Effective operators and continuity revisited - Semantic Scholar

Effective Operators and Continuity Revisited Dieter Spreen Fachbereich Mathematik, Theoretische Informatik Universit~it-GH Siegen HSlderlinstr. 3, D-W 5900 Siegen Germany Email." spreen@hrz, uni-siegen, dbp. de

Abstract

In programming language semantics different kinds of semantical domains are used, among them Scott domains and metric spaces. D. Scott raised the problem of finding a suitable class of spac~ which should include Scott domains and metric spaces such that effective mappings between these spaces are continuous. It is well known that between spaces like effectively given Scott domains or constructive metric spaces such operators are effectively continuous and vice versa. But, as an example of Friedberg shows, effective mappings from metric spaces into Scott domains are not continuous in general. In a joint paper P. Young and the author presented a condition which under fairly general effectivity assumptions forces effective mappings between separable countable topological T0-spaces to be effectively continuous. In this paper the condition is weakened. Moreover, a large class of separable countable T0-spaces is given, and it is proved that a mapping between spaces of the class is effectively continuous, iff it is effective and satisfies the condition. A modification of Friedberg's example shows that the result is false without the extra condition. Among others the class of spaces contains all reeursively separable recursive metric spaces in which one can effectively pass from convergent normed recursive Cauehy sequences to their limits and all Scott domains that can be obtained via product and function space constructions from fiat domains with at least three elements. The topology of the spaces in this class is effectively equivalent to the topology generated by those elements in the distributive lattice of all completely enumerable subsets of the space which possess a pseudocomplement and are regular with respect to this operation.

1

Introduction

In programming language semantics different kinds of semantical domains are used, among them Scott domains and metric spaces (cf. e.g., America and Rutten 1988, de Bakker and Zucker 1982, Nivat 1979, Reed and Roscoe 1988, Scott 1972, 1973, 1976, 1982, Scott and Strachey 1971). In his Logic Colloquium '83 talk D. Scott considered the problem of finding a suitable class of spaces which should include Scott domains and metric spaces such that effective mappings between these spaces are continuous. Continuity is an essential property of mappings which appear as the meaning of program constructs such as procedures: since each converging computation can use only a finite amount of information about its input, it follows that if the value of a computable map with respect to a given input can be found, it must be determined by some finite approximation of the input. If one studies the behaviour of procedures in a system where execution is based on rewriting, then the meaning of a procedure is a map that transforms program code. Maps between semantic domains that are determined by computable operations on the (syntactic) code are called effective. There has been a long interest in logic and constructive mathematics in the question of whether effective maps are continuous: Myhill and Shepherdson (1955) showed that on the set of all partial recursive functions each effective operator is effectively continuous and vice versa. Kreisel, Lacombe and Shoenfield (1959) obtained an analogous result with respect to the set of all total recursive functions. The first result has been lifted to effectively given Scott domains by various authors (cf. Egli and Constable 1976, Er~ov 1977, Weihrauch and Deil 1980) and

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the essential implication of the second result, i.e. the effective continuity of effective maps, has independently been generalized to constructive metric spaces by Ceftin (1962) and Moschovakis (1963, 1964). General versions of the theorems by Myhili and Shepherdson, and Kreisel, Lacombe and Shoenfield respectively have also been investigated by Lachlan (1964). As follows from several examples, effective operators are not effectively continuous in general (cf. Friedberg 1958, Helm 1971, Pour-El 1960, Young 1968, Young and Collins 1983). Friedberg's example shows t h a t effective mappings between metric spaces and Scott domains are not necessarily continuous. In a joint paper P. Young and the author (1984) showed under fairly general effectivity assumptions t h a t an effective map between separable countable topological T0-spaces is effectively continuous, if it has a witness for noninclusion, which means: if some basic open set in the domain is not mapped into some basic open set in the range, then we must be able to effectively find a witness for this, i.e. an element of the basic open set in the domain which is mapped outside the basic open set in the range. In two subsequent papers (1990, 1991) the author proved t h a t this continuity result follows very naturally from a characterization of effectively given topologies by topologies which are generated by certain completely enumerable subsets of the space. Such topologies are called Mal'cev topologies. In this paper we present a class of spaces, called SITS, such t h a t for maps between them also the converse of the continuity result is true, i.e., a map is effectively continuous, iff it is effective and has a witness for noninclusion. The spaces we consider are countable semi-regular separable T0-spaces with a countable basis on which a relation of strong inclusion is defined such t h a t the property of being a basis holds with respect to this relation. Moreover, some effectivity requirements have to be satisfied, which among others imply t h a t the dense subset of the space is effectively enumerable and every basic open set as well as its exterior are completely enumerable. This is done in section 4. In section 2 the general framework is set up. Then, in section 3 Mal'cev topologies are studied. As we shall see, the topology of the spaces in S R S is effectively equivalent to the topology generated by those elements in the distributive lattice of all completely enumerable subsets of the space which possess a pseudoeomplement and are regular with respect to this operation. In the remaining sections it is shown t h a t this class contains all recursively separable recursive metric spaces in which one can effectively pass from convergent normed rccursive Cauchy sequences to their limits, and all Scott domains t h a t can be obtained via product and function space constructions from fiat domains with at least three elements. Recursive metric spaces are considered in section 5 and Scott domains in section 6. In section 7, finally, a modification of Friedberg's example is given, which shows t h a t the continuity result in section 4 is false without the condition of having a witness for noninclusion.

2

Strongly Effective Spaces

In what follows, let ( , ) : w 2 --* w be a recursive pairing function, let p(n) (R(,~)) denote the set of all n-ary partial (total) recursive functions, and let Wi be the domain of the ith partial recursive function 7~i with respect to some G6del numbering ~. We let ~ i ( a ) l mean t h a t the computation of ~ ( a ) stops. Now, let T -- (T, r ) be a countable topological To-space with a countable basis/3. If r/is any topology on T, then we also write ~ = (C) to express t h a t C is a countable basis of 7/. Moreover, for any subset X of T, i n t r ( X ) , cl~(X) and e x t , ( X ) respectively are the interior, the closure and the exterior of X. An open set X is regular open, if X = intr(clT(X)), and T is semi-regular, if all X E B are regular open. In the special cases we have in mind, a relation between the basic open sets can be defined which is stronger than usual set inclusion, and one has to use this relation in order to derive the result we talked about in the introduction. We call a relation -< on 13 sLrong inclusion, if for all X, Y E 13, from X -< Y it follows t h a t X C Y. Furthermore, we say t h a t B is a strong basis, if for all z e T and X, Y e 13 with z e X A Y there is a V e 13 such t h a t z E V, V -< X and V -< Y. If one considers basic open sets as vague descriptions, then strong inclusion relations can be considered as "definite refinement" relations. Strong inclusion relations t h a t satisfy much stronger requirements have also been used in Smyth's work on topological foundations of programming language semantics (cf. Smyth 1987, 1988). Compared with these conditions, the

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above requirements seem to be rather weak, but as we go along, we shall meet a further requirement, and it is this condition which in applications prevents us from choosing -~ to be ordinary set inclusion. For what follows we assume that -~ is a strong inclusion on B a n d / 3 is a strong basis. Let x: w ~ T (onto) and B: to ~ B (onto) respectively be (partial) indexings of T and/3 with domains dora(x) and dora(B). T is recursively separable, if there is some recursively enumerable (r.e.) set E C dom(x) such t h a t {x~ I i 9 E } is dense in T, i.e., it intersects every basic open set. A subset X of T is completely enumerable, if there is an r.e. set W,~ such t h a t x~ E X iff i 9 Wn, for all i 9 dom(x). Set M~ = X in this case, and let Mn be undefined, otherwise. Then M is an indexing of the class C E of all completely enumerable subsets of T. We say t h a t B is computable, if there is some r.e. set L such t h a t for all i 9 dora(x) and n 9 dom(B), (i, n) 9 L iff xl E Bn. Furthermore, the space T is called strongly effective, if B is a total indexing and the property of being a strong basis holds effectively, which means t h a t there exists a function p 9 p(s) such t h a t for i E dom(x) and n, m 9 w with x~ 9 B,~ n B,~, p(i, m, n)J., z~ E Bp(~. . . . ), Bp0 . . . . ) -~ Bin, and Bp0,,~,n ) -~ B,,. As it is shown in (Spreen 1990), 7- is strongly effective, if B is computable and total, and { (m, n) [ B,~ -~ Bn } is r.e. As it is well known, each point of a T0-space is uniquely determined by its neighbourhood filter and/or a base of it. If B is computable, a base of basic open sets can effectively be enumerated for each such filter. An enumeration (Bl(a))ao~ with f : to --, to such t h a t range(f) C_ dom(B) is said to be normed, if it is decreasing with respect to -r If f is recursive, it is also called recursive and any G6del number of f is said to be an index of it. In case (Bi(a)) enumerates a base of the neighbourhood filter of some point, we say it converges to t h a t point. The following result is proved in (Sprcen 1990): L e m m a 2.1 Let 7- be strongly effective and B be computable. Then there is a function q 9 p(1) such that for all i 9 dom(x), q(i) is an index of a normed rccursive enumeration of basic open

sets converging to xi. B is said to allow effective limit passing, if there is a function pt 9 pO) such that, if rn is an index of a normed reeursive enumeration of basic open sets which converges to some point y E T, then pt(m)~, pt(m) 9 dom(x) and xpt(,~) = y. If B allows effective limit passing and is computable, then we call B acceptable. By definition each open set is the union of certain basic open sets. In the context of effective topology one is only interested in such open sets where the union is taken over an effectively enumerable class of basic open sets. They are called Lacombe sets. Set L~ = U { Ba [ a 9 Wn }, if Wn C_dom(B), and let L~ be undefined, otherwise. Then L* is an indexing of the Laeombe sets of 7-. If .7 is a further topology on T with countable basis C and C: to ~ C (onto) is an indexing of C, then 7- is Lacombe finer than .7 and .7 is Lacombe coarser than 7-, if C < • L r, i.e., if there is some function f E p 0 ) such t h a t f ( n ) l , f(n) 9 dom(L r) and C,~ = L T for all f(~), n E dom(C). If *7 is both Lacombe finer and Lacombe coarser than 7-, then *7 and 7- are called

Lacombc equivalent. There is also another possibility to effectively compare *7 and 7-. 7- is said to be reeursively finer than *7 and ~/ recursively coarser than 7-, if there is some function g 9 p(2) such t h a t g(i,m)l, g(i,m) E dom(B) and x~ E Bao,,, 0 C_ Urn, for all i E dom(x) and m 9 dom(C) with xi E Urn. For computable B, *7 is recursively coarser than 7-, if it is Lacombe coarser than 7-.

3

Mal'cev Topologies

A topology ~/on T is a Mal'cev topology, if it has a basis of completely enumerable subsets of T. Any such basis is called a Mal'cev basis. s = (CE) is called Er$ov topology. As it is shown in (Spreen 1990), all Mal'cev bases on T have a canonical computable indexing. We will assume in this paper t h a t any Mal'cev basis is indexed in a computable way. Beside the Error topology there are other important classes of Mal'cev topologies. Obviously, C E is a distributive lattice with respect to union and intersection. For U E CE, let U* denote its pseudocomplement, i.e. the greatest completely enumerable subset of T \ U, if it exists. U is called regular, if U* and U** both exist and U** = U. We say t h a t a Mal'cev

462 topology is regular based, if it has a basis of regular sets. Since the class R E G of all regular subsets of T is closed under intersection, it also generates a regular-based Mal'cev topology on T, which we denote by 7~. Let r/be a regular-based Mal'cev topology on T with regular basis C and C:o~ ~ C (onto) be a numbering of C. We say that C is *-computable, if there is some r.e. set L' such that for all i E dora(z) and rn E dora(C), (i, rn) E L ' iff x~ E C~,. Similar to the general ease of all Mal'cev bases all regular bases on T can be indexed in a canonical computable and *-computable way (el. Sprecn 1990). In what follows, we assume that regular bases are always indexed, both computably and ,-computably, and that R E G is indexed by the canonical indexing. Then 7~ is the Lacombe finest regular-based Mal'cov topology on T. The reason for introducing regular-based Mal'cev topologies is that in certain cases one need not only be able to enumerate each basic open set, hut to a certain extend also its complement. In general, one cannot expect that the whole complement of a basic open set is completely enumerable. A subset X of T is called weakly decidable, if both its interior and its exterior are completely enumerable. We say that a Mal'cev topology r/with Mal'cov basis C is complemented, if all of its basic open sets are weakly decidable. Let C: w ~ C (onto) be a numbering of C. C is called co-computable, if there is an r.e. set L ~ such that for all i E dom(x) and m E dom(U), li, rn I E L' if[ x~ E ext~(C,n). In (Spreen 1991) it is shown that each complemented Mal'cev topology has a canonical computable and co-computable indexing of its basis. In the ease of regular-based Mal'cev topologies we required that the complement of each basic open set contains a largest completely enumerable subset. If this subset is also opeja, it is the exterior of the basic open set, which is therefore weakly decidable. Conversely, we have: L e m m a 3.1 Let T be strongly effective and recursively separable. Fur~ermore, let B be acceptable. Then, for every tceakly decidable basic open set Bn, one has that extr(Bn) --- B*.

Hence, Bn is re@ular iff it is regular open. If, in addition, B is co-computable, then it is also .-computable. Moreover, ~" is regular based iff it is semi-regular. This result corrects and extends Lemma 9 in (Sprcen 1991). The first statement follows with the proof given there. The other properties are immediate consequences of it. In (Sprsen 1990) strongly effective topological spaces are characterized in terms of certain Mal'cev topologies. The central requirement a Mal'cev topology must fulfill is that of being compatible with the given topology. For X C T, let

hi(X) = ~ { 0 e V 1(30')X C_O' ~ 0}. Moreover, let ~/= (C) be a further topology on T, and C: w -~ C (onto) be an indexing of C. Then ~/is said to be compatible with ~-, if there are functions s E p(2) and r E p(3) such that for all i E dom(x), n E dom(B) and rn E dora(C) the following hold: 1. I f x i ECm, then s(i, rn)l, s(i, ra) E dom(M) and xi E M,(~,,n) C Crn. 2. If moreover Bn ~ Cra, then also r(i, n, ra)[, r(i, n, m) E dom(x) and xr(~,n,m) E hl(Bn) \

M,(~,~). This condition is weaker than the corresponding one in (Spreen 1990). Nevertheless, with only minor changes in the proof one can show: P r o p o s i t i o n 3.2 Let T be strongly effective and B be acceptable. Then any Mal'cev topology

that is compatible ~ith ~" is recursivcly coarser than ~. If T is also reeursively separable, than any such topology is even Lacombe coarser than ~-. Under the assumption that T is strongly effective and recursively separable and B is acceptable, it is shown in (Spreen 1991, Theorem 6) that every regular-based Mal'cev topology on T is compatible with r. With Lemma 3.1 we thus obtain: T h e o r e m 3.3 Let Tbe strongly effective, rccursively separable and serni-reoular. Moreover, let

B be acceptable and co-computable. Then r is Lacombe equivalent urith T~.

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As we have seen in Proposition 3.2, any Mai'cev topology that is compatible with r is recursively coarser than r. Conversely, we have for every topology ~/= (C) on T with an indexing C: to ~ C (onto) of its basis:

1. Let B be eompubable and 11 be Lacombe coarser than -c. Then C is also computable. ~. Let T be strongly effective, reeursively separable and semi-regular. Moreover, let B be acceptable and co-computable. Then, if17 is recursively coarser than r, it is also compatible with ~'.

P r o p o s i t i o n 3.4

P r o o f : Statement (1) is obvious, since C 3, 0~ is a type symbol denoting the fiat domain with i elements. 2. ~ is a type symbol denoting the fiat domain w• with eountably infinitely many elements. 3. If a and fl respectively are type symbols denoting Sa and ~q0~then a • fl and a --* respectively are type symbols denoting So x,,q~ and [Sa --*c ~q~]. Obviously, the basis of each fiat domain is finitely bounded-complete and ~ k I e(D~) is bounded } is recursive. Moreover, these properties are hereditary to the product and the function domain (cf. Weihrauch and Dell 1980). Thus, all the domains ,q~ exist. Let T p be the set of all of them. P r o p o s i t i o n 6.4 For any type symbol a, 5a is strongly constructive and i~s compact elements satisfy condition (R). Hence T p C.C_SI'tS.

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P r o o f : The proof is done by induction on the complexity of the type symbols. Note that [791 ~ c (792 xc 793)] and [791 "4c [792 --4r P3]] respectively are recursively isomorphic to [791 --*c 792] Xc IT'1 ~ c 793] and [(791 xc 792) -% 793], which means that there are monotone bijections between the bases of the domains such that both the map and its inverse are effective with respect to the indexings of the basis elements. These bijectious can be extended in a canonical way to bijections between the domains which are computable in both directions. Thus, it suffices to consider only those ,S~ E T p which are either flat, a product domain, or a function domain with a flat domain as target. If 3a is flat, we have already seen that it is strongly constructive. Since it has at least cardinality 3, all of its elements satisfy condition

Ca).

Obviously, these properties are hereditary to product domains. Hence, it remains to consider the case that 8~ = [3al --% Sa2] with •a2 being flat. As we have already seen, ,qo is algebraic. We now show that it is also strongly constructive. Because Zal is dense in Sai, it follows that c{ T / e S iff there is some z e Z / s u c h that for all y E Z m with eli(y) :~ _L~,, or eS(y ) :~ J.,,, we have that eli(y) ---a2 z(y) and eS(y ) U_a, z(y), which since Sa, is flat and hence also finitely bounded-complete is the case, iff for all y E Z~t with eli(y) ~ l a 2 and c;(y) ~ .l.o, we have that eli(y) = eSj(y). { k [ e{(eg') ~ / a , } and { k [ e~(eg') ~ .l.a, } are finite sets, whose D-indices can be computed from i and j respectively. As we have seen above, { (a, b) [ e~'~ ---a2 c~2 } is recursive. Thus, also { (i,j) l eli t ! cf } is recursive. Finally, we have to show that all elements in Z ! satisfy condition (R). Obviously, it suffices to consider only elements of the form u --* v. Let z -- sup{ui --* v~ [ i E I } E Z ! with u --* v {Zf z. Without restriction we assume that vi ~ / ~ 2. Then, for each J C_ I, { vj [ j E J } is hounded, if { uj [ j E J } is bounded. But since ~qa2 is flat, { v j ] j E J } is bounded, only if there is some c E Za2 such that vj = c, for all j E J. Hence, there are a partition J1 . . . . , Jr of I and pairwise different elements vi~, . . . , vi~ E { vi[ i E I } such that z = sup { sup{ uj --* v~k [ j E Jk } ] 1 < k < r }. Moreover, if for J C I { uj ] j E J } is bounded, then there is some k with 1 < k < r such that J C_ Ja. As it is easy to see, u --* v [~l z iffv 17-a2 z(u), which is the case iffv [~a2 sup{ vi I ui __.~ u }. Since { ui ] ui ---m u } is bounded, there is some k with 1 < k < r such that { i E I I ui _Ca~ u } C_ Jk- Then sup{ v~ I ui ---m u } = v~k. Hence, u -4 v ~ ! z iff v IZ~2 v~. Because the elements of Za2 satisfy condition (R), there exists an element c E Za2 with v~h E a 2 c and v ~'a~ c. Define s = s u p { sup{ u~ -~ c IJ E J/~}, sup{sup{uj --* v,~ IJ E Jk' } I 1 _< k' < r, k' ~ k} }. Now, let y E Sa~. If there is no i 6 I with u~ E_m y, then z(y) = .l-a~ = ~.(y). In the other case there is some k' with 1 < k' < r such that { i E I [ u~ E m y } C Jk,. If k' ~ k, it follows that z(y) = vik, = ~(y), and if k' = k, we have that z(y) = vi~ ~_a~ c = s Thus z U_! ~. Next, assume that there is some w ~ Z ! with u --* v __.1w and ~ _~I v. Then v ~-a2 w(u) and ~.(u) U_a~ w(u). But s = c and e ~'~a v. Hence u -~ v X! $. Note that this result is false, if one starts from nonflat domains. Obviously, T p contains the constructive part of the type hierarchy of all partial continuous functionals over the natural numbers, i.e. the Error-Scott higher type partial computable functionals. From results in (Sprecn 1991) we obtain that every effective map from a domain in T p into a space in S R S has a witness for noninclusion. With Propositions 4.1 and 4.3 we thus have: P r o p o s i t i o n 6.5 Let S E T p and T ~ SITS. Then a map F: S 4.4 T is effectively continuous,

iff it is effective. This extends the well-known Myhill-Shepherdson theorem on the effective continuity of effective operators on the partial recursive functions.

7

A Counterexample

From the results presented so far, it follows that effectivemappings between domains in Tp, between recursively separable recursive metric spaces and from domains in T p into recursively separable recursive metric spaces always have a witness for noninclusion. But as a modification of an example of Friedberg (cf.Friedberg 1958, Rogers 1988) shows, Theorem 4.4 is falsewithout the condition of having a witness for noninclusion.

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P r o p o s i t i o n 7.1 There is a rvcursively separable recursive metric space .M, a flat domain {2 with three elemenls, and a map F: M -* Q which is effective, but not continuous, and hence has no witness for noninelusion. P r o o f : Let M be the set RO) of all total recursive functions. As is well known, RO) with the Baire metric 6 defined by 8(f,g) = 0, if f -- 9, and 5(f,9) = 2 -t'n:$(n)#Kn), otherwise, is a recursive metric space. The set of all functions with value 0 almost everywhere is an effectively enumerable dense subset. Moreover, let Q be the fiat domain with the elements { J_, uo, Ul }, and let io and ii respectively be indices of uo and Ul with respect to an appropriate indexing e. Define h E pO) by

io, hCi) =

il, undefined,

if

[(Va < i)~pi(a) ---- O] V (::::ic)[~oe(c) r 0 ^ (va < : ) ~ ( a ) = 0,', (3j < :)(Vb < e)~,(b) = ~Ab)], if (3c < i)[~(c) # 0 ^ (Va < c)~,(a) = 0 ^ (Vj < c)(3b _< e)~oj(b)~ A ~oe(b) # ~ojCb)], otherwise.

As it is readily verified, for all ~oi,~oj E R (1) with ~o~ = ~oj one has that h(i) = h(j). Let x be an admissible indexing of Q. Then there is a function d E R (1) with xd(O = sup e(W~), for all i E w such that e(Wi) is directed. L e t p E R (1) with Wp(i) -- {h(i) }, and set t = d o p . We define the effective mapping F: R (1) ~ Q by F(~oi) = xt(0. Then F(~o~) = uo and F ( ~ ) = ul respectively, if the first or the second condition in the definition of h holds. In any other case F(~o~) = _L.. Now, assume that F is continuous. Since F(An.0) = uo and { uo } is open in the Scott topology on Q, by the definition of the Baire metric there is some m > 0 such that for all g E R0) with g(a) = O, for a < m, we have that F(g) = uo. Set k = max{~o~(m) + 1 [ i < m, ~oi E R0) } and define f 0, ira=tim, ~(a) k, otherwise. Then ~ E R0). Since ~(a) ----0, for all a < m, we have that F(~) = uo. On the other hand, because for j E w with ~ = ~oj it follows that j > m, and since ~(m) ~ 0, we obtain from the definition of F that F(~) = ul or F(~) = .L. Thus, F cannot be continuous.

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