TOWARDS A THEORY OF REPRESENTATIONS. Christoph Kreitz and 'klaus Weihrauch 1)

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1 TOWARDS A THEORY OF REPRESENTATIONS Christoph Kreitz and 'klas Weihrach 1) 1. Introdction Definitions of Type 2 comptability, i.e. comptability on sets with cardinality not greater than that of the contin:n, have been given in several ways (see e.g. Egli/Constable [1], Scott [6], Rogers l5]). Most of these definitions are eqivalent or at least dependent from each other bt there is no generally accepted approach as in the case of comptability on denmerable sets. This paper presents the concept of representations as a fondation for a nified Type 2 comptability theory. Its basic idea is that real world compters cannot operate on abstract elements of a set M bt only on names. We have chosen the set IF of seqences of natral nmbers as a standard set of names and have defined comptability on IF explicitly (see Weihrach [7]). Comptability on other set~ M then can be derived from comptability on IF by means of representations i.e. (partial) mappings from IF onto M. The same comptability theory cold be obtained by sing sets like P oo as standard sets bt considering the applications of or theory IF seems to be the better one. For exa~ple infinite objects are often defined by seqences of finite objects (e.g. Cachy seqences, chains etc.) and not by sets of finite objects. Frthermore the comptation model for fnctions on IF is easy to nderstand and allows stdying comptational complexity. Comptable fnctions trn ot to be continos in general and in most cases fnctions which are not comptable are not even continos. Hence topological considerations are fndamental for Type 2 theory and continity w.r.t. representations will also be stdied. Therefore two versions of Type 2 theory are developed simltaneosly, a topological (t-) and a comptable (c-) one. We assme the reader is familar with ordinary recrsion theory and same basic \ properties of nmberings (MaI'r.p.v[4], Rogers [5], Ershov [2]). By IN we denote the set of all natral nmbers by W (IN) the set of all finite words over IN e is the empty word and 19(w) is the length of the word w. If we W(lN) OSiSn. By A to B, instead of and W= x o x 1 x n (where xi E IN) then we define w(i):= xi for f: A ---. B (with dotted arrow) we denote a partial fnction from where "partial" means dom f.sa. As sal we write (n)(" i) where 1[(n) :lnn-'ln is Cantor's 1[ ~1' n <i 1, in> bijection. 1) Fernniversitat, Postfach 940, Hagen, West-Germany 73

2 (j) d,enotes the standard nmbering Iilf the nary partial recrsive fnctions. Define IF:= {p : m - m} and m :=W( IN) U IF. For a,b, E m define a '.: b : ~ a is a prefix of b. For p E IF and i E m let p[i] : =:,p (0) p(i-l) E W(IN) and conversely for v E W(IN) let [ v] : = {p E IF I v p}. on m we consider the topology defined lly the basis {O IvEw(IN)} where 0:= {b Emlv!:b}. The indced topology on IF is the well v v known Baire's topology. on m we consider the discrete topology. 2. Type 2 Recrsion theory Unlike to ordinary (Type 1) recrsion theory for Type 2 recrsion theory there is no generally accepted formalism. We now otline a nified approach which is formally similar to the Type 1 formalism. More details can by fond in Weihrach's paper [7']. We start with the definition of a standard representation IjJ of [IF - m], the set of continos fnctions from IF to m. From IjJ we derive representations of certain continos fnctions from IF to IF and from IF to m. The constrction of IjJ rests on the following property. Let be isotone (w.r.t. E). Then the fnction Y: IF-m, defined by y(p):= sp{y(w) IWEP}, is continos. And for any continos fnction r:lf-lb, r = Y of P E IF for some isotone y: W (IN)-W (IN). 'Phe fnction y sffiently many prefixes is' called comptable, iff comptable fnctions r: IF - m which on inpt can of rep) = yep) specifies, how from prefixes can be determined. A fnction r= y for some comptable fnction y. The be easily characterized by oracle Tring machines p EIF from time to time read a vale p(i) and from time to time write one'of the vales q(o), q(l),. (in this order) of the reslt qem. For 'transforming n-ary fnctions on IF to nary ones, the following tpling fnctions n(n) : IF - IF are sed: n (1) (p) := p, if x= 2i,P l(i) n+ if Also (j)-ary tpling is possible: n(lil) (Po,Pl, )<i,j >:=Pi(j). The fnctions n(n) and new) are homeomorphisms w.r.t. the prodct topologies. The projections of their inverse are comptable. The definition of y is effective in the following sense. There is a comptable (by an oracle TUring machine) operator r : IF - m,with the following property. on inpt p,q it determines y(q) if y:='v~'v N ('V N is a bijective standard nmbering of W(lN» is isotone, r(q) for some continos r: IF - m otherwise. Then by II! (q)':= ljj(p) (q):= r <p,q> a representation ljj:lf - [IF -m] of the continos p fnctions from IF to m is defined, which satisfies the "niversal Tring machine theorem" and the "smn-theorem". \ 14

3 THEO~Ef.1 (1) (2 ) ~p(q) = r<p,q> for some comptable re [IF - lb] ljj<q,r>=\li,,< >(r) for some comptable EE[lF-m] with range p L. p,q (E)SlF. Notice that r and E are not only continos bt even comptable. Similar to Type 1 recrsion theory the tm-theorem and the smn-theorem charaterize the representation W niqely p to (comptable) eqivalence (see Chapter 3). More interesting than IjJ itself are two representations derived from 1jJ. DEFINITION (1) Define a set [IF - :IN] of partial fnctions from IF to IN and a representation X: IF -[IF - IN] by: ~(q) :=x(p) (q) the first n~er of the seqence IjJ (q) otherwise). p := (div if IP. (q)= ElB,, P (2) Define a set [IF - IF] of partial fnctions from IF to IF and a representation div otherwise). 1jJ: IF - [IF ---,. IF] by III (q) := ljj(p) (q) := (IP. (q) if IjJ (q) E IF, P P P This definition extends well known concepts of comptable operators and fnctionals to a niform topological description, where the elements comptable w.r.t. a given representation are those with comptable names. The fnctions from [IF - IN] and from [IF - IF] have natral domains (c.f. domains of partial recrsive fntions). Bt the set of domains is sfficently rich sch that any continos fnction is essentially considered. Theorem (1) [IF - 'IN j is the set of all continos fnctions E: IF ---- IN sch that dom (E) is open. For any continos fnction r: IF ---~ IN there is some E E [IF - IN] which extends r. (2) A valid statement is obtained by sbstitting ''IN'' by ''IF'' and "open" by "G 6 -sbset" in (1). Also the representations X and IjJ satisfy the tm- and the smn-theorem. This leads to a rich theory fo continity and of comptability. which is formally similar to Type 1 recrsion theory. From the above theorem we conclde that by w' (p~ := dom(~) a representation w' of the open sbsets of IF is defined, which corresponds to the nmbering i ~ dom(~.) of the r.e. sbsets of IN. We l. call a sbset A,: IF t-open (c-open) iff A= w' (p) for some (comptable) p. A is t-clopen (c-clopen), iff A and IF\ A are t-open (c:'open). The t-open (c-open) sets are exactly the projections of the t-clopen (c-clopen) sets. The self applicability and the halting problem of X can be formlated. They are c-open, not t-clopen, c-complete and c-prodctive. Also effective inseperability can be defined. The sets {pix (p) = O} and {pix (p) = 1} are c-effectively p P, inseperable. This property can be sed in the stdy of precomplete representations. 75

4 Many other properties can be proved easily bt more qestions are still nsolved in this theory of continity and comptability on IF. 3. Theory of representations In order to define comptability and constrctivity on a set M with cardinality not greater than that of the continm, we represent M by a srjective mapping 0: IF --- M, called representation of M. Some examples for representations are the enmeration representation 1M: IF - P with 1M(p):= {i li+1e range p}, the III representation 0cf of P w by characteristic fnctions with 0cf(P):= hlp(i) = a}, and the representations \1.1: IF -[IF-lB], \1.1: IF-[lF-IF], X: IF -[IF' -IN], 00': IF - {x slflx is open} introdced in chapter 2. Effectivity properties of theorems, fnctions, sets, predicates etc. can be expressed by effectivity of correspondences (i.e. mltivaled fnctions) which are triples f= (M,M',P) where P.s,Mx M'. Definition Let 0,0' be representations of M resp. M' and let f= (M,M',P) be a correspondence. f is called weakly (0,0')- t- (c-) effective iff there is some (comptable) r' E [IF IF] sch that (oq, O'r'q) E P for all -1 qeo dom(f). f is called (0,0') - t- (c-) effective, iff in addition r'(q) is ndefined for all -1 - q E 5 (M\dom f) (o,v)-effectivity of a correspondence f= (M,S,P) where v is a nmbering of S is defined accordingly 11sing [IF - IN] instead of [IF - IF]. For convenience we shall say "continos" instead of "t.;..effective" and "comptable"" instead of "ceffective" Since a partial fnction is a single valed correspondence the above definition is applicable to fnctions. A sbset A..s 1M "can either be charactem:ed as the domain of a partial fnction or by its characteristic fnction. A set ASM is called 0 - (c-) open iff d A := (M, IN,A xm) is (O,id m ) t - (c-)effective. A is called 0 - (c-) clopen iff CA:=M,lN,{(X,O) I xea} U {(y, 1) Iy M\ A}) is (O,id ln ) t - (c-) effective. Usally we say "provable" instead of "c-open" and "decidable" instead of "c-open". The 0 -effectivity on M strongly depends on the representation O. Consider the two qestions whether complementation on P is effective and whether contable W nion on P w is effective. There is no absolte answer bt only one relative to the considered representation: Complementation is (Ocf;Ocf)-comptable bt not even (lm,d4)-continos, contable nion is comptable w.r.t. 1M bt not even weakly continos w.r.t. 0cf(se n(w) for formalization). This difference can be explained sing the intitive concept of finitely (or continosly) accessible 76

5 (f.a.) j.nformation. Every.tre information n ElM(p) is f.a. from p, no tre information; n $lmp is f.a. from p. Bt every tre information ne 0cf(P) or m $ 0Cf(P) is f.a. from p. Representations may be changed in a certain way withot changing the indcedeffectiviqr. For any two representations 0.0' of M resp. M' define 0<0' -t 0==0' t :<=> M. M' and id, M,M :<=> os 0' and o's O. t, t c- redcibility (Sc) and c- eqivalence to show that 0 fs lm and that lm and c c is (0,0') t-effective, (==) is defined accordingly. It is easy c 0cf are not t- eqivalent. Since effective fnctions are closed nder composition two representations are t- (c-) eqivalent if and only if the define the some continity (comptability) theory. Theorem Let 0,0' be representations of M. Then (1),(2) and (3) are eqivalent. (1) (2) (3) o S 0' t For any representation 0l:IF f (weakly) (0 1,0) t-effective For any representation O 2 : IF g (weakly) (0;0 2 ) t-effective --- Ml and any correspondence f= (M,M l,p): ~ f (weakly) (0'1,0') t-effective M2 and any correspondence g = (M 2,M,P) : ~ g (weakly) (0,0 2 ) t-effective. Every representation 0;: IF --- M indces a topology on M by x E.0 :<=> -1 o x= A () dom 0 for some open sbset A S IF o is called the final- topology of 0 and it consists exactly of all the o-open sbsets of M. For example.1m' the final topology of the enmeration representation of P is determined by the w basis {Oe\e SlN, finite} where 0e:={x SlNle S x}. Clearly t- eqivalent representations have the same final topologies bt the converse does not hold in general (a conterexampel is presented in [a]). If on M already a topology is given then =.0 shold hold for any "reasonable" representation of M. (In some special cases there might be reasons for choosing.0 *. ) For seperable To -spaces representations eqivalent to a standard- representation defined as follows seem to be the most natral ones. Definition Let (M,.) be a seperable To-space and let U be a nmbering of some basis of For x E M let (x):= {ilx E i}' A standard - representation dom 0 :=lm-1 (M) and o of (M,.) is defined by o (p) := -l1m(p) whenever p E dom 0' 77

6 A standard - properties representation ( 1 ) o is continos and open, especially =.1:, U U (2) For any space (M',.') and any H: M --- M' (3) Ho,O is continos <=> H is continos, 1: ~ 0 for any continos 1:: IF --- M. t An immediate conseqence is that all the standard - o of a seperable T -space has some remarkable o ' representations of the same space are t- eqivalent and therefore the eqivalence class {olo= O} does not depend t on the nmbering U. Since every representation eqivalent to 0 indces the same continity -theory we call a representation 0 (or admissible) iff 0= 0. t tions 1M and of p w real nmbers is not (see [a). for some standard - are admissible. The decimal - of a seperable To-space t-effective representation o. The representa representation of the For admissible representations of a space (M,.) the final topology is identical with. Frthermore topological continity and continity w.r.t. these representations are closely related. Theorem Let (Mi'.i) be seperable To-spaces and let 0i: IF Mi be admissible representations (i = 1,2). Let F: Ml M 2, then: (1) F (.1'.2)-continos <=> F weakly (01'02)-continos, For some representations the converse of (2) also holds (e.g. for the representation p of IR by normed Cachy-seqences - see [7]). There are many other aspects of representations which shold be stdied, for example recrsion-theoretic properties, comptable elements, the strctre of eqivalence degrees, closre properties etc. There are also natral representations the final topologies of which are not seperable. See Kreitz & Weihr~ch [3] for frther discssion.

7 References, [ 1] EGLI, H., CONSTABLE, R.L.: Comptability co~cepts for programming langages semantics. TCS ~ (1976),133' [ 2] ERSHOV, JU.L.: Theorie der Nmerierngen I. Z.f. math. Logik ~ (1973), [ 3] KREITZ, C., WEIHRAUCH, K.: Theory of representations (to appear) towards a theory of representations. Informatik-Berichte Nr. 40, FernUniversit&t - Gesamthochschle - Hagen (1983). [4] MAL'CEV, A.I.: Algorithms and Recrsice Fnctions. Wolters Noordhoff, Groningen [5] ROGERS, H. jr.: Theory of Recrsive Fnctions and Effective (bmptability. Mc Graw-Hill, New York 1967 [6] SCOTT, D.: Data Types as lattices. SIAM J. Compo ~ (1976), ] WEIHRAUCH, K.: Type 2 recrsion theory (to be pblished). 8] WEIHRAUCH, K./ KREITZ, C.: Representations of the real nmbers and of the open sbsets of the set of real nmbers (to appear). 79