nal topology, we obtain m = m0 .
2
If we replace, for example, rational numbers by nite binary fractions or by nite decimal fractions in the de nition of the set Int and in De nition 1, we obtain an equivalent representation with the same nal topology. If we replace the relation \ < " in De nition 1 by \ ", \ > " or \ ", we obtain rerpesentations which violate Lemma 3. Remember that by de nition, the topology m has the subbase = fUr;J j r 2 and J 2 Int g where Ur;J = f 2 M j r < (J )g . We prepare the proof of the theorem by two Lemmas. First, we consider the cases \r (J )", \r (J )". Q I
Lemma 2.3 For Q let (Q) be the topology on M generated by the subbase (Q) := fUr;J j r 2 Q; J 2 Intg, where Ur;J = f 2 M j r (J )g. Then (P ) 6 (Q), if t 2 P n Q for some t 2 (0; 1) (for all P; Q ). The statement holds accordingly, if \ " is replaced by \ ". Proof: The set U := f 2 M j t [0; 1)g is in (P ). For x 2 (0; 1) de ne x 2 M by x f0g := x; x f1g := 1 ? x . Then t 2 U . Assume U 2 (Q). Then there are V1 ; : : : ; Vk 2 (Q) with t 2 V1 \ : : : \ Vk U . For i = 1; : : : ; k there are ri 2 Q and Ji 2 Int with Vi = f j ri (Ji )g. If 1 2= Ji , from t 2 Vi we obtain ri t (Ji ), hence ri < t (since t 2= Q). Let r := max(fri j 1 2= Ji g [ f0g). Then r < t. If 0; 1 2= Ji , then ri t (Ji ) = 0 = r (Ji ), if 0 2 Ji and 1 2= Ji then ri r r (Ji ), if 0 2= Ji and 1 2 Ji then ri t (Ji ) = 1 ? t < 1 ? r = r (Ji ), and if Ji = [0; 1] then ri t (Ji ) = 1 = r (Ji ). We obtain r 2 V1 \ : : : \ Vk , but t > r = r [0; 1), hence r 2= U . This is a contradiction. IR
IR
4
The case \ " can be proved accordingly.
2
The next lemma considers the case \r > (J )".
Lemma 2.4 For D (0; 1) let Int(D) := f(a; b); [0; a); (a; 1]; [0; 1] j a; b 2 D; 0 < a < b < 1g. Let (D) be the topology on M generated by the subbase (D) := fUr;J j r 2 ; J 2 Int (D)g where Ur;J = f 2 M j r > (J )g. Then Q I
D E () (D) (E ) (for all D; E (0; 1) ):
Proof: \ =) ": (obvious) \ (= ": Assume D 6 E . Then there is some d 2 D n E . The set U := f 2 M j 1 > [0; d)g is an element of (D). For any x , 0 x 1, de ne a measure x 2 M by x (A) := (1 if x 2 A; 0 otherwise). In particular, we have 1 > 0 = d [0; d), hence d 2 U . We assume U 2 (E ). Then there are V1 ; : : : ; Vk 2 (E ) with d 2 V1 \ : : : \ Vk U . For i = 1; : : : ; k there are ri > 0 and Ji 2 Int (E ) with Vi = f j ri > (Ji )g. De ne e0 := max(fsup Ji j Ji < dg [ finf Ji j d 2 Ji g). Then e0 d, hence e0 < d since d 2= E . De ne e := (e0 + d)=2. For i = 1; : : : ; k we have d 2 Ji () e 2 Ji , hence d(Ji ) = e(Ji ) and d 2 Vi () e 2 Vi , consequently e 2 V1 \ : : : \ Vk . But e[0; d) = 1 since e < d, hence e 2= U . This is a contradiction. 2
Theorem 2.5 If in De nition 2.1 the relation \u < (Iv )" is replaced by \u (Iv )"; \u (Iv )" or \u < (Iv )", the resulting representations m violate the stability theorem (Theorem 2).
Proof:
By Lemma 3, in the cases \ " and \ " a change of the set Q of bounds changes the topology m , by lemma 4, in the case \ > " a change of the set D of interval boundaries changes the topology m . Since the equivalent representations have the same nal topology, each of the modi cations produces a non{equivalent representation. 2 By De nition 1 and Lemmata 3 and 4, many di erent more or less natural representations and hence computability theories for the set M of probability measures on [0; 1]; B) can be introduced. The \user" has to decide, which of them is adequate for his application. The stable representation m from De nition 1 is certainly the most important one, since its computability theory will occur most frequently. We shall study it in the following exclusively. As a simple consequence of De nition 1, all rational lower bounds of (J ) can be obtained from any m {name of and any I {name of J . This property characterizes the representation m except for equivalence: The representation m is {complete in the set of all representations of M, for which (; J ) 7! (J ) is (; I; < ){computable.
Theorem 2.6 For any representation of M: m () (; J ) 7! (J ) is (; I; < ){computable.
Proof:
Consider m . By de nition there is some computable function f : ! ?! ! with (p) = m f (p) 5
for all p 2 dom(). For p 2 dom(), f (p) is a list of all (u; v) with u < m f (p)(Iv ). There is a Turing machine, which for any inputs p 2 dom() and v 2 dom(I ) determines internally the sequence f (p) and simultaneously writes all words u such that (u; v) is listed by f (p). Therefore, (; J ) 7! (J ) is (; I; < ){computable. Consider, on the other hand, that (; J ) 7! (J ) is (; I; < ){computable. Then there is a machine M1 which for any p 2 dom() and v 2 Int produces a list of all u 2 with u < (p)(Iv ) From this machine another machine M2 can be constructed, which for any p 2 dom() constructs a list of all (u; v) with u < (p)(Iw ). This machine translates into m .
2
Notice, that in particular (; J ) 7! (J ) is (m ; I; < ){computable. Computing only lower rational bounds does not seem to be satisfactory. We would like to compute also arbitrarily close upper bounds of (Iv ). We prove a negative and a positive answer. For any x 2 [0; 1] de ne x 2 M by x(A) := (1 if x 2 A; 0 otherwise). For any good and useful representation of M it should be possible to determine a {name of the measure x e ectively from a name of x. Let M 0 := fx j x 2 [0; 1]g.
Theorem 2.7 For any representation of M, for which x 7! x is (; ){continuous on (0; 1), 7! [0; 1=2) is not (; > ){continuous on M 0 . m is such a representation.
Proof:
Assume, 7! [0; 1=2) is (; > ){continuous on M 0 . Then f : x 7! x [0; 1=2) is (; > ){continuous, hence ( IR ; > ){continuous by the main theorem for admissible representations. Since (?1; 1=2) 2 >, f ?1 (?1; 1=2) 2 IR , but f ?1(?1; 1=2) = [1=2; 1) 2= IR . We prove now that g : x 7! x is (; m ){ continuous, i.e. ( IR ; m ){continuous by the main theorem for admissible representations. Let Ur;J := f 2 M j r < (J )g, r 2 , J 2 Int, be some arbitrary subbase element of m . We have g(x) 2 Ur;J , i r < x (J ). For r 1, g?1 Ur;J = ;, for r < 0, g?1 Ur;J = (0; 1) and for 0 r < 1, g?1 Ur;J = J . Therefore, ?1 g Ur;J is IR {open in dom(g) = (0; 1). This shows that g is ( IR ; m ){continuous. Q I
2
Therefore, for reasonable representations of M, in particular for our standard representation m , arbitrarily close rational upper bounds of measures of open intervals cannot be computed. Although this contradicts intuition at rst glance, it has to be accepted as a matter of fact. Notice, that for proving Lemma 2.3, Lemma 2.4 and Theorem 2.7 we have used measures 2 M with fxg > 0 for some x 2 . Since the arguments have been purely topological without reference to computability, we have also shown that the nal topology m of the representation m , which formalizes a concept of \approximation" on the set M of measures, is quite natural. If we exclude measures with fxg > 0 for some x 2 [0; 1], (; J ) 7! (J ) becomes (m ; I; ){computable. Let M0 := f 2 M j 8x 2 [0; 1]:fxg = 0g IR
Theorem 2.8 The function (; J ) 7! (J ) is (m ; I; ){computable for J 2 Int and 2 M0 .
Proof:
By Theorem 2.6, (; J ) 7! (J ) is (m ; I; < ){computable. It suces to show, that this function is (m ; I; > ){computable for 2 M0 . The algorithm branches into four cases depending on the type of the interval J 2 Int. We discuss only the branch for J = (a; b) with 0 < a < b < 1. For 2 M0 we have 1 = [0; a) + (a; b) + (b; 1]. If we can approximate [0; a) and (b; 1] from below, we can approximate (J ) from above. Consider m (p) = 2 M0 and Iw = (a; b) 2 Int, 0 < a < b < 1. 6
Then for any r 2 we have: r > (a; b), i there are rational numbers sl ; sr with sl < [0; a) and sr < (b; 1] such that r > 1 ? sl ? sr . There is a machine, which for input p and w writes the rational number r to the output , i it has found numbers sl and sr with the above properties by reading the input p. 2 Q I
3 Computable Functions on Measures In this section we prove computability of some interesting functions on probability measures. By the next theorem, the linear combination of measures is computable in all variables.
Theorem 3.1
The function (a; ; 0 ) 7! a + (1 ? a)0 is (; m ; m ; m ){computable for 0 a 1
Proof:
Consider m (p) = , m (p0 ) = 0 and (pa ) = a. Then p is a list of all (r; J ) with r < (J ), and p0 is a list of all (r; J ) with r < 0 (J ) (r 2 ; J 2 Int). For r 2 and J 2 Int we have r < a(J )+(1 ? a)0 (J ), i there are r; r0 ; s; s0 2 with r < sr + (1 ? s0 )r0 , r < (J ) and r0 < 0 (J ), and s < a < s0 . Therefore, there is a Turing machine, which for any input (p; p0 ; pa ), (p; p0 2 dom(m ); 0 (pa ) 1) produces some list q 2 dom(m ) of all (r; J ) 2 Int for which there are r; r0 ; s; s0 2 such that the above properties hold or are listed by p, p0 or pa , respectively. Q I
Q I
Q I
Q I
Q I
2
By Theorem 2.6, (; J ) 7! (J ) is (m ; I; < ){computable on M Int. We extend this result to I0R = fU \ [0; 1] j U 2 IRg, the set of all open subsets of [0; 1]. First we need a representation of this topology. For the set IR of open subsets of , the following information structure ( IR ; ; ) and its derived representation o and topology o are natural (see Example 4.2 and Section 6 in [Wei95a]): For any U 2 IR and u; v 2 let U 2 (ucj v) i [u; v ] U . Consequently, o (p) = U i p is a list of all closed intervals with rational boundaries contained in U . We de ne our standard representation of I0R accordingly: o0 (p) = U : () p is a list of all w 2 with cls(Iw ) U (p 2 ! ; U 2 IR0 ). Let L 2 M be the Lebesgue measure on ([0; 1]; B). IR
Theorem 3.2
(1) (; U ) 7! (U ) for 2 M and U 2 IR0 is (m ; o0 ; < ){computable. (2) (; U ) 7! (U ) for = L and U 2 IR0 is not (m ; o0 ; > ){continuous.
Proof:
Consider m (p) = and o0 (q) = U . Then p lists all (r; J ) 2 Int with r < (J ), and q lists all K 2 Int such that cls(K ) U . For s 2 we have s < (U ), i there are intervals J1 ; : : : ; Jk 2 Int with cls(Ji ) \ cls(Jj ) = ; for 1 i < j k and rational numbers r1 ; : : : ; rk such that: s < r1 + : : : + rk , ri < (Ji ) and cls(Ji ) U for i = 1; : : : ; k. Therefore, there is some machine, which from any p 2 dom(m ) and q 2 dom(o0 ) computes some q0 2 ! with ){continuous, hence ( 0 ; > ){continuous, where 0 is the nal topology of o0 de ned by the subbase = fVJ j J 2 Intg, where U 2 VJ i cls(J ) U . Since F [0; 1=2) 2 (?1; 3=4) 2 > , by continuity of F there are intervals J1 ; : : : ; Jk 2 Int such that [0; 1=2) 2 VJ1 : : : VJ F ?1 (?1; 3=4). We obtain [0; 1] 2 VJ for i = 1; : : : ; k, but F [0; 1] = 1 2= (?1; 3=4), a contradiction. Q I
Q I
k
2
7
i
For uniform formulations in the next theorems we need a standard representation ! of the set C [0; 1] of continuous functions f : [0; 1] ?! . We de ne ! and the corresponding nal topology ! by the following information structure (C [0; 1]; ; ): f 2 (ucj vcj w) : () u < f (cls Iv ) < w for all f 2 C [0; 1] and u; v; w 2 . Properties of ! are discussed in [Wei87, Wei95b, Wei95a]. In particular, ! is the compact{open topology on C [0; 1], which is also generated by the metric d(f; g) := maxfjf (x) ? g(x)jj0 x 1g on C [0; 1]. For any measure 2 M and any continuous function f : [0; 1] ?! [0; 1] de ne the measure Tf () by Tf ()(A) := f ?1 (A) for every Borel set A [0; 1] (see [Bau74], page 42). IR
Theorem 3.3
The function (f; ) 7! Tf () for continuous f : [0; 1] ! [0; 1] and 2 M is (! ; m ; m ){ computable.
Proof:
An easy consideration shows that (J; f ) 7! f ?1 (J ) is (I; ! ; o0 ){computable. By Theorem 3.2, (; U ) ?! (U ) is (m ; o0 ; < ){computable. Therefore, (; f; J ) ?! Tf ()(J ) is (m ; ! ; I; < ){ computable, i.e. there is a computable function h such that m (p)(! (q))?1 Iv = < h(p; q; v). There is a machine which for any input (p; q; v) produces a list of all u such that u < < h(p; q; w). Therefore there is a machine, which for inputs p and q with m (p) = and ! (q) = f produces a list of all (u; v) such that u < f ?1 (Iv )) = Tf ()(Iv ). This means that (f; ) 7! Tf () is (! ; m ; m ){computable.
2
We apply this theorem to iterated function systems with probabilities [Bar93]. An interated function system (IFS) on [0; 1] with probabilities is a tuple S = ([0; 1]; f1 ; : : : ; fk ; p1 ; : : : ; pk ) where f1 ; : : : ; fk : [0; 1] ?! [0; 1] are continuous functions and p1 ; : : : ; pk are positive real numbers with p1 + : : : + pk = 1. With S one associates the function TS : M ?! M de ned by
TS () :=
k X i=1
piTf () i
Corollary 3.4 Let S = ([0; 1]; f1 ; : : : ; fk ; p1 ; : : : ; pk ) be an IFS such that f1 ; : : : ; fk are ! {computable and p1 ; : : : ; pk are {computable. Then TS : M ?! M is (m ; m ){computable. Proof:
Since computable functions transform computable elements to computable elements, by Theorem 3.3 Tf : M ?! M is is computable for i = 1; : : : ; k. Since the composition of computable functions is computable, by Theorem 3.1 the operator TS is computable. i
2
Therefore, for any computable iterated function system with S with probabilities, the associated measure transformation TS : M ?! M is is a (m ; m ){computable function. We shall show that integration of continuous functions is computable in both arguments. The integral of a continuous function can be de ned via summations over nite partitions. Consider 2 M and f 2 C [0; 1]. Let Part be the set of all nite partitions Z of [0; 1] into intervals with rational boundaries. For Z 2 Part de ne s+(Z ) := P (J ) sup f (x); J 2Z
x2J
f (x): s?(Z ) := P (J ) xinf 2J J 2Z
8
R
Since f is continuous, we have sup s? (Z ) = Z 2inf s (Z ) =: fd. Part + Z 2Part The following lemma is the key to the next proof.
Lemma 3.5
For any ; > 0 there are a nite set T Int of (pairwise disjoint) open intervals and a nite set L of closed intervals such that T [ L 2 Part, length(J ) < for every J 2 T and (S L) < . (L can be chosen, such that each J 2 L has length 0.)
Proof:
S
De ne Xn := fx j fxg > 2?n g (n 2 !) and X := Xn . Since (X ) is nite and (X ) = sup (Xn ), there is some n such that (X n Xn ) < . Since Xn is nite, there are a nite set T S Int of pairwise disjointS neighboured intervals with 8J 2 T:length(J ) < , L := f[a; a]ja 2 [0; 1] n T g is nite and Xn T . For each y 2 S L we have y 2 X n Xn or fyg = 0, hence (S L) < . 2
Theorem 3.6
R
The function (f; ) 7! fd for f 2 C [0; 1] and 2 M is (! ; m ; ){computable.
Proof:
P
For any T Int let s? (T ) := f(J ) inf f (J ) j J 2 T g. Consider f 2 C [0; 1] and " > 0. By uniform continuity of f there is some > 0 such that jx ? yj < =) jfx ? fyj < "=4. Let M := maxfjf (x)jj0 x 1g, choose := "=(4(1 3.5 there is some set T Int of S + M )). By8JLemma pairwise disjoint intervals such that 1 ? < T 1 and 2 T:length (J ) < . Furthermore, there P are zJ 2 such that zJ < (J ) for J 2 T and 1 ? ){continuous on M. IR
Proof: R fd = [0; 1=2), apply Theorem 2.7. 2
4 Further representations on measures In De nition 2.1 we have used atomic J 2 Int for identifying R properties r < (J ) with r 2for and measures. By Theorem 3.6, (f; ) 7! fd is (! ; m ; ){computable continuous functions. In the R R following we indentify measures by atomic properties r < td or r < td < s, where r; s 2 and t is from a set of simple continuous \test functions". Q I
Q I
De nition 4.1 For n 2 ! and 0 m 2n de ne the triangle function tnm 2 C [0; 1] by 8 >< x ? (m ? 1)2?n if (m ? 1)2?n x m 2?n tnm (x) := > (m + 1)2?n ? x if m 2?n < x (m + 1) 2?n :0 otherwise. Let m0 and m00 be the standard representartion of M induced by the information structures (M; 0 ; 0 ) and (M; 00 ; 00 ), respectively, de ned as follows:
Z
2 0(0n cj 0m cj u) : () u < tnm d; 2 00 (0n cj 0m cj ucj v) : ()
Z
u < tnm d < v
for all 2 M, n 2 !, 0 m < 2n and u; v 2 dom(Q ). We have not yet shown, that the systems 0 and 00 from De nition 4.1 identify points, i.e. m0 and m00 may still be representations of partitions of M which are coarser than ffg j 2 M g.
Theorem 4.2
m0 and m00 are representations of M such that m m0 m00 : 10
Proof:
For any interval J 2 Int let cJ : [0; 1] ?! be the characteristic function of J . Let T be the set of linear combinations with positive integer coecients of functions tnm (n 2 !; 0 m 2n ). An easy consideration shows, that there is a sequence u0 ; u1 ; : : : of functions from T such that 8x; k:uRk (x) R uk+1 (x) and 8x:cJ (x) = sup uk (x). By a basic property of integrals [Bau74], cJ d = sup uk d, k Rk R i.e. r < (J ) () r < cJ d () R 9k:r < uk d. Therefore r < (J ), i there is some function u 2 T with 8x:u(xR) cJ (x) and r < ud. In particular, for each J 2 Int, (J ) is de ned uniquely by the set of all tnm d, therefore 0 and 00 from De nition 4.1 identify points and m0 and m00 are representations of M. There is a machine which fransforms anyR p 2 ! to some q 2 ! with the following property. If p is a list of all (n; m; u) such that u < tnm d for some 2 M, then q is a list of all (u; v) such that vR< (Iv ). This proves m0 m . On the other hand, we know from Theorem 3.6, that (f; ) 7! fd is (! ; m ; ){computable. Furthermore, (n; m) 7! tnm is (Q ; Q ; ! ){computable on ! !. Therefore there is a machine which transforms any p 2 ! to some q 2 ! with the following property. RIf p is a list of all (u; v) with u < (Iv ) for some 2 M, then q is a list of all (n; m; u; v) with u < tnm d < v. This proves m m00 . An easy consideration shows m00 m0 IR
2
By de nition, the weak topology w on the R set M of probability measures on ([0; 1]; B) is the coarsest, i.e. smallest, topology , such that 7! fd is ( ; IR ){continuous for every f 2 C [0; 1] [Bau74]. As a corollary of Theorem 4.2 we obtain:
Corollary 4.3 The weak topology w is the nal topology m of the representation m .
Proof:
R
By Theorem 3.6, 7! fd is (m ; ){continuous, i.e. ( m ; IR ){continuous for all f 2R C [0; 1]. We obtain w m byR de nition of the weak topology. Consider the functions Fnm : 7! tnm d. Let ?1 (r; 1), hence Vnmr 2 w . Vnmr = f j r < tnm dg be a subbase element of m0 . Then Vnmr = Fnm This shows m0 w . From Theorem 4.2 we know m = m0 , hence m w .
2
The weak topology w on ([0; 1]; B) can be generated by a metric [Bau74].
De nition 4.4 (Hutchinson metric) Let Lip := ff 2 C [0; 1] j f (x) = 0 and 8x; y:jf (x) ? f (y)j jx ? yjg. De ne dH : M M ?! by Z Z dH (; 0 ) := supfj fd ? fd0 jjf 2 Lipg: IR
The metric dH is called the Hutchinson metric [Hut81, Bar93].
Lemma 4.5
dH is a metric on M.
Proof:
R
R
For any f 2 Lip; 8x:jf (x)j 1, hence fd 1d = [0; 1] = 1. Therefore, dH (; 0 ) 2 is We have dH (; ) = 0 by de nition. Consider dH (; 0 ) = 0. Then by tnm 2 Lip, Rwell-de ned. R fnm d = tnm d0 for all n and 0 m 2n , hence = 0 since the set 0 from De nition 4.1 11
IR
R
R
R
R
R
R
identi es points. For any f 2 Lip we have j fd ? fd0 j j fd ? fd00 j + j fd00 ? fdj. Taking sups rst on the right hand side and then on the left hand side, we obtain the triangle inequality for dH .
2
Theorem 4.6 dH : M M ?! is (m ; m ; ){computable. Proof: IR
We introduce a simple dense subset of Lip. Let Sn be the set of all polygon functions f 2 C [0; 1] with break points m2?n , 0 m 2n , such that f (0) = 0 and f (m2?n ) ? f ((m ? 1)2?n ) 2 f?2?n ; 0; 2?n g. For all g 2 Lip Sand all n 2 ! there is some f 2 Sn with d(f; g) 2?n . Let be some standard notation of S := Sn. Then ! . It suces, to describe a method for determining dH (; 0 ) with n error < 2?n from p 2 m?1 (), pR0 2 m?1 (R0 ) and n 2 !. From p; p0 and n determine for all f 2 Sn+2 some rf 2 such that jrf ? j fd ? fd0 jj < 2?n?2 . Let r := maxfrf j f 2RSn+2g. RWe show jr ? dH (; 0 )j < 2?n . Let f 2 Sn+2 such that r = rf . FirstR of all, rR ? 2?n?2 < j fd ? fd0j dH (; 0 ). There is some g 2 Lip with dH (; 0) ? 2?n?2 < j gd ? gd0 j. There is some h 2 Sn+2 with d(g; h) 2?n?2 . We obtain: R R dH (; 0 ) < j R gd ? R gd0 j + 2R?n?2 R j gd ? R hdj + Rj hd ? hd0 j + j R hd0 ? R gd0 j + 2?n?2 2?n?2 + j hd ? hd0 j + 2?n?2 + 2?n?2 3 2?n?2 + rh + 2?n?2 < 2?n + r: Q I
R
Therefore, jr ? dH (; 0 )j < 2?n . Since ! and (f; ) 7! fd is (! ; m ; ){computable, there is a machine which determines some r with the above properties form p; p0 and n.
2
By Lemma 2.1 from [Wei93], the metric space (M; dH ) has a countable dense subset. By Corollary 45.4 from [Bau74], the discrete measures are dense. We shall use the discrete measures determined by rational numbers as a dense subset. Let Md be the set of all probability measuresP 2 M such that there arePa nite set K and rational numbers rk ; sk 2 [0; 1] for all k 2 K such that fsk j k 2 K g = 1 and = sk r , where x (A) = (1 if x 2 A; 0 otherwise). Let d be a standard notation of Md . A computable metric space is a quadruple (M; d; A; ) such that (M; d) is a metric space, A is a dense countable subset and is a notation : ?! A of A such that the set f(u; v; w; x) j u < d( (v); (w)) < xg is r.e. [Wei93]. For a computable metric space (M; d; A; ), the Cauchy rerpesentation C [Wei87, Wei95a] is de ned as follows (we assume w.l.o.g. dom( ) ( n f]g) ) : (u ): C (p) = x : () p = u0 ]u1 ] : : : such that 8i > k d( (ui ); (uk )) < 2?k and x = ilim !1 i k
Theorem 4.7 1. d m 2. (M; dH ; Md ; d ) is a computable metric space.
Proof:
3. The Cauchy representation mC for this space is equivalent to m .
1. This can be proved easily. 2. From (1) and Theorem 4.6 we conclude, that u < dH (d (v); d (w)) < x is r.e. It remains to show, that M is dense in M. Consider 2 M and n 2 !. By Lemma 3.5, there are a nite set T and 12
a nite set of closed intervals L such that T 0 := T [ L is a Spartition of [0; 1], length (J ) < 2?n?3 S for all J 2 T and L < P 2?n?3 . Since 1 ?P2?n?3 < T 1, there are rational numbers ? n ?3 tJ < (J ) with 1 ? 2 < ftJ j J 2 T g < f(J ) j J 2 T g 1. De ne S := PftJ j J 2 T g. De ne 0 2 Md by K := T , rJ := the center of J , sJ := tJ =S for all J 2 T . For any f 2 Lip we obtain:
j R fd ? s?(T 0 )j jPs+(T 0) ? s?(T 0)j fj sup f (J ) ? inf f (J )j(J )jJ 2 T 0g 2P?n?3 js?(T 0) ? s?(T )j = Sfj inf f (J )j(J )jJ 2 Lg L ?n?3 2P R js?(T ) ? fd0j = jP finf f (J )(J ) ? f (rJ )sJ jJ 2 T gj fj inf f P (J )(J ) ? f (rJ )(J )j + jf (rJ )(J ) ? f (rJ )sJ jjJ 2 T g 2?n?3 + Pfj(J ) ? sJ jjJ 2 T g P 2?n?3 + fjP(J ) ? tJ jjJ 2 T g + fjtJ ? tJ =S jjJ 2 T g 2 2?n?3 + fjtJ j J 2 T g (1=S ? 1) < 4 2?n?3 (since 7=8 < S 1): R R Combining the three inequalities, we obtain j fd ? fd0 j < 2?n . Therefore M is dense in M. Pfr j k 2 K g, 3. For each function t from De nition 4.1 and each 2 M with = nm s R t d = P r t (s ) 2 . Obviously, (n; m; ) 7! R t d is ( ; ; ; k){computable nm nm Q Q d Q k nm k R t dfor? nR 2 !, 0 m 2n and 2 M . Let mC (p) = with p =R u0 ]u1 ]u2 ] : : :. We have j R t dnm(u )? v < t d () 9 k:v < tnmdd (uk )j dH (; (uk )) 2?k for all k, therefore nm nm d k R 2?k . By the above observation, v < tnm dd (w) < 2?k is decidable in v; w; n; m; k . Therefore, R from p a list of all (n; m; v) can be computed such that v < tnm d. This shows mC m0 . We prove m mC . It sucies to show, that there is a machine, which for any p 2 dom(m ) and n 2 ! determines some v 2 such that dH (m (p); d (v)) < 2?n . The method is already k
Q I
outlined in (2) above. By de nition, p is a list of all (u; J ) with u < (J ) ( := m (p)). Compute v as follows:
By exhaustive search determine a nite set T Int of pairwise disjoint intervals and rational numbers P tJ (J 2 T ) such that length (J ) < 2?n?3 and tJ < (J ) for all J 2 T and 1 ? 2?n?3 < ft j J 2 T g. J
DeterminePv such that d(v) is the measure determined by the numbers rJ ; sJ (J 2 T ) with sJ = tJ = ftJ j J 2 J g. The existence of T and the numbers tJ (J 2 T ) has been shown in (2). Also the property dH (; d (v)) < 2?n has been proved in (2). Therefore, m mC . 2 Since m m0 m00 mC , these four representations of the probability measures M on the space ([0; 1]; B) induce the same computability theory. In measure theory not only probability measures but arbitrary measures : B ?! [f1g are studied. Let Mb be the set of all measures : B ?! , i.e. all bounded measures on ([0; 1]; B). Let < be the representation of M b obtained from De nition 2.1, where M is replaced by Mb . While m (p)[0; 1] = 1, < (p)[0; 1] may be any non{negative real number. An easy proof shows that 7! [0; 1] is only (< ; < ){computable and not (< ; ){continuous. This means, that informations about upper bounds of < (p)[0; 1] are not available from pre xes of p. As a consequence, Theorem 3.6 on integration fails for < . Only the following weak version can be proved: IR
IR
13
R
(f; ) 7! fd for non{negative f 2 C [0; 1] and 2 Mb is (! ; < ; < ){computable. We can, however, include informations about upper bounds of [0; 1] in the names. Let b be the representation of Mb de ned by the following notation of atomic informations: 2 (ucj vcj w) () u < (Iv ) and [0; 1] < w. Then the theorems we have proved for m hold accordingly for b , in particular Theorem 3.6 on integration. The connection to m is given by the following lemma.
Lemma 4.8
The function 7! [0; 1] on Mb is (b ; ){computable, and the function 7! =[0; 1] is (b ; m ){computable for 2 M b ; [0; 1] 6= 0.
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