Effectivity in Spaces with Admissible Multirepresentations

Math. Log. Quart. 48 (2002) Suppl. 1, 78 – 90

Mathematical LogicQuarterly

c© 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Effectivity in Spaces with Admissible Multirepresentations

Matthias Schroder

FernUniversitat Hagen, D-58084 Hagen, Germany1)

Abstract. The property of admissibility of representations plays an important role inType–2 Theory of Effectivity (TTE). TTE defines computability on sets with continuumcardinality via representations. Admissibility is known to be indispensable for guaranteeingreasonable effectivity properties of the used representations.

The question arises whether every function that is computable with respect to arbritraryrepresentations is also computable with respect to closely related admissible ones. We definethree operators which transform (multi–) representations into admissible ones in such a waythat relative computability of functions is preserved. Thus the use of admissible (multi–)representations rather than of non–admissible ones does not decrease the class of relativelycomputable functions.

Keywords: Type–2 Theory of Effectivity, Multirepresentations, Admissibility, SequentialTopological Spaces, (Weak) Limit Spaces, Equilogical Spaces.

1 Introduction

Type–2 Theory of Effectivity (TTE) supplies a computational framework for uncount-able sets which are equipped with a natural approximation structure (cf. [16, 8, 13,14, 15]). The basic idea is to represent the points of a set X by elements of theBaire space B := N

N and to perform the actual computation on these names. Thecorresponding partial surjection δ :⊆ B → X mapping every name p ∈ dom(δ) to itsencoded element δ(p) is called a representation of the set X.

In this paper we consider the generalized concept of multirepresentations. Theyallow, in contrast to representations, a name to represent more than one element. Amultirepresentation δ :⊆ B⇒ X of a set X is a surjective correspondence

(B, X, G(δ)

)(cf. Subsection 1.1). Surjectivity means that every element of X has a name, i.e.

(∀x ∈ X)(∃p ∈ B)x ∈ δ[p] :={y ∈ X

∣∣ (p, y) ∈ G(δ)} .If δ is a multirepresentation of X, then we call (X, δ) a represented space.

Computability for functions between represented spaces is introduced by transfer-ring the usual computability notion for functions on the Baire space (cf. [13, 14, 16])via the considered multirepresentations. Generalizing the concept of relative com-putability w.r.t. representations, we define f :⊆ X → Y to be computable with respect

1)e-mail: [email protected]

c© 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0942-5616/02/4811-0078 $ 17.50+.50/0

Effectivity in Spaces with Admissible Multirepresentations 79

to multirepresentations δ :⊆ B ⇒ X and γ :⊆ B ⇒ Y (for short (δ, γ)–computable),iff there is a computable function g :⊆ B → B realizing f w.r.t. δ and γ, which means

(1)(∀p ∈ B)

(∀x ∈ δ[p]

)f(x) ∈ γ[g(p)] .

Since computable functions on B are topologically continuous w.r.t. Baire’s topology,the following notion of continuous realizability is a natural topological generalizationof relative computability. The function f is called continuously realizable w.r.t. δand γ, iff f has a continuous realizer g satisfying (1). Relative computability andcontinuous realizability of multivariate functions are defined accordingly.

A multirepresentation induces three types of approximations structures on therepresented set (cf. Section 3), namely a topology, the convergence relation of a limitspace and the convergence relation of a weak limit space (cf. Section 2). Weak limitspaces are a generalization of limit spaces and thus of topological spaces. Relativelycomputable functions turn out to be continuous w.r.t. each of these approximationstructures (cf. Proposition 3.1).

Most multirepresentations of a space lead to an unsatisfactory class of relativelycomputable functions. In many cases they are unsuitable simply for topological rea-sons. A well–known example is the ordinary decimal representation of R which doesnot admit continuous realizability of multiplication by 3. Admissible multirepresen-tations are in a topological sense well–behaved (cf. [8, 11]). Admissibility is definedin such a way that exactly the continuous functions between the represented spacesare continuously realizable w.r.t. admissible multirepresentations (cf. Section 3).

We define three operators on multirepresentations. Each of them transforms amultirepresentation δ into an admissible multirepresentation of one of the three afore-mentioned kinds of quotient spaces generated by δ (cf. Section 5). We prove that theseoperators preserve relative computability (cf. Section 6). Thus the use of admissiblemultirepresentations actually increases the class of relatively computable functions,and not only the class of continuously realizable functions (cf. [11]).

1.1 Preliminaries

A correspondence (or multi–valued function) F :⊆ X ⇒ Y between sets X, Y is atriple (X, Y, G(F )), where G(F ) is a relation G(F ) ⊆ X × Y . For x ∈ X and M ⊆ X,we denote by F [x] the set {y ∈ Y | (x, y) ∈ G(F )}, by F [M ] the set

⋃{F [a] | a ∈M},

by dom(F ) the set {a ∈ X |F [a] �= ∅} and by F−1 :⊆ Y ⇒ X the inverse of F . Weview any (partial) function f :⊆ X → Y as a correspondence f :⊆ X ⇒ Y satisfying(y1 ∈ f [x] ∧ y2 ∈ f [x]) =⇒ y1 = y2 for every x ∈ X.

For p ∈ B, n ∈ N and a word w ∈ N∗, we denote by p<n the prefix p(0) . . . p(n−1)

of p of length n and by wB the set {p ∈ B | (∃n ∈ N) p<n = w}.

2 Limit Spaces and Related Notions

We define a convergence space to be a pair X = (X,→X), where X is a set and →X

is a subset of XN ×X called the convergence relation of X. A sequence (xn)n ∈ XN

is said to converge to an element x∞ ∈ X w.r.t. →X, iff (xn)n →X x∞ holds. In thissituation we also say that the (generalized) sequence (xn)n≤∞ (which formally is afunction from N

+ := N ∪ {∞} = {0, 1, . . .} ∪ {∞} to X) is a convergent sequence ofthe convergence space X = (X,→X).

80 Matthias Schroder

For every topological space Z = (Z, τ), we denote by →τ the convergence relationinduced by the topology τ (cf. [5, 6, 12]). If no misunderstandings can occur, wewill sometimes write Z for the convergence space (Z,→τ ). On the set N

+ we willalways use the convergence relation →

N+ induced by the countably–based topology

τN+ := {O ⊆ N+ |∞ ∈ O =⇒ (∃n0 ∈ N)(∀n ≥ n0)n ∈ O} and on B the one induced

by Baire’s countably–based topology τB :={⋃

{wB |w ∈W}∣∣W ⊆ N

∗}.Let Y = (Y,→Y) be a further convergence space. We define a correspondence

F :⊆ X ⇒ Y to be continuous with respect to →X and →Y (for short (→X,→Y)–continuous), iff F preserves convergent sequences, i.e. (xn)n →X x∞ and (yn)n≤∞ ∈∏n∈N+ F [xn] implies (yn)n →Y y∞. For functions, this notion of continuity is equal

to the usual one of sequential continuity (cf. [4, 5, 6, 7]). Continuity of multivariatefunctions is defined accordingly. Note that a sequentially continuous function betweentopological spaces need not be topologically continuous (cf. [5, 6]).

We define X = (X,→X) to be a weak limit space (cf. [11]), iff X satisfies Axioms(L1), (L4), (L5). The pair X is called a limit space, iff Axioms (L1), (L2), (L3) aresatisfied (cf. [4, 9, 10]):(L1) (x)n →X x;(L2) if (yn)n →X x and ϕ : N→ N is strictly increasing, then (yϕ(n))n →X x;(L3) if (yn)n �→X x, then there is a strictly increasing function ϕ : N → N such that

(yϕψ(n))n �→X x for all strictly increasing function ψ : N → N;(L4) if (xn)n →X x∞, then the function x : N

+ → X is (→N+ ,→X)–continuous;

(L5) if (yn+1)n →X x and y0 ∈ X, then (yn)n →X x.Every limit space is a weak limit space (cf. [11]). It is well–known that every topo-logical space Z is a limit space.

Let X = (X,→X) be a weak limit space. We define L(X) to be the limit space whoseconvergence relation →L(X) is the finest (smallest) one containing →X and satisfyingAxiom (L3). A subset O ⊆ X is called sequentially open in X, iff every sequence(xn)n that converges to an element of O is eventually in O. The family seq(→X) of allsequentially open sets of X turns out to be a sequential topology2) on X. We denotethe sequential topological space (X, seq(→X)) by T (X) and its convergence relationby →T (X).

P r o p o s i t i o n 2.1. Every continuous function f : X×Y → Z between weak limitspaces X = (X,→X), Y = (Y,→Y), and Z = (Z,→Z) is also continuous w.r.t. →T (X),→T (Y) and →T (Z). The analogue holds for the operator L.

P r o o f. Let (xn)n≤∞ and (yn)n≤∞ be convergent sequences of T (X) and Y,respectively. Let V be a sequentially open set of Z containing f(x∞, y∞). Since(f(x∞, yn))n converges in Z to f(x∞, y∞) by (→X,→Y,→Z)–continuity of f , there issome n1 ∈ N with (∀n ≥ n1) f(x∞, yn) ∈ V . We define U by

U :={x ∈ X

∣∣ (∀n ∈ {n1, . . . ,∞})f(x, yn) ∈ V}.

Suppose for contradiction that U is not sequentially open. Then there is a sequence(an)n in X \ U that converges in X to some a∞ ∈ U . For every n ∈ N there issome kn ∈ {n1, . . . ,∞} with f(an, ykn) /∈ V . Since (yn)n≤∞ is a convergent sequence

2)A topological space is called sequential ([5, 6]), iff every sequentially open set is open.


of Y, there is some strictly increasing ϕ : N → N and some b ∈ {n1, . . . ,∞} with(ykϕ(n))n →Y yb. By continuity of f , (f(aϕ(n), ykϕ(n)))n converges to f(a∞, yb) inZ. As f(a∞, yb) ∈ V , this implies that (f(aϕ(n), ykϕ(n)))n is eventually in V , acontradiction.

Since (xn)n converges to x∞ in T (X), (xn)n is eventually in U . Hence (f(xn, yn))nis eventually in V . We conclude that (f(xn, yn))n converges to f(x∞, y∞) in T (Z) andthat f is (→T (X),→Y,→T (Z))–continuous. From this result we obtain by an analogousconsideration that f is (→T (X),→T (Y),→T (T (Z)))–continuous. As T (T (Z)) = T (Z),this concludes the proof for T .

We omit the straightforward proof for L. �By WeakLim (Lim, Seq) we denote the category of weak limit spaces (limit space,

sequential topological spaces) and of total sequentially continuous functions. Theoperators T and L can be extended to functors from WeakLim to, respectively, Seqand Lim by defining T (f) := L(f) := f . By Proposition 2.1, T and L preserve finiteproducts. Moreover, T and L can be shown to be left–adjoints to the correspondinginclusions of Seq and Lim into WeakLim.

3 Quotient Spaces and Admissible Multirepresentations

Let X = (X,→X) be a weak limit space. Any multirepresentation δ :⊆ B⇒ X inducestwo convergence relations→δ,�δ and a sequential topology τδ on the represented setX. They are defined by

(xn)n →δ x∞ :⇐⇒(∃(pn)n≤∞

)((pn)n →B p∞ ∧ (∀n ∈ N

+)xn ∈ δ[pn])

�δ := →L(X,→δ) and τδ :={O ⊆ X

∣∣ (∃U ∈ τB)(δ−1[O] ⊆ U ∧ δ[U ] ⊆ O)}. One

can easily verify that (X,→δ) is a weak limit space and that the final topology τδ isequal to seq(→δ). We call (X,→δ) the WeakLim–quotient generated by δ, (X,�δ) theLim–quotient generated by δ, and (X, τδ) the topological quotient generated by δ. Wesay that δ is a WeakLim–quotient multirepresentation of X = (X,→X), iff →δ =→X.

P r o p o s i t i o n 3.1. Let δ :⊆ B ⇒ X and γ :⊆ B ⇒ Y be multirepresentations,and let f :⊆ X → Y be continuously realizable w.r.t. δ and γ. Then f is (→δ,→γ)–continuous. If f is total, then f is (τδ, τγ)–continuous and (�δ,�γ)–continuous.

P r o o f. Let (xn)n≤∞ be a convergent sequence of (X,→X) contained in dom(f),and let g :⊆ B → B be a computable and thus continuous realizer of f . Thereis a convergent sequence (pn)n≤∞ of (B,→B) with (xn)n≤∞ ∈

∏n∈N+ δ[pn]. Then

(g(pn))n≤∞ is a convergent sequence with (f(xn))n ∈∏n∈N+ γ[g(pn)]. Since γ is

(→B,→γ)–continuous, (f(xn))n converges to f(x∞) in (Y,→γ). The second statementfollows from the first and from Proposition 2.1. �

Generalizing the definition in [11] for representations, we define δ to be an admis-sible multirepresentation of the weak limit space X, iff δ satisfies (a) and (b):

(a) δ is continuous w.r.t.→B and →X and(b) for every continuous correspondence φ :⊆ B ⇒ X there is continuous function

g :⊆ B → B which translates φ to δ, i.e. (∀p ∈ dom(φ))φ[p] ⊆ δ[g(p)].


The property (b) is called the universality of δ: an admissible multirepresentation of Xcontains in a way every continuous multirepresentation of X. This definition general-izes the one of admissible representations of countably–based T0–spaces in [8, 14, 16].By AdmWeakLim (AdmLim, AdmSeq) we denote the category whose objects are theweak limit spaces (limit spaces, sequential topological spaces) having an admissiblemultirepresentation and whose morphisms are the total continuous functions.

Admissibility guarantees the equivalence of continuous realizability and continuity.Th e o r em 3.2. Let δ be a WeakLim–quotient multirepresentation of a weak limit

space X = (X,→X), and let γ be an admissible one of a weak limit space Y = (Y,→Y).Then every function f :⊆ X → Y is continuous w.r.t.→X and →Y, if and only if f iscontinuously realizable w.r.t. δ and γ. The analogue holds for multivariate functions.

P r o o f. Similar to [11, Theorem 6]. �P r o p o s i t i o n 3.3. Every admissible multirepresentation δ of a weak limit space

X = (X,→X) satisfies →δ =→X and τδ = seq(→X).P r o o f. Similar to [11, Lemma 5 and Proposition 8]. �

4 Products and Exponentials

Let X = (X,→X), Y = (Y,→Y) and Z = (Z,→Z) be weak limit spaces. We equip theCartesian product X × Y with the convergence relation →X⊗Y defined by

(xn, yn)n →X⊗Y (x∞, y∞) :⇐⇒ (xn)n →X x ∧ (yn)n →Y y∞ .

One easily verifies that X⊗Y := (X×Y,→X⊗Y) is a weak limit space. Note that a func-tion f :⊆ X × Y → Z is (→X⊗Y,→Z)–continuous, iff it is (→X,→Y,→Z)–continuous.

By C(X,Y) we denote the set of all total continuous functions from X to Y. Weequip C(X,Y) with the convergence relation� of continuous convergence: a sequence(fn)n of continuous functions is said to converge continuously to f∞ ∈ C(X,Y) (forshort (fn)n � f∞), if and only if the transpose

ft : N+ ×X → Y, ft(n, x) := fn(x)

is (→N+ ,→X,→Y)–continuous. One can easily verify that in the case of limit spaces

this definition of continuous convergence is equivalent to the ordinary one (cf. [10]),i.e. (fn)n � f∞, iff (xn)n →X x∞ implies (fn(xn))n →Y f∞(x∞). From [11] weknow that the function space C(X,Y) := (C(X,Y),�) is actually a weak limit space.Furthermore, C(X,Y) is an exponential of X and Y in the category WeakLim. HenceWeakLim is cartesian–closed.

Now let δ :⊆ B⇒ X and γ :⊆ B⇒ Y be multirepresentations.We define the multirepresentation δ ⊗ γ of X × Y by

δ ⊗ γ[〈p, q〉] := δ[p]× γ[q]

for p, q ∈ B, where 〈·, ·〉 : B× B → B denotes the computable bijection defined by

〈p, q〉 := p(0)q(0)p(1)q(1)p(2)q(2) . . .

In other words, r ∈ B is a δ⊗ γ–name of (x, y) ∈ X × Y , iff π1(r) is δ–name of x andπ2(r) is γ–name of y, where π1 := pr1 ◦ 〈·, ·〉−1 and π2 := pr2 ◦ 〈·, ·〉−1 are the twocomputable projections associated with the pairing function 〈·, ·〉.


For defining an admissible multirepresentation of C(X,Y), we equip Fωω, theset of all continuous functions g :⊆ B → B having a Gδ–domain, with an effectiverepresentation η : B → Fωω satisfying the following properties:

(1) the universal function (p, q) �→ ηp(q) is computable (utm–Theorem);(2) for every computable function g :⊆ B

2 → B there is some computable functionh : B → B with (∀p, q ∈ B) ηh(p)(q) = g(p, q) (computable smn–Theorem);

(3) for every continuous function g :⊆ B2 → B there is some continuous function h :

B → B with (∀(p, q) ∈ dom(g)) ηh(p)(q) = g(p, q) (continuous smn–Theorem).

These properties guarantee that B and the application function (α|β) := ηα(β) forma partial combinatory algebra (cf. [2, 3]). Such a representation η can be found in[13, 14, 2, 3] or constructed from one for the Cantor space (e.g. the one in [16]).

With the help of η, we define the multirepresentation [δ → γ] of the set

C(δ, γ) := {f : X → Y | f is continuously realizable w.r.t. δ and γ}by

f ∈ [δ → γ][p] :⇐⇒ ηp realizes f with respect to δ and γ .

Since every continuous function g :⊆ B → B has an extension in Fωω, [δ → γ] isindeed surjective. If γ is an admissible multirepresentation of Y and δ generates X asits WeakLim–quotient, then [δ → γ] is a multirepresentation of C(X,Y) by Theorem3.2.

A multirepresentation δ is said to be computably translatable to another multirep-resentation δ′ (for short δ ≤cp δ′), iff there is a computable translator g :⊆ B → B

with (∀p ∈ dom(δ)) δ[p] ⊆ δ′[g(p)]. By δ ≡cp δ′ we denote computable equivalencedefined by δ ≤cp δ′ ≤cp δ. Similar to [16], where only single–valued representationsare considered, the following effectivity properties of the operators ⊗ and [· → ·] canbe shown.

P r o p o s i t i o n 4.1. Let δ, δ′ :⊆ B ⇒ X, γ, γ′ :⊆ B ⇒ Y and ζ :⊆ B ⇒ Z bemultirepresentations.

(1) A function f : X × Y → Z is computable w.r.t. δ ⊗ γ and ζ, iff f is (δ, γ, ζ)–computable, and iff the transposed function Λ(f) : X → C(γ, ζ) defined byΛ(f)(x)(y) := f(x, y) is computable w.r.t. δ and [γ → ζ].

(2) If δ ≤cp δ′ and γ ≤cp γ′, then δ ⊗ γ ≤cp δ′ ⊗ γ′, [δ′ → γ] ≤cp [δ → γ′], and(δ′, γ)–computability implies (δ, γ′)–computability.

(3) A function f : X → Y is computable w.r.t. δ and γ, iff there is a computablefunction s : N→ N with f ∈ [δ → γ][s].

Now we prove that the operators ⊗ and [· → ·] preserve admissibility.

P r o p o s i t i o n 4.2. Let X = (X,→X) and Y = (Y,→Y) be weak limit spaces.

(1) If δ and γ are admissible multirepresentations of X and Y, respectively, thenδ ⊗ γ is an admissible multirepresentation of X⊗Y.

(2) If δ is a WeakLim–quotient multirepresentation of X, and γ is an admissible oneof Y, then [δ → γ] is an admissible multirepresentation of C(X,Y).


(3) If γ is an admissible multirepresentation of Y and h : X → Y reflects convergentsequences3), then h−1 ◦ γ is4) an admissible multirepresentation of X.

P r o o f. (1) Clearly, δ ⊗ γ is continuous. For every continuous correspondenceφ :⊆ B ⇒ X × Y , the projections pr1 ◦ φ and pr2 ◦ φ are continuous as well.By admissibility of δ and γ, there are continuous functions g1, g2 :⊆ B → B

translating pr1 ◦φ to δ and pr2 ◦φ to γ, respectively. Obviously, the continuousfunction g :⊆ B → B defined by g(p) := 〈g1(p), g2(p)〉 translates φ to δ ⊗ γ.

(2) Continuity: Let (pn)n≤∞ be a convergent sequence of (B, τB), and let (fn)n≤∞ ∈∏n∈N+ [δ → γ][pn]. Let (kn)n≤∞ and (xn)n≤∞ be convergent sequences of

(N+,→N+) and X, respectively. Since →δ =→X, there is a convergent sequence

(qn)n≤∞ of (B, τB) with (xn)n≤∞ ∈∏n∈N+ δ[qn]. For every n ∈ N

+, we havefkn(xn) ∈ γ[ηpkn

(qn)], because ηpknrealizes fkn w.r.t. δ and γ. By continu-

ity of (p, q) �→ γ[ηp(q)], (fkn(xn))n converges to fk∞(x∞). We conclude thatthe “universal” function of (fn)n≤∞ is continuous and that (fn)n convergescontinuously to f∞.Universality: Let φ :⊆ B⇒ C(X,Y) be continuous. We define ψ :⊆ B⇒ Y by

ψ[r] :={f(x)

∣∣ f ∈ φ[π1(r)] and x ∈ δ[π2(r)]}.

Let (rn)n≤∞ be a convergent sequence of (B, τB) and (yn)n≤∞ ∈∏n∈N+ ψ[rn].

For every n ∈ N+, there are fn ∈ φ[π1(rn)] and xn ∈ δ[π2(rn)] with fn(xn) = yn.

By continuity of φ and δ, we have (fn)n � f∞ and (xn)n →X x∞. Thus(yn)n →Y y∞. Hence ψ is continuous. By admissibility of γ, there is a contin-uous function g :⊆ B → B with (∀r ∈ dom(ψ))ψ[r] ⊆ γ[g(r)]. The continuoussmn–Theorem yields a continuous function s : B → B with ηs(p)(q) = g〈p, q〉 forall 〈p, q〉 ∈ dom(g) ⊇ dom(ψ).Let p, q ∈ B, f ∈ φ[p] and x ∈ δ[q]. Then f(x) ∈ ψ[〈p, q〉] ⊆ γ[g〈p, q〉] =γ[ηs(p)(q)]. This implies f ∈ [δ → γ][s(p)]. Hence s translates φ continuouslyto [δ → γ].

(3) Continuity: Let (pn)n≤∞ be a convergent sequence of (B, τB), and let (xn)n≤∞ ∈∏n∈N+ h−1 ◦ γ[pn]. By continuity of γ, (h(xn))n converges to h(x∞). Since h

reflects convergent sequences, (xn)n converges to x∞.Universality: Let φ :⊆ B ⇒ X be continuous. By continuity of h, h ◦ φ is alsocontinuous and thus continuously translatable to γ by some continuous functiong :⊆ B → B. One easily verifies that g also translates φ to h−1 ◦ γ.

�

5 The Multirepresentations δWe, δLi and δSi

We will now define three operators which transform multirepresentations δ into admis-sible ones of the WeakLim–quotient space, the Lim–quotient space and the topologicalquotient space generated by δ, respectively.

3)This means that h satisfies (xn)n →X x∞ ⇐⇒ (h(xn))n →Y h(x∞) for all sequences (xn)n≤∞.4)The multirepresentation h−1 ◦ γ :⊆ B⇒ X maps any p ∈ B to {x ∈ X | h(x) ∈ γ[p]}.


The idea is to embed a given space X = (X,→X) into the space C(C(X,U),U),where U = (U,→U) is an appropriate space equipped with an admissible multirep-resentation 'U. Proposition 4.2 tells us that [[δ → 'U] → 'U] is an admissiblemultirepresentation of C(C(X,U),U), if δ is a multirepresentation generating X asits WeakLim–quotient. If additionally the obviously continuous transpose of the eval-uation function

eX,U : X → C(C(X,U),U) , defined by eX,U(x)(h) := h(x),

reflects convergent sequences3) , then by Proposition 4.2(3) the multirepresentationδU :⊆ B⇒ X defined by

(2) δU[p] :={x ∈ X

∣∣ eX,U(x) ∈ [[δ → 'U]→ 'U][p]}

is an admissible multirepresentation of X.Obviously, the Sierpinski–space Si has the property that, for all sequential spaces

X, eX,Si reflects convergent sequences. Its underlying set is Si := {⊥,"} and itsconvergence relation →Si is induced by the topology τSi :=

{∅, {"}, {⊥,"}

}.

We define the limit space Li := (Li,→Li) and the weak limit space We :=(We,→We) by Li := {⊥, ↑ ,"}, We := Li ∪ {M |M ⊆ N}, and

(bn)n →Li b∞ :⇐⇒ b∞ �= " or (∃m ∈ N)(∀n ≥m) bn ∈ {↑,"}(bn)n →We b∞ :⇐⇒ b∞ �= " or (∃j, l ∈ N)(∀n ≥ l)(j ∈ bn ∨ bn ∈ {↑ ,"}) .

Admissible multirepresentations 'We : B ⇒ We, 'Li : B ⇒ Li, and 'Si : B → Si ofthe spaces We, Li, and Si can be defined by

'We[p] :={ {

↑ ,"}∪

{M ⊆ N

∣∣ p(min{i | p(i) �= 0})− 1 ∈M}

if p �= 0ω{⊥, ↑

}∪

{M

∣∣M ⊆ N}

else,

'Li[p] := 'We[p] ∩ Li and 'Si[p] := 'We[p] ∩ Si.

Th e o r em 5.1. Let δ be a multirepresentation of a set X.(1) δSi is an admissible multirepresentation of the sequential space (X, τδ).(2) δLi is an admissible multirepresentation of the limit space (X,�δ).(3) δWe is an admissible multirepresentation of the weak limit space (X,→δ).

P r o o f. For every continuous φ :⊆ B⇒We, the function g : B → B defined by

g(p) :={

0np(min{j ∈ N |φ[p<npB] ⊆ Uj

}+ 1

)0ω if np <∞

0ω else,

where Uj := {↑ ,"} ∪ {M ⊆ N | j ∈ M} and np := min({∞} ∪ {n ∈ N | (∃j ∈ N)

φ[p<nB] ⊆ Uj}), is a continuous translator from φ to 'We. Obviously, 'We is contin-

uous. Hence 'We is an admissible multirepresentation of We. In the same way, theadmissibility of 'Li and 'Si can be shown.(1) Let X := (X, τδ). Then eX,Si reflects convergent sequences, since we have

(xn)n →τδx∞ ⇐⇒

(∀h ∈ C(X,Si)

)(h(xn))n →Si h(x∞) .

Proposition 2.1 implies C((X,→δ),Si) = C(X,Si) and hence e(X,→δ),Si = eX,Si.Thus δSi is an admissible multirepresentation of (X, τδ) by Proposition 4.2.


(2) Let X := (X,�δ). We show that eX,Li reflects convergent sequences between Xand C(C(X,Li),Li). Let (xn)n≤∞ be a non–convergent sequence of X. There isa subsequence (x′n)n of (xn)n such that no subsequence of (x′n)n converges tox∞ by Axiom (L3). We define hn, h∞ : X → Li by

hn(y) :={⊥ if y = x′n↑ else and h∞(y) :=

{" if y = x∞↑ else

for n ∈ N and y ∈ X. Then (hn)n can be shown to converge to h∞ in C(X,Li).Since eX,Li(x′n)(hn) = ⊥ holds for every n, (eX,Li(x′n))n and hence (eX,Li(xn))ndo not converge to eX,Li(x∞). Thus eX,Li, being the continuous transpose ofthe evaluation function, reflects convergent sequences.Proposition 2.1 implies C((X,→δ),Li) = C(X,Li), since L(Li) = Li. Hencee(X,→δ),Li is equal to eX,Li. By Proposition 4.2, δLi is an admissible multirep-resentation of (X,�δ).

(3) By implicitly using the fact that X := (X,→δ) has a countable “limit base”(cf. [11]), we prove that eX,We reflects convergent sequences between X andC(C(X,We),We). Let (xn)n≤∞ be a non–convergent sequence of X. We choosea sequence of words w0, w1, . . . ∈ N

∗ with{wi

∣∣ i ∈ N}=

{w ∈ N

∗ ∣∣ x∞ ∈ δ[wB] and (∀m ∈ N)(∃n ≥ m)xn /∈ δ[wB]}

and define h : X →We by

h(x∞) := " and h(y) :={i ∈ N

∣∣ y ∈ δ[wiB]}for y �= x∞.

We show that h is continuous. Let (yn)n be a sequence that converges in X to x∞(this is the only critical case). By definition of→δ, there is some p ∈ δ−1[{x∞}]such that, for all k ∈ N, (yn)n is eventually in δ[p<kB]. From the continuity ofδ and Axiom (L5) it follows that there is some k ∈ N such that δ[p<kB] doesnot contain xn for infinitely many n (cf. [11, Lemma 1]). For l with wl = p<k,(h(yn))n is eventually in {"} ∪ {M ⊆ N | l ∈M}. Hence (h(yn))n converges toh(x∞) = ". We obtain h ∈ C(X,We).Since for every j ∈ N there are infinitely many n ∈ N with h(xn) /∈ {↑ ,"} ∪{M ⊆ N | j ∈ M}, (eX,We(xn)(h))n does not converge to eX,We(x∞)(h). Thisimplies (eX,We(xn))n �� eX,We(x∞). Thus the continuous function eX,We re-flects convergent sequences.From Proposition 4.2 we conclude that δWe = (eX,We)−1 ◦ [[δ → 'We] → 'We]is an admissible multirepresentation of X = (X,→δ).

�Note that each of these operators maps some representations to non single–valued

multirepresentations (namely if and only if the corresponding quotient space has notthe T0–property). This is the reason why we have to consider multirepresentations.

6 Computability Properties of δWe, δLi and δSi

We prove that the operators δ �→ δWe, δ �→ δLi, and δ �→ δSi preserve relativecomputability of total functions.

Let U = (U,→U) be a weak limit space equipped with an admissible multirepre-sentation 'U :⊆ B ⇒ U . For weak limit spaces X = (X,→X), Y = (Y,→Y), and


Z = (Z,→Z) and for continuous functions f ∈ C(X⊗Y,Z), h ∈ C(Z,U) and y ∈ Y wedefine the function HU(f) : Y × C(Z,U)→ C(X,U) by

HU(f)(y, h) :=(x �→ h(f(x, y))

).

Since for all convergent sequences (hn)n≤∞ of C(Z,U) and (yn)n≤∞ of Y the function(n, x) �→ hn(f(x, yn)) is (→N

+ ,→X,→U)–continuous, (HU(f)(yn , hn))n≤∞ is a conver-gent one of C(X,U). Consequently, the operator HU maps f to a continuous functionbetween the spaces Y⊗ C(Z,U) and C(X,U) and can also be applied to HU(f).

L emma 6.1. Let 'U :⊆ B ⇒ U be an admissible multirepresentation of a weaklimit space U = (U,→U). Let δ :⊆ B ⇒ X, γ :⊆ B ⇒ Y , and ζ :⊆ B ⇒ Z bemultirepresentations, and let X := (X,→δ), Y := (Y,→γ), Z := (Z,→ζ) be the weaklimit spaces generated by these multirepresentations.(1) For all functions f ∈ C(X⊗Y,Z), x ∈ X and y ∈ Y ,(

HU ◦ HU ◦ HU(f))(eX,U(x), eY,U(x)

)= eZ,U

(f(x, y)

).

(2) The function HU : C(δ ⊗ γ, ζ) → C(Y ⊗ C(Z,U),C(X,U)

)is computable w.r.t.

[δ ⊗ γ → ζ] and [γ ⊗ [ζ → 'U]→ [δ → 'U]].P r o o f. (1) H2

U(f) is a continuous function between C(Z,U), C(C(X,U),U) andC(Y,U), and H3

U(f) is a continuous one between C(C(X,U),U), C(C(Y,U),U)and C(C(Z,U),U). For every h ∈ C(Z,U) we have

H3U(f)

(eX,U(x), eY,U(y)

)(h) = eY,U(y)

(H2

U(f)(h, eX,U(x)))

= eY,U(y)(b �→ eX,U(x)(HU(f)(b, h))

)= eY,U(y)

(b �→ h(f(x, b))

)= h(f(x, y)) = eZ,U(f(x, y))(h) .

Thus H3U(f)

(eX,U(x), eY,U(y)

)= eZ,U(f(x, y)).

(2) By admissibility of 'U, Propositions 4.2, 3.3 and the fact that the ⊗–operator onmultirepresentations preserves the property of being a WeakLim–quotient mul-tirepresentation, [γ⊗ [ζ → 'U]→ [δ → 'U]] is an admissible multirepresentationof the function space C

(Y⊗ C(Z,U),C(X,U)

).

Two applications of the utm–Theorem and the computable smn–Theorem yielda computable function gH : B → B with

(∀q, r, s, t ∈ B) ηηgH(t)〈s, q〉(r) = ηq(ηt〈r, s〉

).

Let q, r, s, t ∈ B, f ∈ [δ⊗ γ → ζ][t], y ∈ γ[s], h ∈ [ζ → 'U][q] and x ∈ δ[r]. Then

HU(f)(y, h)(x) = h(f(x, y)

)∈ 'U

[ηq(ηt〈r, s〉)

]= 'U[ηηgH(t)〈s, q〉(r)]

and hence

HU(f)(y, h) ∈ [δ → 'U][ηgH(t)〈s, q〉] .Since (y, h) ∈ γ ⊗ [ζ → 'U]〈s, q〉, this implies

HU(f) ∈[γ ⊗ [ζ → 'U]→ [δ → 'U]

][gH(t)

].

Thus gH realizes HU w.r.t. [δ ⊗ γ → ζ] and[γ ⊗ [ζ → 'U]→ [δ → 'U]

].

�By this lemma, we can prove the following computable translation properties.


P r o p o s i t i o n 6.2. Under the assumptions of Lemma 6.1 we have:(1) δ ≤cp δ

U ≡cp (δU)U and 'U ≡cp ('U)U.(2) [δ → ζ] ≤cp [δU → ζU] and [δ ⊗ γ → ζ] ≤cp [δU ⊗ γU → ζU].(3) (δ ⊗ γ)U ≡cp δ

U ⊗ γU.(4) δWe ≤cp δ

Li ≤cp δSi.

P r o o f. (1) By the utm–Theorem and the computable smn–Theorem, there isa computable function g1 : B → B with (∀p, q ∈ B) ηg1(p)(q) = ηq(p).Let p, q ∈ B, x ∈ δ[p] and h ∈ [δ → 'U][q]. Then

eX,U(x)(h) = h(x) ∈ 'U[ηq(p)] = 'U[ηg1(p)(q)]

implying eX,U(x) ∈ [[δ → 'U]→ 'U][g1(p)] and thus x ∈ δU[g1(p)], i.e. δ ≤cp δU.

Since the identical function idB : B → B is computable, there is some com-putable qid : N → N with ηqid = idB (cf. [16]). Let p ∈ B and x ∈ ('U)U[p].Since qid is a ['U → 'U]–name of the identical function idU : U → U , we have

x = eU,U(x)(idU) ∈ 'U[ηp(qid)] .

Thus the function p �→ ηp(qid), being computable by the utm–Theorem, trans-lates ('U)U computably to 'U. Hence ('U)U ≤cp 'U.

It remains to prove (δU)U ≤cp δU. By (2) we have [δ → 'U] ≤cp [δU →('U)U]. Proposition 4.1(2) and ('U)U ≤cp 'U imply [δ → 'U] ≤cp [δU → 'U], i.e.there is a computable function g2 :⊆ B → B with

(∀q ∈ B) [δ→ 'U][q] ⊆ [δU → 'U][g2(q)] .

By the utm–Theorem and the computable smn–Theorem, there is a computablefunction g3 : B → B with (∀p, q ∈ B) ηg3(p)(q) = ηp(g2(q)).Let p, q ∈ B, x ∈ (δU)U[p] and h ∈ [δ → 'U][q]. Since h ∈ [δU → 'U][g2(q)] andthus h ∈ C((X,→δU),U), we have

eX,U(x)(h) = h(x) = e(X,→δU),U(x)(h) ∈ 'U

[ηp(g2(q))

]= 'U[ηg3(p)(q)] .

This implies eX,U(x) ∈ [[δ → 'U]→ 'U][g3(p)], x ∈ δU[g3(p)] and (δU)U ≤cp δU.

(2) Let gH : B → B be the computable function constructed in the proof of Lemma6.1(2). Let r, s, t ∈ B, f ∈ [δ ⊗ γ → ζ][t], x ∈ δU[r] and y ∈ γU[s]. Threeapplications of Lemma 6.1(2) show that H3

U(f) is contained in[([[δ → 'U]→ 'U]⊗ [[γ → 'U]→ 'U])→ [[ζ → 'U]→ 'U]

][g3H(t)

],

hence ηg3H(t) realizes H3U(f) w.r.t. [[δ → 'U] → 'U] ⊗ [[γ → 'U] → 'U] and

[[ζ → 'U]→ 'U]. Therefore

eZ,U(f(x, y)) = H3U(f)(eX,U(x), eY,U(y)) ∈ [[ζ → 'U]→ 'U][ηg3H(t)〈r, s〉]

by Lemma 6.1(1). This implies f(x, y) ∈ ζU[ηg3H(t)〈r, s〉] and therefore

f ∈ [δU ⊗ γU → ζU][gH ◦ gH ◦ gH(t)] ,

showing [δ ⊗ γ → ζ] ≤cp [δU ⊗ γU → ζU]. The first statement follows from thesecond.


(3) δU ⊗ γU ≤cp (δ ⊗ γ)U: The computable qid from the proof of (1) is a [δ ⊗ γ →δ ⊗ γ]–name of the identical function idX×Y . From the proof of (2) we knowthat the computable function ηg3H(qid) realizes idX×Y w.r.t. δU⊗γU and (δ⊗γ)U .This means that ηg3H(qid) translates δU ⊗ γU computably to (δ ⊗ γ)U.(δ⊗γ)U ≤cp δ

U⊗γU : The projection pr1 : X×Y → X is computable w.r.t. δ⊗γand δ (a computable realizer is π1 : B → B from Section 4). From [δ⊗γ → δ] ≤cp

[(δ⊗ γ)U → δU] and Proposition 4.1(3) it follows the existence of a computablefunction g1 :⊆ B → B realizing pr1 w.r.t. (δ⊗γ)U and δU. By the same argumentthere is a computable function g2 :⊆ B → B realizing the other projection w.r.t.(δ ⊗ γ)U and γU. One easily verifies that the computable function g :⊆ B → B

defined by g(p) :=⟨g1(π1(p)), g2(π2(p))

⟩translates (δ ⊗ γ)U computably to

δU ⊗ γU.(4) This follows from 'Li[p] = 'We[p] ∩ Li and 'Si[p] = 'Li[p] ∩ Si for all p ∈ B.

�Propositions 6.2 and 4.1 imply our main result: every relatively computable total

function is also computable w.r.t. the corresponding admissible multirepresentationsgenerated by each of the operators defined in Section 5.

Th e o r em 6.3. Let δ :⊆ B ⇒ X and γ :⊆ B ⇒ Y be multirepresentations, andlet f : X → Y be a total function. Then the following implications hold:

f is (δ, γ)–computable =⇒ f is (δWe, γWe)–computable

=⇒ f is (δLi, γLi)–computable =⇒ f is (δSi, γSi)–computable.

The analogue holds for multivariate functions.

We define EffWeakLim (EffLim, EffSeq) to be the category which has pairs (X, δ) asobjects, where X is a weak limit space (a limit space, a sequential topological space)and δ :⊆ B⇒ X is an admissible multirepresentation of X with δ ≡cp δ

We (δ ≡cp δLi,

δ ≡cp δSi); the morphisms between two objects (X, δ) and (Y, γ) are the total (δ, γ)–

computable functions between the underlying sets of X and Y. Propositions 4.1, 4.2,and 6.2 imply that these categories are cartesian closed.

Let TE : EffWeakLim → EffSeq and LE : EffWeakLim → EffLim be the functorsthat are defined by

TE(X, δ) := (T (X), δSi) , LE(X, δ) := (L(X), δLi) , TE(f) := LE(f) := f

for objects (X, δ) and (Y, γ) and for morphisms f : (X, δ)→ (Y, γ). From Proposition6.2 one can conclude by [1, Theorem 5.3.2] that the functors TE and LE are left–adjoints to the corresponding inclusion functors of EffSeq and EffLim into EffWeakLim.By Theorem 6.3, TE and LE preserve finite products (and also exponentials).

7 Final Remarks

Theorem 6.3 can be generalized to partial functions in the case of the operatorsδ �→ δWe and δ �→ δLi, but not in the topological case. This reflects the fact that thecategory Seq of sequential topological spaces is not locally cartesian closed in contrastto category Lim of limit spaces (cf. [10]).


AdmSeq and AdmLim turn out to be equal to the categories PQ and PQL, re-spectively, which have been introduced by M. Menni and A. Simpson in [10]. Themain idea of the proofs of the equalities PQ = AdmSeq and PQL = AdmLim is dueto A. Bauer (cf. [2, 3]). M. Menni and A. Simpson characterize PQ as the largestcommon full subcategory of Seq and ωEqu, the category of countably–based equilog-ical spaces, and PQL as the largest common full subcategory of ωEqu and Lim (cf.[10]).

Theorem 5.1 implies that the objects of AdmSeq are exactly the topological quo-tients of countably–based spaces. Moreover, the objects of AdmLim (AdmWeakLim)are precisely the Lim–quotients (the WeakLim–quotients) of countably–based spaces.Thus the class of spaces which can be equipped with a reasonable computabilitynotion is very natural.

References

[1] Asperti, A., and G. Longo, Categories, Types, and Structures: An Introduction toCategory Theory for the Working Computer Scientist. MIT–Press 1991.

[2] Bauer, A., The Realizability Approach to Computable Analysis and Topology. Ph.D.thesis, Carnegie Mellon University 2000. Available at http://andrej.com/thesis

[3] Bauer, A., A Relationship between Equilogical Spaces and Type Two Effectivity.Mathematical Logic Quarterly 48 (2002), (this issue).

[4] Dudley, R.M., On Sequential Convergence. Transactions of the American Mathemat-ical Society 112 (1964), 483–507.

[5] Engelking, R., General Topology. Heldermann, Berlin (1989).[6] Franklin, S.P., Spaces in which sequences suffice. Fundamenta Mathematicae 57

(1965), 107–115.[7] Kisynski, J., Convergence du Type L. Colloquium Mathematicum 7 (1960), 205–211.[8] Kreitz, C., and K. Weihrauch, Theory of representations. Theoretical Computer

Science 38 (1985), 35–53.[9] Kuratowski, K., Topology, Vol. 1. Academic Press, New York 1966.

[10] Menni, M., and A. Simpson, Topological and Limit-space Subcategories of Countably–based Equilogical Spaces. To appear in Math. Struct. in Comp. Sci. 12 (2002). Availableat http://www.dcs.ed.ac.uk/home/als

[11] Schroder, M., Admissible Representations of Limit Spaces. In: Computability andComplexity in analysis (J. Blanck, V. Brattka, and P. Hertling, eds.), LectureNotes in Computer Science 2064, Springer, Berlin 2001, 273–295.

[12] Smyth, M.B., Topology. In: Handbook of Logic in Computer Science, Vol. 1, Oxfordscience publications 1992, 641–761.

[13] Weihrauch, K., Type 2 Recursion Theory. Theoretical Computer Science 38 (1985),35–53.

[14] Weihrauch, K., Computability. Springer, Berlin 1987.[15] Weihrauch, K., A Foundation for Computable Analysis. EATCS Bulletin 57 (1995),

167–182.[16] Weihrauch, K., Computable Analysis. Springer, Berlin 2000.

(Received: January 4, 2002; Revised: April 25, 2002)

Effectivity in Spaces with Admissible Multirepresentations

Documents

Transcript of Effectivity in Spaces with Admissible Multirepresentations