LLogical Methods in Computer ScienceVolume 15, Issue 1, 2019, pp. 35:1–35:21https://lmcs.episciences.org/ Submitted Jan. 31, 2018Published Mar. 29, 2019
MOSCHOVAKIS EXTENSION OF REPRESENTED SPACES
DIMITER SKORDEVSofia University, Faculty of Mathematics and Informatics, Sofia, Bulgaria e-mail address : [email protected]fia.bg
Abstract.
Given a represented space (in the sense of TTE theory), an appropriaterepresentation is constructed for the Moschovakis extension of its carrier (with payingattention to the cases of effective topological spaces and effective metric spaces). Someresults are presented about TTE computability in the represented space obtained in this way.For single-valued functions, we prove, roughly speaking, the computability of any functionwhich is absolutely prime computable in some computable functions. A similar resultholds for multi-valued functions, but with an analog of absolute prime computability. Theformulation of this result makes use of the notion of computability in iterative combinatoryspaces – a notion studied by the author in other publications.
Introduction
By [We00, Theorems 3.1.6 and 3.1.7], TTE computability of functions is preserved undercomposition and primitive recursion. In the case of composition, multi-valued functionsare also admitted, but their composition is different from the usual one ( i.e. from the onereducible to composition of relations). In [We08] such closedness is proved with respect toflowchart programs in the general case of multi-valued functions in sets with generalized multi-representations. There are reasons to expect that all natural kinds of relative computabilityvia deterministic programs using given single-valued functions can be obtained as instancesof the notion of absolute prime computability introduced in [Mo69]. The notion in questionmakes sense also in the case of multi-valued given functions, but ceases in this case to beadequate for such a kind of study concerning TTE computability (this is due, in particular,to the fact that the usual composition of multi-valued functions does not preserve TTEcomputability ). Fortunately, the general notion of computability in iterative combinatoryspaces introduced by the present author ( cf. e.g. [Sk92]) embraces as particular instancesnumerous kinds of computability via programs (including absolute prime computability),and there is one among them which is appropriate for the case of programming throughgiven multi-valued TTE computable functions. Key words and phrases:
Moschovakis extension, represented space, effective topological space, effectivemetric space, TTE, partial function, multi-valued function, realization, abstract first order computability,computable, absolutely prime computable, combinatory space, iteration, first recursion theorem. The TTE computability of multi-valued functions considered here is of the kind from [We00]. It isreferred to as weak one in [Br03], where another computability notion for such functions is studied (seeDefinition 7.1 there).
LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.23638/LMCS-15(1:35)2019 c (cid:13)
Dimiter Skordev CC (cid:13) Creative Commons
Dimiter Skordev
Vol. 15:1
The definition of absolute prime computability given in [Mo69] uses a constructionwhich extends any given set by adding an additional object and building the closure withrespect to ordered pairs. The extended set built in this way is used as a universe where thecomputations are carried. In Section 1 of the present paper, two iterative combinatory spaces S ( X ) and S ( X ) are considered for any set X . Each of them is based on a semigroup offunctions in the Moschovakis extension of X . In the case of S ( X ), this is the semigroup ofthe unary single-valued partial functions with the composition operation as multiplication.In the case of S ( X ), the semigroup consists of the unary multi-valued partial functions, andthe other kind of composition mentioned above is used as multiplication. The computabilityin S ( X ) is equivalent to absolute prime computability (restricted to single-valued partialfunctions). The computability in S ( X ) is an analog of absolute prime computability, butit is different from it.TTE computability is usually considered for functions between represented spaces. InSection 2, a way is considered for turning into a represented space the Moschovakis extension X ∗ of the carrier X of any given represented space. Some particular cases are studied – forinstance, the case when the given represented space is the one generated by an effectivetopological space with carrier X or the one generated by an effective metric space withcarrier X . In these cases, it is shown that the corresponding represented space with carrier X ∗ can be generated by an effective space of the same kind with carrier X ∗ .In the last section, Section 3, the combinatory spaces S ( X ) and S ( X ) from Section 1are considered in the case when the given set is the carrier X of a represented space. It isshown that S ( X )-computability and S ( X )-computability preserve TTE computabilityin the case of single-valued functions and in the case of multi-valued ones, respectively.This allows applying the First Recursion Theorem from the general theory of iterativecombinatory spaces for establishing TTE computability of some concrete functions. Twoexamples are given in which X is the set of the real numbers and the usual computability ofreal numbers and functions is considered.It is worth noting that the considerations in Sections 2 and 3 can be generalized formulti-represented spaces. Remark.
Since many definitions of TTE notions in different sources, although usuallyessentially equipollent, diverge in some details, the versions used in this paper of mostdefinitions of such notions are explicitly indicated. Moschovakis extensions and some combinatory spaces of functions in it
The notion of Moschovakis extension.
Let X be an arbitrary set. A Moschovakisextension X ∗ of X is constructed by taking an object o not in X and building the closureof X ∪ { o } under the formation of ordered pairs, assuming the ordered pair operation ischosen in such a way that no element of X ∪ { o } is an ordered pair (see [Mo69], wherethe set X and the element o are denoted by B and 0, respectively). Finite sequences ofelements of X ∗ can be encoded by elements of X ∗ as follows: we use the element o as acode of the empty sequence, and, for any k ∈ N , we encode the sequence z , z , . . . , z k bythe ordered pair ( z , z ), where z encodes the sequence z , . . . , z k (for nonempty sequences, The bold symbol o used here should not be confused with the normal text symbol o used elsewhere bythe author for least elements of iterative combinatory spaces. ol. 15:1 MOSCHOVAKIS EXTENSION OF REPRESENTED SPACES 35:3 this encoding coincides with the one introduced in [Mo69, (1.9)]). For any n ∈ N , let n ∗ be the code of the n -term sequence of o ’s. The element n ∗ will be regarded also as a codeof the natural number n , thus the numbers 0 , , , , . . . will be encoded by the elements o , ( o , o ) , ( o , ( o , o )) , ( o , ( o , ( o , o ))) , . . . of X ∗ . Let N ∗ = { n ∗ | n ∈ N } , where N , as usual, isthe set of the natural numbers. Having in mind the above or similar encodings, we mayrepresent partial functions in X ∗ of any number of arguments by unary partial functionsin X ∗ , and partial functions of any number of arguments from X ∗ to N by unary partialfunctions from the set X ∗ to its subset N ∗ . Partial functions of any number of argumentsfrom N to X ∗ and such ones in N can be similarly represented by unary partial functionsin X ∗ .1.2. Combinatory spaces. The spaces S ( X ) and S ( X ) . We will recall below thedefinition of the notion of combinatory space from [Sk92] (Definition 1 on pp. 45–46 there).For the sake of ease of comprehension, we, however, will first give an example of combinatoryspace (in this example which is an instance of Example 1 on p. 48 of [Sk92], it would beappropriate to read ( x, y ) ∈ θ as follows: “ y is an image of x under θ ”). Let X be a set, andlet S = ( F , I, C , Π , L, R, Σ , T, F ) , (1.1)where F is the set of all binary relations in X ∗ considered with the composition operation ϕψ = { ( x, z ) | ∃ y (( x, y ) ∈ ψ & ( y, z ) ∈ ϕ ) } and the partial ordering by inclusion, I is the diagonal relation on X ∗ , C is the set of allrelations of the form X ∗ × { c } , where c ∈ X ∗ , Π (juxtaposition) is the binary operation in F defined by Π( ϕ, ψ ) = { ( z, ( x, y )) | ( z, x ) ∈ ϕ & ( z, y ) ∈ ψ } ,L = { (( x, y ) , x ) | x, y ∈ X ∗ } , R = { (( x, y ) , y ) | x, y ∈ X ∗ } , Σ (branching) is the ternaryoperation in F defined byΣ( χ, ϕ, ψ ) = { ( x, z ) | ∃ y (( x, y ) ∈ χ & (( y ∈ X ∗ \ ( X ∪ { o } ) & ( x, z ) ∈ ϕ ) ∨ ( y ∈ X ∪ { o } & ( x, z ) ∈ ψ )) } ,T = X ∗ × { ( o , o ) } , F = X ∗ × { o } . Then S is a combinatory space in the sense of theabove-mentioned definition which reads as follows (up to some changes in the notations):a combinatory space is a 9-tuple (1.1), where F is a partially ordered semigroup, I is anidentity of this semigroup, C is a subset of F , Π and Σ are, respectively, a binary anda ternary operation in F , L, R, T, F are elements of F , and the following conditions areidentically satisfied when ϕ, ψ, θ, χ range over F and ξ, η range over C : ∀ ξ ( ϕξ ≥ ψξ ) ⇒ ϕ ≥ ψ, (1.2)Π( ξ, η ) ∈ C , L Π( ξ, η ) = ξ, R Π( ξ, η ) = η, (1.3)Π( ϕ, ψ ) ξ = Π( ϕξ, ψξ ) , (1.4)Π( I, ψξ ) θ = Π( θ, ψξ ) , (1.5)Π( ξ, I ) θ = Π( ξ, θ ) , (1.6) T (cid:54) = F, T ξ ∈ C , F ξ ∈ C , (1.7) Moschovakis uses, so to speak, the mirror-image of this encoding of the natural numbers – in the presentnotation, the elements o , ( o , o ) , (( o , o ) , o ) , ((( o , o ) , o ) , o ) , . . . are the ones used by him ( cf. [Mo69, (1.3)]). Dimiter Skordev
Vol. 15:1 Σ( T, ϕ, ψ ) = ϕ, Σ( F, ϕ, ψ ) = ψ, (1.8) θ Σ( χ, ϕ, ψ ) = Σ( χ, θϕ, θψ ) , (1.9)Σ( χ, ϕ, ψ ) ξ = Σ( χξ, ϕξ, ψξ ) , (1.10)Σ( I, ϕξ, ψξ ) θ = Σ( θ, ϕξ, ψξ ) , (1.11) ϕ ≥ ψ & θ ≥ χ ⇒ Σ( I, ϕ, θ ) ≥ Σ( I, ψ, χ ) . (1.12)It easily follows from the definition that not only the multiplication in F is monotonicallyincreasing, but so are also the operations Π and Σ.Obviously, (1.6) is the particular instance with ϕ = I of the following equality:Π( ϕξ, I ) θ = Π( ϕξ, θ ) . The combinatory space S is called symmetric if this equality holds for all ϕ, θ in F and all ξ in C (the combinatory space from the above example is a symmetric one). We will define now two different symmetric combinatory spaces S ( X ) = ( F , I , C , Π , L , R , Σ , T , F ) , S ( X ) = ( F , I , C , Π , L , R , Σ , T , F )with semigroups F and F consisting of functions in the Moschovakis extension X ∗ of a anarbitrary given set X .The combinatory space S ( X ) is an instance of the ones considered in [Sk92, Example 2on p. 48]. We take as F the partially ordered semigroup of all partial mappings of X ∗ into X ∗ (including the total ones) – with multiplication defined as composition, i.e. ϕψ ( z ) = ϕ ( ψ ( z )) , where dom( ϕψ ) = ψ − (dom( ϕ )), and partial order defined by ψ ≥ ϕ ⇔ ψ is an extension of ϕ. The identity mapping of X ∗ onto X ∗ is taken as I , and L , R ∈ F are defined in thefollowing way: L ( o ) = R ( o ) = o ,L ( z ) = R ( z ) = ( o , o ) if z ∈ X,L ( z ) = x, R ( z ) = y if z = ( x, y )( L and R are the functions π and δ from [Mo69]). The set of all constant mappings of X ∗ into X ∗ is taken as C , and T , F are those of them whose values are ( o , o ) and o ,respectively. The operations Π and Σ are defined in F as follows:Π ( ϕ, ψ )( z ) = ( ϕ ( z ) , ψ ( z )) , Σ ( χ, ϕ, ψ )( x ) = (cid:40) ϕ ( x ) if x ∈ χ − ( X ∗ \ ( X ∪ { o } )) ,ψ ( x ) if x ∈ χ − ( X ∪ { o } ) , In publications of the author prior to [Sk92] (from 1975 on), a somewhat different notion of combinatoryspace is used. Roughly speaking, the combinatory spaces considered in these publications are the symmetriccombinatory spaces such that T and F belong to C . Such kind of combinatory spaces actually would besufficient for the present paper. We note, however, that some changes ought to be done in the description ofthe combinatory spaces S ( X ) and S ( X ) introduced next if we would want to be in concordance with thepublications in question: one should exchange o and ( o , o ) in the definition of T and F , as well as X ∪ { o } and X ∗ \ ( X ∪ { o } ) in the definition of Σ , and a similar change would be needed in the definition of Σ . ol. 15:1 MOSCHOVAKIS EXTENSION OF REPRESENTED SPACES 35:5 where dom(Π ( ϕ, ψ )) = dom( ϕ ) ∩ dom( ψ ) , dom(Σ ( χ, ϕ, ψ )) = (cid:0) χ − ( X ∗ \ ( X ∪ { o } )) ∩ dom( ϕ ) (cid:1) ∪ (cid:0) χ − ( X ∪ { o } ) ∩ dom( ψ ) (cid:1) . The second combinatory space can be regarded as an extension of the first one. Theelements of F will be the partial functions from X ∗ to the set of the non-empty subsetsof X ∗ , and any θ ∈ F will have as its counterpart in F the function ˜ θ transforming each x ∈ dom( θ ) into the one-element set { θ ( x ) } . The multiplication in F will be defined in theway indicated in the last sentence on p. 289 of [Sk92], namelydom( ϕψ ) = { x ∈ dom( ψ ) | ψ ( x ) ⊆ dom( ϕ ) } , ∀ x ∈ dom( ϕψ ) (cid:16) ϕψ ( x ) = (cid:91) { ϕ ( y ) | y ∈ ψ ( x ) } (cid:17) (compare with [We00, p. 11] and the second composition operation introduced in [We08,Definition 6]). The partial order in F will be as in Remark 9 on p. 293 of [Sk92]: ϕ ≥ ψ ⇔ dom( ϕ ) ⊇ dom( ψ ) & ∀ x ∈ dom( ψ )( ϕ ( x ) ⊆ ψ ( x ))(compare with [We08, Definition 7]). As one immediately sees, the above-mentioned mapping θ (cid:55)→ ˜ θ of the partially ordered semigroup F into the partially ordered semigroup F is anisomorphic embedding. We set further C = { ˜ ξ | ξ ∈ C } , and we take as L , R , T , F the images of L , R , T , F , respectively, under the mapping θ (cid:55)→ ˜ θ . Finally we set Π ( ϕ, ψ )( z ) = { ( x, y ) | x ∈ ϕ ( z ) & y ∈ ψ ( z ) } , Σ ( χ, ϕ, ψ )( x )= { z | ( χ ( x ) \ ( X ∪ { o } ) (cid:54) = ∅ & z ∈ ϕ ( x )) ∨ ( χ ( x ) ∩ ( X ∪ { o } ) (cid:54) = ∅ & z ∈ ψ ( x )) } with dom(Π ( ϕ, ψ )) = dom( ϕ ) ∩ dom( ψ ) , dom(Σ ( χ, ϕ, ψ )) = { x ∈ dom( χ ) | ( χ ( x ) \ ( X ∪ { o } ) (cid:54) = ∅ ⇒ x ∈ dom( ϕ )) & ( χ ( x ) ∩ ( X ∪ { o } ) (cid:54) = ∅ ⇒ x ∈ dom( ψ )) } ( cf. [Sk92, the text on pp. 290–291 between Remark 2 and Remark 3, as well as Remark 6on p. 292]). One verifies that, for all χ, ϕ, ψ ∈ F , the elements Π ( ˜ ϕ, ˜ ψ ) and Σ ( ˜ χ, ˜ ϕ, ˜ ψ ) of F are the images of Π ( ϕ, ψ ) and Σ ( χ, ϕ, ψ ), respectively, under the mapping θ (cid:55)→ ˜ θ .1.3. Iterative combinatory spaces.
If a combinatory space (1.1) is given, and σ, χ areelements of its component F , then the notion of iteration of σ controlled by χ is introduced.Its definition makes use of certain other notions. Let A be any subset of C . By definition,two elements ϕ and ψ of F satisfy the inequality ϕ ≥ A ψ if ∀ ξ ∈ A ( ϕξ ≥ ψξ ). The set A issaid to be invariant with respect to an element σ of F if ϕ ≥ A ψ ⇒ ϕσ ≥ A ψσ for all ϕ, ψ ∈ F .By [Sk92, Definition 1 on p. 74], the iteration of σ controlled by χ is an element ι of F suchthat ι = Σ( χ, ισ, I ) and, for all τ, ρ ∈ F and each subset A of F which is invariant with Dimiter Skordev
Vol. 15:1 respect to σ , the inequality τ ≥ A Σ( χ, τ σ, ρ ) implies the inequality τ ≥ A ρι . By applyingthe last requirement with A = C and ρ = I , one makes the following conclusion: if ι is theiteration of σ controlled by χ then ι is the least τ such that τ ≥ Σ( χ, τ σ, I ). Thus if theiteration of σ controlled by χ exists then it is a uniquely determined element of F . As in[Sk92], this element will be denoted by [ σ, χ ].The combinatory space (1.1) is called iterative if [ σ, χ ] exists for all σ, χ ∈ F . It iseasily seen that the operation of iteration in an iterative combinatory space is monotonicallyincreasing.For any given set X , the combinatory spaces S ( X ) and S ( X ) turn out to be iterative.For a class of combinatory spaces which contains S ( X ), the property of being iterative isindicated on p. 74 of [Sk92]. For any σ, χ ∈ F , the partial function [ σ, χ ] can be characterizedas follows: [ σ, χ ]( z ) = v iff z and v are the initial and the last term, respectively, of a finitesequence of elements of dom( χ ) such that χ ( u ) (cid:54)∈ X ∪ { o } , u (cid:48) = σ ( u ) , whenever u is a term of this sequence and u (cid:48) is its next term, and, additionally, χ ( v ) ∈ X ∪ { o } (thus the function ι = [ σ, χ ] can be represented in Pascal-like notations by means of thedeclaration function ι ( z : X ∗ ) : X ∗ ;var u : X ∗ ;begin u := z ;while χ ( u ) (cid:54)∈ X ∪ { o } do u := σ ( u ); ι := u end;) . The fact that S ( X ) is iterative can be seen by using [Sk92, Proposition 2 on p. 291and Remark 9 on p. 293]. In this combinatory space, the characterization of [ σ, χ ] is morecomplicated. Actually the description of the set [ σ, χ ]( z ) for z ∈ dom([ σ, χ ]) is a naturalmodification of the characterization of the value [ σ, χ ]( z ) in the case of S ( X ) and looks asfollows: v ∈ [ σ, χ ]( z ) iff z and v are the initial and the last term, respectively, of a finitesequence of elements of dom( χ ) such that χ ( u ) \ ( X ∪ { o } ) (cid:54) = ∅ , u (cid:48) ∈ σ ( u ) , (1.13)whenever u is a term of this sequence and u (cid:48) is its next term, and, additionally, χ ( v ) ∩ ( X ∪ { o } ) (cid:54) = ∅ . In the author’s publications before 1980, somewhat stronger requirements are imposed on the iteration.This difference, however, is not important for the considerations in the present paper. Assuming that, for u, u (cid:48) ∈ X ∗ and ϕ ∈ F , the condition u ∈ dom( ϕ ) is taken for granted in statementsof the form ϕ ( u ) = u (cid:48) and in ones of the form u (cid:48) = ϕ ( u ). Unfortunately, there are two misprints in condition (a) of the mentioned proposition, namely the thirdand the fourth M in this condition must be replaced with Q . Assuming that, for u, u (cid:48) ∈ X ∗ and ϕ ∈ F , the condition u ∈ dom( ϕ ) is taken for granted in statementsof the form u (cid:48) ∈ ϕ ( u ). ol. 15:1 MOSCHOVAKIS EXTENSION OF REPRESENTED SPACES 35:7 The complications occur in the description of dom([ σ, χ ]). Let us call a σ, χ -path any sequence(finite or infinite) of elements of X ∗ such that, whenever u is a term of this sequence and u (cid:48) is its next term, then u ∈ dom( χ ) ∩ dom( σ ) and the conditions (1.13) are satisfied. Then(see [Sk92, Remark 4 on p. 292]) dom([ σ, χ ]) consists of the elements z of X ∗ which havethe following two properties:(i) all σ, χ -paths beginning at z are finite;(ii) whenever u is a term of some σ, χ -path beginning at z , then u ∈ dom( χ ) & ( χ ( u ) \ ( X ∪ { o } ) (cid:54) = ∅ ⇒ u ∈ dom( σ )) (1.14)(clearly, we may restrict ourselves in (ii) to the last terms of the σ, χ -paths beginning at z ). Example 1.1.
Let β be the function from F with domain X ∗ such that β ( z ) = { o , ( o , o ) } for any z ∈ X ∗ , and let σ, χ ∈ F be defined as follows: σ = Σ ( β, L , R ) , χ = Σ ( I , β, I ) . Let κ = [ σ, χ ] . Then dom( κ ) = X ∗ and κ ( z ) = (cid:40) { z } if z ∈ X ∪ { o } , { z } ∪ κ ( L ( z )) ∪ κ ( R ( z )) otherwise . One verifies that, for all σ, χ ∈ F , the iteration in S ( X ) of ˜ σ controlled by ˜ χ is theimage under the mapping θ (cid:55)→ ˜ θ of the iteration in S ( X ) of σ controlled by χ .1.4. Computability in iterative combinatory spaces.
Let an iterative combinatoryspace (1.1) be given. By [Sk92, Definition 1 on p. 142], an element θ of F is said to be S -computable in a subset B of F if θ can be obtained from L, R, T, F and finitely manyelements of B by means of multiplication in the semigroup F and the operations Π anditeration . According to [Sk92, Definition 2 on p. 143], a mapping Γ of F n into F is saidto be S -computable in B if, for θ , . . . , θ n ∈ F , the element Γ( θ , . . . , θ n ) is S -computablein B ∪ { θ , . . . , θ n } uniformly with respect to θ , . . . , θ n . Of course, if
B ⊆ B (cid:48) ⊆ F , then S -computability in B implies S -computability in B (cid:48) .As before, X is assumed to be an arbitrary given set. Example 1.2.
Let us use the notations of Example 1.1. As seen from this example, thefunction κ is S ( X ) -computable in the set { β } . The mapping Γ : F → F defined by Γ( θ , θ ) = λ z.θ ( z ) ∪ θ ( z ) (with dom(Γ( θ , θ )) = dom( θ ) ∩ dom( θ ) ) is also S ( X ) -computable in the set { β } . Thisfollows from the fact that Γ( θ , θ ) = Σ ( β, θ , θ ) for all θ , θ ∈ F . In fact, T and F are superfluous in the case of S = S or S = S , since we have the equalities T = L [ L, L ], F = Π( T, T ) in this case. Besides this, whenever the combinatory space S is iterative, thefollowing equalities hold for all χ, ϕ, ψ ∈ F : I = [ ϕ, F ] , Σ( χ, ϕ, ψ ) = R [ (cid:98) F ψR, L ] R [ (cid:98) F ϕR , L ]Π( χ, (cid:98) T ) (1.15)with (cid:98) T = Π( T, I ) and (cid:98) F = Π( F, I ) ( cf. [Sk92, Example 1 on p. 74, Proposition 1 on p. 103 and Proposition 4on p. 104]), thus I is S -computable in ∅ , and the operation Σ preserves S -computability in B . For instance (by the second of the equalities (1.15)), the mapping Σ of F into F is S -computable in ∅ . Dimiter Skordev
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Remark 1.3. If θ ∈ F then the graph of θ is the set { ( z, u ) | z ∈ dom( θ ) & u ∈ θ ( z ) } .A mapping of F into F also related to unions is the one which transforms any pair offunctions from F into the function whose graph is the union of their graphs. This mapping,however, is not monotonically increasing with respect to the partial order in S ( X ) , thereforeit is S ( X ) -computable in no subset of F . If B is a subset of F , and B is the image of B under the mapping θ (cid:55)→ ˜ θ , it is easyto see that, for any element θ of F , the S ( X )-computability of θ in B is equivalent to the S ( X )-computability of ˜ θ in B .From [Sk92, Theorem 1 on pp. 192–193] it is seen that, for any subset B of F , the S ( X )-computability in B of an element θ of F is equivalent to absolute prime computabilityof the function θ in some finite subset of B . The proof of this theorem makes use of [Sk92,Theorem 1 on p. 170] (the First Recursion Theorem for iterative combinatory spaces).According to it, if Γ , . . . , Γ l are mappings of F l into F which are S -computable in a subset B of F then the system of inequalities θ r ≥ Γ r ( θ , . . . , θ l ) , r = 1 , . . . , l, (1.16)has a least solution, and the components of this solution are S -computable in B (actually,the mentioned inequalities are satisfied as equalities by this solution). A parameterizedversion [Sk92, Theorem 2 on pp. 170–171] of the theorem in question concerns systems ofthe form θ r ≥ Γ r ( θ , . . . , θ l , θ l +1 , . . . , θ l + m ) , r = 1 , . . . , l, where Γ , . . . , Γ l are mappings of F l + m into F which are S -computable in B . The existenceof mappings ∆ , . . . , ∆ l of F m into F S -computable in B is asserted such that, for anychoice of θ l +1 , . . . , θ l + m in F , the l -tuple (∆ ( θ l +1 , . . . , θ l + m ) , . . . , ∆ l ( θ l +1 , . . . , θ l + m ) is theleast solution of the above system with respect to θ , . . . , θ l (the inequalities are satisfied asequalities again).Here is an example of an application to S ( X ) of the First Recursion Theorem foriterative combinatory spaces. Example 1.4.
Let Γ be the mapping θ (cid:55)→ Σ ( I , Π ( θR , θL ) , I ) of F into F . Clearlythis mapping is S ( X ) -computable in ∅ . By the definition of the operations in F , we have Γ( θ )( z ) = (cid:40) z if z ∈ X ∪ { o } , ( θ ( R ( z )) , θ ( L ( z ))) otherwisefor all θ ∈ F and all z ∈ X ∗ . Using this, it is easy to see directly the existence of a uniquefixed point of Γ (this fixed point is the total function in X ∗ which transforms any elementof X ∗ into its “mirror-image”). By the First Recursion Theorem for iterative combinatoryspaces, the fixed point of Γ is S ( X ) -computable in ∅ (hence absolutely prime computablein ∅ ). To conclude that the least fixed point in the above example is absolutely prime com-putable in ∅ , one could of course use also the First Recursion Theorem for prime com-putability (for the details concerning it, see [Mo69, p. 460]). The notion of absolute primecomputability makes sense also for multi-valued functions. However, no obvious way is For the definition of absolute prime computability, cf. [Mo69] (to apply this definition to a function θ from F , one must actually consider the partial multiple-valued function which transforms any element z of X ∗ into the one-element set { θ ( z ) } if z ∈ dom( θ ) and into the empty set otherwise). ol. 15:1 MOSCHOVAKIS EXTENSION OF REPRESENTED SPACES 35:9 seen for applying the First Recursion Theorem for prime computability to problems about S ( X )-computability, because the S ( X )-computability of a function from F in some finitesubset B of F can turn out to be not equivalent to absolute prime computability in B (thus S ( X )-computability is an analog to absolute prime computability, but it is different fromit). Example 1.5.
Let β be the same as in Example 1.1. Making use of K˝onig’s Lemma, onesees that the values of the functions S ( X ) -computable in the set { β } are always finite sets.On the other hand, the function transforming all elements of X ∗ into the infinite set N ∗ is absolutely prime computable in β – this can be shown by using [Sk92, Theorem 1 onpp. 192–193] and the equality N ∗ = ι ( o ) , where ι is the iteration of the function z (cid:55)→ ( o , z ) controlled by β in the simpler sense relevant for the situation. Hence absolute primecomputability in β of a unary function in X ∗ does not imply its S ( X ) -computability in β .The converse implication does not hold either. To show this, let us note that the notionof recursive enumerability makes sense for subsets of the closure of { o } under the pairingoperation. It can be shown that, whenever a partial multiple-valued unary function θ in X ∗ isabsolutely prime computable in β , the intersection of { z ∈ X ∗ | θ ( z ) (cid:54) = ∅ } with the closure inquestion is recursively enumerable. On the other hand, an S ( X ) -computable in β function θ can be constructed such that dom( θ ) is a subset of this closure without being recursivelyenumerable. Moschovakis extension of a represented space
A general construction.
As in [Br03, Definition 2.1], a representation of a set is apartial mapping of N N onto this set (where, of course, N N is the set of all functions from N to N ), and a represented space is an ordered pair whose terms are a set and a representationof this set.Let ( X, (cid:37) ) be a represented space, and let a Moschovakis extension X ∗ of X be con-structed in the way from Section 1. We choose a computable bijection J from N to N suchthat the inequality J ( m, n ) ≥ max( m, n ) holds for all m, n ∈ N , and we define a partialmapping (cid:37) ∗ of N N into X ∗ in the following inductive way, where k ranges over N :(i) For any q ∈ dom( (cid:37) ), λ k. q ( k ) + 2 ∈ dom( (cid:37) ∗ ) and (cid:37) ∗ ( λ k. q ( k ) + 2) = (cid:37) ( q ).(ii) λ k. ∈ dom( (cid:37) ∗ ) and (cid:37) ∗ ( λ k.
0) = o .(iii) If q, r ∈ dom( (cid:37) ∗ ) then λ k. J ( q ( k ) , r ( k )) + 1 ∈ dom( (cid:37) ∗ ) and (cid:37) ∗ ( λ k. J ( q ( k ) , r ( k )) + 1) = ( (cid:37) ∗ ( q ) , (cid:37) ∗ ( r )) . Of course, when applying the definition of absolute prime computability to a function θ from F one mustactually consider the extension of θ as a partial multiple-valued function whose value is empty everywhere in X ∗ \ dom( θ ). If σ, χ ∈ F then this kind of iteration of σ controlled by χ is the function ι ∈ F defined as follows:(i) dom( ι ) consists of the initial terms of the finite σ, χ -paths whose last term belongs to the set { v ∈ dom( χ ) | χ ( v ) ∩ ( X ∪ { o } ) (cid:54) = ∅ } ;(ii) for any z ∈ dom( ι ), the set ι ( z ) consists of the last terms of those of the above-mentioned sequenceswhose initial term is z .This is the iteration of σ controlled by χ in an iterative combinatory space not essentially different from S m ( M X ), where S m ( M B ) is the combinatory space considered in the proof of the mentioned Theorem 1. Dimiter Skordev
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Clearly the range of the partial mapping (cid:37) ∗ is the whole X ∗ , hence ( X ∗ , (cid:37) ∗ ) is a representedspace. We will regard it as a Moschovakis extension of ( X, (cid:37) ).If the mapping (cid:37) is injective then so is (cid:37) ∗ . On the other hand, (cid:37) ∗ is surely not total(even in the case when (cid:37) is total). For instance, (cid:37) ∗ is not defined for the elements of N N which have both zero and nonzero values or have both even and odd values.Of course, the representation (cid:37) ∗ depends on the choice of the computable bijection J .However, if J (cid:48) is an arbitrary computable bijection from N to N such that we have J (cid:48) ( m, n ) ≥ max( m, n ) for all m, n ∈ N then the corresponding representation (cid:37) ∗(cid:48) is connectedwith (cid:37) ∗ in a simple way, namely a computable bijection h from N to N exists such thatdom( (cid:37) ∗(cid:48) ) = h − (dom( (cid:37) ∗ )) and, for any p ∈ dom( (cid:37) ∗(cid:48) ), the equality (cid:37) ∗(cid:48) ( p ) = (cid:37) ∗ ( hp ) holds(where, of course, hp = λ k.h ( p ( k ))). In fact, we may define h in the following inductive way: h (2 l ) = 2 l, h (2 J (cid:48) ( m, n ) + 1) = 2 J ( h ( m ) , h ( n )) + 1 . In the next two subsections, we will indicate two cases when properties of a representedspace (
X, (cid:37) ) are preserved by the formation of its Moschovakis extension ( X ∗ , (cid:37) ∗ ), namelythe case when ( X, (cid:37) ) is generated by some effective topological space and the case when(
X, (cid:37) ) is generated by some effective metric space.2.2.
The case of represented space generated by an effective topological space.
An effective topological space can be regarded as an ordered pair ( X, U ), where X isa set, and U = {U i } i ∈ N is a base for a T topology in X . In this situation, let us set U − ( x ) = { i ∈ N | x ∈ U i } for any x ∈ X . The represented space corresponding to ( X, U )has as its second term the partial mapping (cid:37) of N N into X defined as follows: (cid:37) ( p ) = x iffrng( p ) = U − ( x ). Let us define a family U ∗ = {U ∗ j } j ∈ N of subsets of X ∗ by means of theequalities U ∗ = { o } , U ∗ i +2 = U i , U ∗ J ( m,n )+1 = U ∗ m × U ∗ n . Proposition 2.1.
The ordered pair ( X ∗ , U ∗ ) is an effective topological space, and ( X ∗ , (cid:37) ∗ ) is the represented space generated by it. In addition, if the effective topological space ( X, U ) is computable then the effective topological space ( X ∗ , U ∗ ) is also computable.Proof. The statement that ( X ∗ , U ∗ ) is an effective topological space means that U ∗ is a basefor a T topology in X ∗ .Let Z = (cid:83) ∞ i =0 U ∗ i . We will show that Z = X ∗ by showing that X ∪ { o } ⊆ Z and Z is closed under formation of ordered pairs. Indeed, if z ∈ X then z ∈ U i = U ∗ i +2 ⊆ Z forsome i , the element o also belongs to Z because o ∈ U ∗ ⊆ Z , and if u, v ∈ Z then u ∈ U ∗ m and v ∈ U ∗ n for some m and n , therefore ( u, v ) ∈ U ∗ J ( m,n )+1 ⊆ Z .The next thing we have to show is the following statement: if z ∈ U ∗ i ∩ U ∗ i then thereis some i such that z ∈ U ∗ i ⊆ U ∗ i ∩ U ∗ i . Let Z be the set of all elements z of X ∗ suchthat the above implication holds for all i , i . We will show that Z = X ∗ by showing that X ∪ { o } ⊆ Z and Z is closed under formation of ordered pairs. Indeed, if z ∈ X and z ∈ U ∗ i ∩ U ∗ i then U ∗ i = U l , U ∗ i = U l for some l , l , hence z ∈ U l ∩ U l , therefore we have z ∈ U l ⊆ U l ∩ U l for some l , i.e. z ∈ U ∗ i ⊆ U ∗ i ∩ U ∗ i with i = 2 l + 2. If o ∈ U ∗ i ∩ U ∗ i then i = i = 0, hence o ∈ U ∗ i ⊆ U ∗ i ∩ U ∗ i with i = 0. Thus X ∪ { o } ⊆ Z . Suppose nowthat u, v ∈ Z and ( u, v ) ∈ U ∗ i ∩ U ∗ i . Then U ∗ i = U ∗ m × U ∗ n , U ∗ i = U ∗ m × U ∗ n for some m , n , m , n . Clearly u ∈ U ∗ m ∩ U ∗ m and v ∈ U ∗ n ∩ U ∗ n , therefore u ∈ U ∗ m ⊆ U ∗ m ∩ U ∗ m and v ∈ U ∗ n ⊆ U ∗ n ∩ U ∗ n for some m, n , hence ( u, v ) ∈ U ∗ i ⊆ U ∗ i ∩ U ∗ i with i = 2 J ( m, n ) + 1. ol. 15:1 MOSCHOVAKIS EXTENSION OF REPRESENTED SPACES 35:11 Up to here, we showed that U ∗ is a base for some topology in X ∗ . We have to showthat the topology in question is a T one. We will achieve this goal by showing that any z ∈ X ∗ has the following property: whenever z (cid:48) ∈ X ∗ and z (cid:48) (cid:54) = z , then there is some i suchthat U ∗ i contains one of z and z (cid:48) but not the other. Let Z be the set of the elements z of X ∗ with this property. The elements of X belong to Z . Indeed, suppose z ∈ X , z (cid:48) ∈ X ∗ and z (cid:48) (cid:54) = z . If z (cid:48) ∈ X , there is some i such that U i contains one of z and z (cid:48) but not theother, and if z (cid:48) (cid:54)∈ X then no U i contains z (cid:48) and therefore some U i contains z but not z (cid:48) . Inboth cases there is some i such that U ∗ i +2 contains one of z and z (cid:48) but not the other. Theelement o also belongs to Z because U ∗ contains o , but does not contain other elementsof X ∗ . So X ∪ { o } ⊆ Z . Suppose now u, v ∈ Z , z (cid:48) ∈ X ∗ and z (cid:48) (cid:54) = ( u, v ). If z (cid:48) ∈ X ∪ { o } then z (cid:48) ∈ Z and therefore there is some i such that U ∗ i contains one of z (cid:48) and ( u, v ) butnot the other. Let z (cid:48) (cid:54)∈ X ∪ { o } . Then z (cid:48) = ( u (cid:48) , v (cid:48) ) for some u (cid:48) , v (cid:48) ∈ X ∗ , and some of theinequalities u (cid:48) (cid:54) = u , v (cid:48) (cid:54) = v holds. Suppose u (cid:48) (cid:54) = u . Then there is some m such that U ∗ m contains one of u and u (cid:48) but not the other. If u ∈ U ∗ m then we consider some n such that v ∈ U ∗ n , and we observe that U ∗ J ( m,n )+1 contains ( u, v ) but does not contain z (cid:48) . If u (cid:48) ∈ U ∗ m then we consider some n such that v (cid:48) ∈ U ∗ n , and we observe that U ∗ J ( m,n )+1 contains z (cid:48) butdoes not contain ( u, v ). The reasoning in the case v (cid:48) (cid:54) = v is similar.So far, we have established that ( X ∗ , U ∗ ) is an effective topological space. As such one,it generates a representation σ of its carrier X ∗ . By definition, σ is the partial mapping of N N onto X ∗ defined by the equivalence σ ( p ) = z ↔ p ∈ N N & rng( p ) = U ∗− ( z ) . We will prove that σ = (cid:37) ∗ . To this end, we will consider the set Z of the elements z of X ∗ such that σ ( p ) = z ⇔ (cid:37) ∗ ( p ) = z for all p ∈ N N , and we will show that Z = X ∗ . If z ∈ X then U ∗− ( z ) = (cid:8) i + 2 | i ∈ U − ( z ) (cid:9) , hence (cid:110) p ∈ N N (cid:12)(cid:12)(cid:12) rng( p ) = U ∗− ( z ) (cid:111) = (cid:110) λ k. q ( k ) + 2 (cid:12)(cid:12)(cid:12) q ∈ N N & rng( q ) = U − ( z ) (cid:111) = { λ k. q ( k ) + 2 | (cid:37) ( q ) = z } = { p | (cid:37) ∗ ( p ) = z } , thus z ∈ Z . The element o also belongs to Z because U ∗− ( o ) = { } , hence (cid:110) p ∈ N N (cid:12)(cid:12)(cid:12) rng( p ) = U ∗− ( o ) (cid:111) = { λ k. } = { p | (cid:37) ∗ ( p ) = o } . Let now z = ( u, v ) with u, v ∈ Z . Then U ∗− ( z ) = (cid:110) J ( m, n ) + 1 | m ∈ U ∗− ( u ) , n ∈ U ∗− ( v ) (cid:111) , hence (cid:110) p ∈ N N (cid:12)(cid:12)(cid:12) rng( p ) = U ∗− ( z ) (cid:111) = (cid:110) λ k. J ( q ( k ) , r ( k )) + 1 (cid:12)(cid:12)(cid:12) q, r ∈ N N & rng( q ) = U − ( u ) & rng( r ) = U − ( v ) (cid:111) = { λ k. J ( q ( k ) , r ( k )) + 1 | (cid:37) ∗ ( q ) = u & (cid:37) ∗ ( r ) = v } = { p | (cid:37) ∗ ( p ) = z } , thus z ∈ Z . Actually the stronger statement holds that some U ∗ i contains z (cid:48) but does not contain ( u, v ). Dimiter Skordev
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Suppose now that the effective topological space ( X, U ) is computable. Then a recursivelyenumerable subset S of N exists such that U i ∩ U i = (cid:91) {U i | ( i , i , i ) ∈ S } for all i , i ∈ N . We define S ∗ ⊆ N in the following inductive way:(i) (2 i + 2 , i + 2 , i + 2) ∈ S ∗ , whenever ( i , i , i ) ∈ S .(ii) (0 , , ∈ S ∗ .(iii) If ( m , m , m ) ∈ S ∗ and ( n , n , n ) ∈ S ∗ then(2 J ( m , n ) + 1 , J ( m , n ) + 1 , J ( m, n ) + 1) ∈ S ∗ . The set S ∗ is recursively enumerable and U ∗ i ∩ U ∗ i = (cid:91) { U ∗ i | ( i , i , i ) ∈ S ∗ } for all i , i ∈ N , hence the effective topological space ( X ∗ , U ∗ ) is computable.2.3. The case of represented space generated by an effective metric space.
Thenotion of effective metric space will be understood as follows: this is a triple (
X, d, α ), where(
X, d ) is a metric space, α is a mapping of N into X and the set rng( α ) is dense in ( X, d ).The represented space generated by (
X, d, α ) has as its second term the partial mapping (cid:37) of N N into X defined as follows: (cid:37) ( p ) = x iff d ( α ( p ( k )) , x ) < − k for all k ∈ N . We willdefine inductively d ∗ : X ∗ × X ∗ → R and α ∗ : N → X ∗ . The function d ∗ : X ∗ × X ∗ → R isdefined as follows:(i) d ∗ ( x, x (cid:48) ) = min( d ( x, x (cid:48) ) ,
1) for all x, x (cid:48) ∈ X .(ii) d ∗ ( o , o ) = 0.(iii) d ∗ ( o , x ) = d ∗ ( x, o ) = 1 for all x ∈ X .(iv) d ∗ ( z, z (cid:48) ) = d ∗ ( z (cid:48) , z ) = 1 whenever z ∈ X ∪ { o } , z (cid:48) ∈ X ∗ \ ( X ∪ { o } ).(v) d ∗ ( z, z (cid:48) ) = max( d ∗ ( L ( z ) , L ( z (cid:48) )) , d ∗ ( R ( z ) , R ( z (cid:48) ))) for all z, z (cid:48) ∈ X ∗ \ ( X ∪ { o } ).The definition of the mapping α ∗ : N → X ∗ is the following one:(i) α ∗ (2 i + 2) = α ( i ) for all i ∈ N .(ii) α ∗ (0) = o .(iii) α ∗ (2 J ( m, n ) + 1) = ( α ∗ ( m ) , α ∗ ( n )) for all m, n ∈ N . Proposition 2.2.
The triple ( X ∗ , d ∗ , α ∗ ) is an effective metric space, and ( X ∗ , (cid:37) ∗ ) is therepresented space generated by it. In addition, if the effective metric space ( X, d, α ) issemicomputable or computable then the effective metric space ( X ∗ , d ∗ , α ∗ ) is, respectively,also semicomputable or computable.Proof. Let Z be the set of the elements z of X ∗ such that0 ≤ d ∗ ( z, z (cid:48) ) = d ∗ ( z (cid:48) , z ) ≤ , d ∗ ( z, z (cid:48) ) = 0 ⇔ z = z (cid:48) (2.1)for all z (cid:48) ∈ X ∗ . Obviously X ∪ { o } ⊆ Z and Z is closed with respect to formation ofordered pairs. Hence Z = X ∗ , i.e. we have (2.1) for all z, z (cid:48) ∈ X ∗ . To prove that d ∗ ( z (cid:48) , z (cid:48)(cid:48) ) ≤ d ∗ ( z (cid:48) , z ) + d ∗ ( z, z (cid:48)(cid:48) ) (2.2)for all z, z (cid:48) , z (cid:48)(cid:48) ∈ X ∗ , we consider the set Z of the elements z of X ∗ such that the inequality(2.2) holds for all z (cid:48) , z (cid:48)(cid:48) ∈ X ∗ . We will show that Z = X ∗ by showing that X ∪ { o } ⊆ Z and Z is closed with respect to formation of ordered pairs. Both statements will be provedby using that the negation of (2.2) implies the inequalities d ∗ ( z (cid:48) , z ) < , d ∗ ( z (cid:48)(cid:48) , z ) < d ∗ ( z (cid:48) , z (cid:48)(cid:48) ) ≤ d ∗ ( z (cid:48) , z ) ≥ d ∗ ( z, z (cid:48)(cid:48) ) ≥ z ∈ X ∪ { o } , and suppose theinequality (2.2) is violated for some z (cid:48) , z (cid:48)(cid:48) ∈ X ∗ , hence we have the inequalities (2.3) forthese z (cid:48) , z (cid:48)(cid:48) . If z ∈ X these inequalities entail that z (cid:48) and z (cid:48)(cid:48) also belong to X , hence thenegation of (2.2) turns into the inequalitymin( d ( z (cid:48) , z (cid:48)(cid:48) ) , > min( d ( z (cid:48) , z ) ,
1) + min( d ( z, z (cid:48)(cid:48) ) , d ( z (cid:48) , z (cid:48)(cid:48) ) > min( d ( z (cid:48) , z ) ,
1) + min( d ( z, z (cid:48)(cid:48) ) , , > min( d ( z (cid:48) , z ) ,
1) + min( d ( z, z (cid:48)(cid:48) ) , . Due to the nonnegativity of min( d ( z (cid:48) , z ) ,
1) and min( d ( z, z (cid:48)(cid:48) ) , d ( z (cid:48) , z ) ,
1) = d ( z (cid:48) , z ) , min( d ( z, z (cid:48)(cid:48) ) ,
1) = d ( z, z (cid:48)(cid:48) ) , but this, together with the first of them, contradicts the inequality d ( z (cid:48) , z (cid:48)(cid:48) ) ≤ d ( z (cid:48) , z ) + d ( z, z (cid:48)(cid:48) ) . If z = o then the inequalities (2.3) entail z (cid:48) = z (cid:48)(cid:48) = o , and the negation of (2.2) turns intothe false inequality 0 > z = ( u, v ), where u and v belong to Z . We willshow that z belongs to Z too. Suppose the inequality (2.2) is violated for some z (cid:48) , z (cid:48)(cid:48) ∈ X ∗ ,hence we have the inequalities (2.3) for these z (cid:48) , z (cid:48)(cid:48) . Since z ∈ X ∗ \ ( X ∪ { o } ), it followsfrom these inequalities that z (cid:48) and z (cid:48)(cid:48) also belong to X ∗ \ ( X ∪ { o } ), hence z (cid:48) = ( u (cid:48) , v (cid:48) ), z (cid:48)(cid:48) = ( u (cid:48)(cid:48) , v (cid:48)(cid:48) ) for some u (cid:48) , v (cid:48) , u (cid:48)(cid:48) , v (cid:48)(cid:48) ∈ X ∗ . Then d ∗ ( z (cid:48) , z (cid:48)(cid:48) ) = max( d ∗ ( u (cid:48) , u (cid:48)(cid:48) ) , d ∗ ( v (cid:48) , v (cid:48)(cid:48) )) ,d ∗ ( z (cid:48) , z ) = max( d ∗ ( u (cid:48) , u ) , d ∗ ( v (cid:48) , v )) ,d ∗ ( z, z (cid:48)(cid:48) ) = max( d ∗ ( u, u (cid:48)(cid:48) ) , d ∗ ( v, v (cid:48)(cid:48) )) ,d ∗ ( u (cid:48) , u (cid:48)(cid:48) ) ≤ d ∗ ( u (cid:48) , u ) + d ∗ ( u, u (cid:48)(cid:48) ) ≤ d ∗ ( z (cid:48) , z ) + d ∗ ( z, z (cid:48)(cid:48) ) ,d ∗ ( v (cid:48) , v (cid:48)(cid:48) ) ≤ d ∗ ( v (cid:48) , v ) + d ∗ ( v, v (cid:48)(cid:48) ) ≤ d ∗ ( z (cid:48) , z ) + d ∗ ( z, z (cid:48)(cid:48) ) , hence we may conclude that (2.2) holds and thus get a contradiction.After we proved above that ( X ∗ , d ∗ ) is a metric space, we will now show that rng( α ∗ )is dense in this metric space. Let ε >
0, and let Z ε be the set of the elements z of X ∗ such that d ∗ ( α ∗ ( k ) , z ) < ε for some k ∈ N . We will show that Z ε = X ∗ by showing that X ∪ { o } ⊆ Z ε and Z ε is closed with respect to formation of ordered pairs. Any z ∈ X belongs to Z ε because d ∗ ( α ∗ (2 i + 2) , z ) = d ∗ ( α ( i ) , z ) = min( d ( α ( i ) , z ) , < ε for any i ∈ N such that d ( α ( i ) , z ) < ε . The element o also belongs to Z ε because d ∗ ( α ∗ (0) , o ) = d ∗ ( o , o ) = 0 . Suppose now that z = ( u, v ), where u and v belong to Z ε . There are m, n ∈ N such that d ∗ ( α ∗ ( m ) , u ) < ε and d ∗ ( α ∗ ( n ) , v ) < ε . Then d ∗ ( α ∗ (2 J ( m, n ) + 1) , z ) = d ∗ (( α ∗ ( m ) , α ∗ ( n )) , ( u, v )) = max( d ∗ ( α ∗ ( m ) , u ) , d ∗ ( α ∗ ( n ) , v )) < ε. Dimiter Skordev
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Hence z ∈ Z ε .Let now ( X ∗ , σ ) be the represented space corresponding to the effective metric space( X ∗ , d ∗ , α ∗ ), i.e. σ is the partial mapping of N N onto X ∗ defined by the equivalence σ ( p ) = z ⇔ ∀ k ∈ N (cid:16) d ∗ ( α ∗ ( p ( k )) , z ) < − k (cid:17) . We will show that σ = (cid:37) ∗ , where (cid:37) is the second term of the represented space correspondingto the metric space ( X, d, α ). Let Z be the set of the elements z of X ∗ such that σ ( p ) = z ⇔ (cid:37) ∗ ( p ) = z (2.4)for any p ∈ N N . We will show that Z = X ∗ by showing that X ∪ { o } ⊆ Z and Z is closedwith respect to formation of ordered pairs. Let z ∈ X . Then the inequality d ∗ ( z (cid:48) , z ) < − k ,where z (cid:48) ∈ X ∗ , k ∈ N , can be satisfied only in the case of z (cid:48) ∈ X (since 2 − k ≤ σ ( p ) = z can be satisfied only if α ∗ ( p ( k )) ∈ X for all k ∈ N , i.e. if all p ( k ) arepositive even numbers. In such a case, d ∗ ( α ∗ ( p ( k )) , z ) = d ∗ ( α ( p ( k ) / − , z ) = min( d ( α ( p ( k ) / − , z ) , , hence d ∗ ( α ∗ ( p ( k )) , z ) < − k ⇔ d ( α ( p ( k ) / − , z ) < − k . Therefore the equality σ ( p ) = z holds iff p ( k ) is a positive even number for any k ∈ N and ∀ k ∈ N (cid:16) d ( α ( p ( k ) / − , z ) < − k ) (cid:17) . (2.5)On the other hand, the equality (cid:37) ∗ ( p ) = z holds iff p ( k ) is a positive even number for any k ∈ N and (cid:37) ( λ k.p ( k ) / −
1) = z . By the definition of (cid:37) , the last equality is equivalent to thecondition (2.5), and thus we see that z ∈ Z . The element o also belongs to Z , becausein the case z = o each of the sides of the equivalence (2.4) is satisfied iff p ( k ) = 0 for all k ∈ N . Suppose now that z = ( u, v ), where u, v ∈ Z . Then the inequality d ∗ ( z (cid:48) , z ) < − k ,where z (cid:48) ∈ X ∗ , k ∈ N , can be satisfied only in the case of z (cid:48) ∈ X ∗ \ ( X ∪ { o } ). Thereforethe equality σ ( p ) = z can be satisfied only if α ∗ ( p ( k )) ∈ X ∗ \ ( X ∪ { o } ) for any k ∈ N , i.e. if all p ( k ) are positive odd numbers. Any positive odd number has a unique representationin the form 2 J ( m, n ) + 1 with m, n ∈ N , hence any function in N N with positive odd valuescan be uniquely represented in the form λ k. J ( q ( k ) , r ( k )) + 1 (2.6)with q, r ∈ N N . For any two such q, r and any k ∈ N , we have d ∗ ( α ∗ (2 J ( q ( k ) , r ( k )) + 1) , z ) = max( d ∗ ( α ∗ ( q ( k )) , u ) , d ∗ ( α ∗ ( r ( k )) , v )) . Therefore d ∗ ( α ∗ (2 J ( q ( k ) , r ( k )) + 1) , z ) < − k ⇔ d ∗ ( α ∗ ( p ( k )) , u ) < − k & d ∗ ( α ∗ ( q ( k )) , v )) < − k for any k ∈ N , hence σ ( λ k. J ( q ( k ) , r ( k )) + 1) = z ⇔ σ ( q ) = u & σ ( r ) = v ⇔ (cid:37) ∗ ( q ) = u & (cid:37) ∗ ( r ) = v ⇔ (cid:37) ∗ ( λ k. J ( q ( k ) , r ( k )) + 1) = z. Since the value of (cid:37) ∗ can be equal to z only at functions of the form (2.6), it is thus shownthat z ∈ Z . ol. 15:1 MOSCHOVAKIS EXTENSION OF REPRESENTED SPACES 35:15 Suppose now (
X, d, α ) is semicomputable. Then the set H = (cid:26) ( i, j, k, l ) ∈ N (cid:12)(cid:12)(cid:12)(cid:12) d ( α ( i ) , α ( j )) < k + 1 l + 1 (cid:27) is recursively enumerable. We will show that ( X ∗ , d ∗ , α ∗ ) is semicomputable by proving therecursive enumerability of the set H ∗ = (cid:26) ( i, j, k, l ) ∈ N (cid:12)(cid:12)(cid:12)(cid:12) d ∗ ( α ∗ ( i ) , α ∗ ( j )) < k + 1 l + 1 (cid:27) . To this end, it is sufficient to indicate that H ∗ can be defined inductively as follows (with i, j, k, l, m, m (cid:48) , n, n (cid:48) ranging over N ):(i) If k > l then ( i, j, k, l ) ∈ H ∗ .(ii) If ( i, j, k, l ) ∈ H then (2 i + 2 , j + 2 , k, l ) ∈ H ∗ .(iii) (0 , , k, l ) ∈ H ∗ .(iv) If ( m, m (cid:48) , k, l ) ∈ H ∗ and ( n, n (cid:48) , k, l ) ∈ H ∗ then (2 J ( m, n ) + 1 , J ( m (cid:48) , n (cid:48) ) + 1 , k, l ) ∈ H ∗ .Suppose ( X, d, α ) is computable. Then a computable function δ : N → Q exists suchthat | δ ( i, j, k ) − d ( α ( i ) , α ( j )) | < − k for all i, j, k ∈ N . We define a function δ ∗ : N → Q by means of the following inductivedefinition, where i, j, k, m, n, m (cid:48) , n (cid:48) range over N :(i) δ ∗ (2 i + 2 , j + 2 , k ) = δ ( i, j, k ).(ii) δ ∗ (0 , , k ) = 0.(iii) δ ∗ (0 , i + 2 , k ) = δ ∗ (2 i + 2 , , k ) = 1.(iv) δ ∗ (2 i, j + 1 , k ) = δ ∗ (2 j + 1 , i, k ) = 1.(v) δ ∗ (2 J ( m, n ) + 1 , J ( m (cid:48) , n (cid:48) ) + 1 , k ) = max( δ ∗ ( m, m (cid:48) , k ) , δ ∗ ( n, n (cid:48) , k )).Then δ ∗ is a computable function from N to Q such that | δ ∗ ( i, j, k ) − d ∗ ( α ∗ ( i ) , α ∗ ( j )) | < − k for all i, j, k ∈ N , hence ( X ∗ , d ∗ , α ∗ ) is computable.3. On the TTE computability in the Moschovakis extensionof a represented space
In this section, an arbitrary represented space (
X, (cid:37) ) will be supposed to be given. TTEcomputability in the represented space ( X ∗ , (cid:37) ∗ ) constructed as in the previous section willbe considered.3.1. Some general properties.
As well known, the values of the second component ofa represented space at computable sequences of natural numbers are called computable elements of the space. The following three statements are easily verifiable, and they entailthe corollary after them:
Proposition 3.1.
The element o is a computable element of ( X ∗ , (cid:37) ∗ ) . Proposition 3.2. If x ∈ X then x is a computable element of ( X ∗ , (cid:37) ∗ ) iff x is a computableelement of ( X, (cid:37) ) . Dimiter Skordev
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Proposition 3.3. If u and v belong to X ∗ then ( u, v ) is a computable element of ( X ∗ , (cid:37) ∗ ) iff u and v are computable elements of ( X ∗ , (cid:37) ∗ ) . Corollary 3.4.
The set of the computable elements of the represented space ( X ∗ , (cid:37) ∗ ) is theclosure under the pairing operation of the set consisting of the element o and the computableelements of the represented space ( X, (cid:37) ) . If ( X , (cid:37) ) and ( X , (cid:37) ) are represented spaces, then, by definition, a ( (cid:37) , (cid:37) ) -realization of a partial function ϕ from X to X is any partial mapping γ of N N into N N such that p ∈ dom( γ ), γ ( p ) ∈ dom( (cid:37) ) and ϕ ( (cid:37) ( p )) = (cid:37) ( γ ( p )) for any p ∈ (cid:37) − (dom( ϕ )); the function ϕ is said to be ( (cid:37) , (cid:37) ) -computable if a computable ( (cid:37) , (cid:37) )-realization of ϕ exists. Clearlythe last condition is satisfied in the particular case when rng( ϕ ) consists of only one elementand it is a computable element of ( X , (cid:37) ). Thus, for instance, the constant functions T and F which are components of the combinatory space S ( X ) are ( (cid:37) ∗ , (cid:37) ∗ )-computable(with the same (cid:37) ∗ as above). Of course, the condition is satisfied also in the case when( X , (cid:37) ) = ( X , (cid:37) ) and ϕ is the corresponding identity mapping (or some restriction of it).A little more care is needed for the case of the components L and R of S ( X ). Proposition 3.5.
The functions L and R are ( (cid:37) ∗ , (cid:37) ∗ ) -computable.Proof. A computable ( (cid:37) ∗ , (cid:37) ∗ )-realization γ of L can be defined by setting γ ( p )( k ) = g ( p ( k )),where the function g : N → N is defined by means of the equalities g (0) = 0 , g (2 i + 2) = 1 , g (2 J ( m, n ) + 1) = m. For R , the construction of a computable ( (cid:37) ∗ , (cid:37) ∗ )-realization is similar – with right-handside n of the third equality in the definition of g .The identity mapping id X of X into X can be regarded also as a function from X to X ∗ , as well as a partial function from X ∗ to X . Proposition 3.6.
The mapping id X is a ( (cid:37), (cid:37) ∗ ) -computable function from X to X ∗ , as wellas a ( (cid:37) ∗ , (cid:37) ) -computable partial function from X ∗ to X .Proof. To define a computable ( (cid:37), (cid:37) ∗ )-realization of id X , we may set γ ( p )( k ) = 2 p ( k ) + 2. Acomputable ( (cid:37) ∗ , (cid:37) )-realization of id X can be defined by setting γ ( p )( k ) = (cid:98) p ( k ) / (cid:99) · − Corollary 3.7.
For any partial function from X to X , its ( (cid:37), (cid:37) ) -computability is equivalentto any of the following three properties: ( (cid:37), (cid:37) ∗ ) -computability as a partial function from X to X ∗ , ( (cid:37) ∗ , (cid:37) ) -computability as a partial function from X ∗ to X , ( (cid:37) ∗ , (cid:37) ∗ ) -computability as apartial function from X ∗ to X ∗ . If ( X , (cid:37) ) is a represented space then a partial function ϕ from X to N is said to be (cid:37) -computable if a computable partial mapping γ of N N into N exists such that p ∈ dom( γ )and ϕ ( (cid:37) ( p )) = γ ( p ) for all p ∈ (cid:37) − (dom( ϕ )); a partial function ψ from N to X is said to be (cid:37) -computable if a computable partial mapping δ of N into N N exists such that n ∈ dom( δ ), δ ( n ) ∈ dom( (cid:37) ) and ψ ( n ) = (cid:37) ( δ ( n )) for all n ∈ dom( ψ ). Proposition 3.8.
A partial function from X to N is (cid:37) -computable iff it is (cid:37) ∗ -computableas a partial function from X ∗ to N . The computability notion for partial mappings of N N into N N is supposed to be introduced in some ofthe well-known mutually equivalent ways. Note that 1 = 2 J (0 ,
0) + 1 because the surjectivity of J and the inequalities J ( m, n ) ≥ m , J ( m, n ) ≥ n imply J (0 ,
0) = 0. ol. 15:1 MOSCHOVAKIS EXTENSION OF REPRESENTED SPACES 35:17
Proof.
Follows from Proposition 3.6.
Proposition 3.9.
The function n (cid:55)→ n ∗ from N to X ∗ and its inverse function are (cid:37) ∗ -computable.Proof. Let f : N → N be defined by means of the equalities f (0) = 0, f ( n + 1) = J (0 , f ( n )),and, for any n ∈ N , let δ ( n ) : N → N N be defined by means of the equality δ ( n )( k ) = f ( n ).Then δ is a computable mapping of N into dom( (cid:37) ∗ ) and n ∗ = (cid:37) ∗ ( δ ( n )) for all n ∈ N , hencethe function n (cid:55)→ n ∗ is (cid:37) ∗ -computable. Let g be the partial function in N defined by meansof the equality g ( m ) = µ n [ f ( n ) = m ], and let γ be the partial mapping of N N into N definedby means of the equality γ ( p ) = g ( p (0)). Then γ is computable, and, whenever p ∈ dom( (cid:37) ∗ )and (cid:37) ∗ ( p ) = n ∗ , where n ∈ N , the equality n = γ ( p ) holds. This shows the (cid:37) ∗ -computabilityof the inverse function of n (cid:55)→ n ∗ . Corollary 3.10.
A partial function ϕ from X ∗ to N is (cid:37) ∗ -computable iff the partial function z (cid:55)→ ϕ ( z ) ∗ from X ∗ to X ∗ is ( (cid:37) ∗ , (cid:37) ∗ ) -computable. Corollary 3.11.
A partial function h from N to N is computable iff the partial function n ∗ (cid:55)→ h ( n ) ∗ from N ∗ to N ∗ is ( (cid:37) ∗ , (cid:37) ∗ ) -computable. Closedness of the set of the TTE computable functions from F under S ( X ) -computability.Theorem 3.12. Let B be the set of all functions from F which are ( (cid:37) ∗ , (cid:37) ∗ ) -computable.Then all functions S ( X ) -computable in B belong to B .Proof. Having in mind footnote 9 and Proposition 3.5, as well as the fact that B is closedunder composition, we must prove that B is closed under the operations Π and S ( X )-iteration. Let ϕ , ϕ ∈ B . Then there are computable ( (cid:37) ∗ , (cid:37) ∗ )-realizations γ and γ of ϕ and ϕ , respectively. To show that Π ( ϕ , ϕ ) ∈ B , we make use of the mapping γ : dom( γ ) ∩ dom( γ ) → N N defined by means of the equality γ ( p )( k ) = 2 J ( γ ( p )( k ) , γ ( p )( k )) + 1 . Let now ι be the S ( X )-iteration of ϕ controlled by ϕ . The proof that ι ∈ B is based onthe following observation (where γ must be considered as denoting the identity mapping of N N into N N ): if p ∈ (cid:37) ∗− (dom( ι )) then a natural number n exists such that p ∈ dom( γ n ), γ n ( p ) ∈ dom( (cid:37) ∗ ), ι ( (cid:37) ∗ ( p )) = (cid:37) ∗ ( γ n ( p )), γ i ( p ) ∈ dom( γ ) for all i ≤ n and γ ( γ i ( p ))(0) mod 2 = (cid:40) i = n, i < n. The mappings γ and γ can be extended to computable operators Γ and Γ in the set ofall unary partial functions in N . If p ∈ (cid:37) ∗− (dom( ι )) then ι ( (cid:37) ∗ ( p )) = (cid:37) ∗ (Γ n ( p )) with n = µ i [Γ (Γ i ( p ))(0) mod 2 = 0] . Let Γ † be the operator transforming unary partial functions in N into binary ones which isdefined by the equality Γ † ( f )( i, k ) = Γ i ( f )( k ) . Dimiter Skordev
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Let the operators Γ † and Γ transforming unary partial functions in N into unary ones bedefined as follows: Γ † ( f )( i ) = Γ (Γ i ( f ))(0) , Γ( f )( k ) = Γ † ( f )( µ i [Γ † ( f )( i ) mod 2 = 0] , k ) . One consecutively shows that Γ † , Γ † , Γ are computable (this can be done, for instance, byusing their continuity and the effectiveness of their restrictions to partial recursive functions).For all p ∈ (cid:37) ∗− (dom( ι )), Γ( p ) ∈ dom( (cid:37) ∗ ) and the equality ι ( (cid:37) ∗ ( p )) = (cid:37) ∗ (Γ( p )) holds.Therefore some restriction of Γ will be a realization of ι . Corollary 3.13.
Let B be the same as in Theorem 3.12, and let Γ , . . . , Γ l be mappings of F l into F which are S ( X ) -computable in B . Then the components of the least solutionof the system of inequalities (1.16) belong to B . Example 3.14 (concerning computability of real numbers and real functions in the usualsense) . Let c be a computable real number. Let α : ( c, + ∞ ) → R and β : ( −∞ , c ) × R → R becomputable. Suppose ϕ : R \ { c, c − , c − , c − , . . . } → R is defined recursively as follows: ϕ ( x ) = (cid:40) α ( x ) if x > c,β ( x, ϕ ( x + 1)) if x < c, x (cid:54)∈ { c − , c − , c − , . . . } . Then ϕ is also computable. To prove this we may consider the combinatory space S ( X ) corresponding to X = R and use the fact that ϕ is the only solution in it of the equation θ = Σ ( χ, α, β Π ( I , θσ )) , where σ : R → R and χ : R \ { c } → { o , ( o , o ) } are defined as follows: σ ( x ) = x + 1 ,χ ( x ) = (cid:40) ( o , o ) if x > c, o if x < c (to see that the mapping θ (cid:55)→ Σ ( χ, α, β Π ( I , θσ )) is B -computable, we make use of footnote10). Remark 3.15.
By footnote 9, the operation Σ preserves S ( X ) -computability in B , hence,by Theorem 3.12, the set B is closed under Σ . It is easy to show this also directly:whenever γ , γ , γ are ( (cid:37) ∗ , (cid:37) ∗ ) -realizations, respectively, of ϕ , ϕ , ϕ ∈ F , we can constructa ( (cid:37) ∗ , (cid:37) ∗ ) -realization γ of Σ ( ϕ , ϕ , ϕ ) by setting γ ( p )( k ) = (cid:40) γ ( p )( k ) if γ ( p )(0) is odd, γ ( p )( k ) otherwisefor all p ∈ dom( γ ) such that either γ ( p )(0) is odd and p ∈ dom( γ ) , or γ ( p )(0) is evenand p ∈ dom( γ ) ; if γ , γ , γ are computable then γ is also computable. ol. 15:1 MOSCHOVAKIS EXTENSION OF REPRESENTED SPACES 35:19 Closedness of the set of the TTE computable functions from F under S ( X ) -computability. Theorem 3.12 has an analog for the ( (cid:37) ∗ , (cid:37) ∗ )-computability of functionsfrom F provided that the following definition is accepted (compare with [We00, Clause 4 ofDefinition 3.1.3]): if ( X , (cid:37) ) and ( X , (cid:37) ) are represented spaces, then a ( (cid:37) , (cid:37) ) -realization of a partial function ϕ from X to the set of the nonempty subsets of X is any partialmapping γ of N N into N N such that p ∈ dom( γ ), γ ( p ) ∈ dom( (cid:37) ) and (cid:37) ( γ ( p )) ∈ ϕ ( (cid:37) ( p ))for any p ∈ (cid:37) − (dom( ϕ )); the function ϕ is said to be ( (cid:37) , (cid:37) ) -computable if a computable( (cid:37) , (cid:37) )-realization of ϕ exists. Theorem 3.16.
Let B be the set of all functions from F which are ( (cid:37) ∗ , (cid:37) ∗ ) -computable.Then all functions S ( X ) -computable in B belong to B .Proof. For any function ϕ ∈ F , the corresponding function ˜ ϕ belongs to B iff ϕ belongsto B . Hence L and R belong to B . The set B is obviously closed under S ( X )-multiplication. Hence it remains to prove that B is closed under the operations Π and S ( X )-iteration. Let ϕ , ϕ ∈ B . Then there are computable ( (cid:37) ∗ , (cid:37) ∗ )-realizations γ and γ of ϕ and ϕ , respectively. The fact that the function Π ( ϕ , ϕ ) belongs to B is shownby using its ( (cid:37) ∗ , (cid:37) ∗ )-realization defined as in the proof of Theorem 3.12. Let now ι be the S ( X )-iteration of ϕ controlled by ϕ . In order to prove that ι also belongs to B , wewill show that the observation used in the mentioned proof holds in the present situationwith the following modification: the equality ι ( (cid:37) ∗ ( p )) = (cid:37) ∗ ( γ n ( p )) must be replaced by thecondition (cid:37) ∗ ( γ n ( p )) ∈ ι ( (cid:37) ∗ ( p )). Let p ∈ (cid:37) ∗− (dom( ι )), and let H p be the set of all naturalnumbers n such that(i) p ∈ dom( γ n );(ii) γ i ( p ) ∈ dom( γ ) and γ ( γ i ( p ))(0) is odd for all i < n .Clearly 0 ∈ H p , and all natural numbers which are less than a number from H p also belongto H p . Let ˙ H p be the set of the numbers n ∈ H p such that γ i ( p ) ∈ dom( (cid:37) ∗ ) for all i ≤ n and the sequence (cid:37) ∗ ( p ) , (cid:37) ∗ ( γ ( p )) , . . . , (cid:37) ∗ ( γ n ( p )) (3.1)is a ϕ , ϕ -path. We will show that actually ˙ H p = H p , i.e. that n ∈ H p implies n ∈ ˙ H p .This will be done by induction. The implication holds for n = 0 since the statement that0 ∈ ˙ H p is trivially true. Before taking the inductive step, we will show that (cid:37) ∗ ( γ n ( p )) ∈ dom( ϕ ) & ( ϕ ( (cid:37) ∗ ( γ n ( p ))) \ ( X ∪ { o } ) (cid:54) = ∅ ⇒ (cid:37) ∗ ( γ n ( p )) ∈ dom( ϕ )) , (3.2) γ n ( p ) ∈ dom( γ ) , γ ( γ n ( p )) ∈ dom( (cid:37) ∗ ) , (cid:37) ∗ ( γ ( γ n ( p ))) ∈ ϕ ( (cid:37) ∗ ( γ n ( p ))) (3.3)for any n ∈ ˙ H p . Suppose n ∈ ˙ H p . Then (cid:37) ∗ ( p ) , (cid:37) ∗ ( γ ( p )) , . . . , (cid:37) ∗ ( γ n ( p )) is a ϕ , ϕ -path whoseinitial term belongs to dom( ι ), therefore the conjunction (1.14) holds with (cid:37) ∗ ( γ n ( p )) , ϕ , ϕ in the roles of u, σ, χ , respectively, i.e. the conjunction (3.2) holds, and the first term ofthis conjunction implies (3.3). Let now n be a natural number such that n ∈ H p implies n ∈ ˙ H p , and suppose that n + 1 ∈ H p . Then n ∈ H p , hence n ∈ ˙ H p and therefore thestatements (3.3) hold. The assumption n + 1 ∈ H p entails that γ ( γ n ( p ))(0) is odd, hence (cid:37) ∗ ( γ ( γ n ( p ))) (cid:54)∈ X ∪ { o } , and therefore ϕ ( (cid:37) ∗ ( γ n ( p ))) \ ( X ∪ { o } ) (cid:54) = ∅ . The second term Note that B is surely not closed with respect to the usual composition. Here is an example showingthis. Let h be a non-computable function from N to N \ { } , and let ϕ , ϕ ∈ F be defined as follows:dom( ϕ ) = N ∗ \{ o } , ϕ ( x ) = { x } for any x ∈ N ∗ \{ o } , dom( ϕ ) = N ∗ and, for any n ∈ N , ϕ ( n ∗ ) = { o , h ( n ) ∗ } .Then ϕ , ϕ ∈ B , but the usual composition ϕ ϕ transforms n ∗ into { h ( n ) ∗ } for any n ∈ N , therefore itdoes not belong to B . Dimiter Skordev
Vol. 15:1 of the conjunction (3.2) implies that (cid:37) ∗ ( γ n ( p )) ∈ dom( ϕ ), and it follows from here that γ n +11 ( p )) ∈ dom( (cid:37) ∗ ), (cid:37) ∗ ( γ n +11 ( p )) ∈ ϕ ( (cid:37) ∗ ( γ n ( p )). Thus n + 1 ∈ ˙ H p .It is easy to see now that H p is a finite set – otherwise there would exist an infinite ϕ , ϕ -path (cid:37) ∗ ( p ) , (cid:37) ∗ ( γ ( p )) , (cid:37) ∗ ( γ ( p )) , . . . , and this is impossible because (cid:37) ∗ ( p ) ∈ dom( ι ).Let n be the maximal number in H p . Then the sequence (3.1) is a ϕ , ϕ -path. Making useof (3.2) and (3.3), we will show that γ ( γ n ( p ))(0) is even and (cid:37) ∗ ( γ n ( p )) ∈ ι ( (cid:37) ∗ ( p )). Suppose γ ( γ n ( p ))(0) is odd. Then (cid:37) ∗ ( γ ( γ n ( p ))) (cid:54)∈ X ∪ { o } , therefore (cid:37) ∗ ( γ n ( p )) ∈ dom( ϕ ). Itfollows from here that γ n ( p ) ∈ dom( γ ) and (cid:37) ∗ ( γ n +11 ( p )) ∈ ϕ ( (cid:37) ∗ ( γ n ( p ))). Thus the sequence (cid:37) ∗ ( p ) , (cid:37) ∗ ( γ ( p )) , . . . , (cid:37) ∗ ( γ n ( p )) , (cid:37) ∗ ( γ n +11 ( p ))turns out to be also a ϕ , ϕ -path, contrary to the maximality of n . The fact that γ ( γ n ( p ))(0)is even entails (cid:37) ∗ ( γ ( γ n ( p ))) ∈ X ∪ { o } , hence (cid:37) ∗ ( γ n ( p )) ∈ ι ( (cid:37) ∗ ( p )).Let Γ , Γ , Γ † , Γ † and Γ be constructed in the same way as in the proof of Theorem 3.12.Then Γ( p ) ∈ dom( (cid:37) ∗ ) and (cid:37) ∗ (Γ( p )) ∈ ι ( (cid:37) ∗ ( p )), whenever p ∈ (cid:37) ∗− (dom( ι )). Corollary 3.17.
Let B be the same as in Theorem 3.16, and let Γ , . . . , Γ l be mappings of F l into F which are S ( X ) -computable in B . Then the components of the least solutionof the system of inequalities (1.16) belong to B . Example 3.18 (concerning computability of real numbers and real functions in the usualsense) . Let c be a computable real number. Let the real-valued functions α and β withdomains [ c, + ∞ ) and ( −∞ , c + 1) × R , respectively, be computable and satisfy the condition α ( x ) = β ( x, α ( x + 1)) for c ≤ x < c + 1 . Suppose the function ϕ : R → R is defined recursively as follows: ϕ ( x ) = (cid:40) α ( x ) if x ≥ c,β ( x, ϕ ( x + 1)) otherwise.Then ϕ is also computable. To prove this, we may consider the combinatory space S ( X ) corresponding to X = R and use the fact that ˜ ϕ is the only solution in it of the equation θ = Σ ( χ, ˜ α, ˜ β Π ( I , θ ˜ σ )) , where σ is the same as in Example 3.14, and χ ∈ F with domain R is defined as follows: χ ( x ) = { ( o , o ) } if x ≥ c + 1 , { o , ( o , o ) } if c < x < c + 1 , { o } if x ≤ c. Remark 3.19.
The statement that the function ϕ considered in the above example iscomputable remains valid if we replace the assumptions about the domain of β and about theinterconnection between α and β with the following ones: dom( β ) ⊇ ( −∞ , c ] × R , α ( c ) = β ( c, α ( c + 1)) . Indeed, let us then consider the function α : [ c − , c ] → R defined by means of the equality α ( x ) = β ( x, α ( x + 1)) . This function is also computable, and the equality α ( c ) = α ( c ) holds.Let ¯ α be the join of α and α , i.e. their common extension with domain [ c − , + ∞ ) . By ol. 15:1 MOSCHOVAKIS EXTENSION OF REPRESENTED SPACES 35:21 [We00, Lemma 4.3.5] , the function ¯ α is computable too. For c − ≤ x < c , the equalities ¯ α ( x ) = β ( x, α ( x + 1)) = β ( x, ϕ ( x + 1)) = ϕ ( x ) hold, therefore we have ϕ ( x ) = (cid:40) ¯ α ( x ) if x ≥ c − ,β ( x, ϕ ( x + 1)) otherwise.for all x ∈ R . Then we may get the needed conclusion by applying the result from Example 3.18with c − , ¯ α and the restriction of β to ( −∞ , c ) × R in the roles of c , α and β , respectively. Remark 3.20.
In view of footnote 9, Theorem 3.16 implies that B is closed with respectto the operation Σ . A direct proof of the last statement can be given similarly to what wasdone in Remark 3.15: if γ , γ , γ are ( (cid:37) ∗ , (cid:37) ∗ ) -realizations, respectively, of ϕ , ϕ , ϕ ∈ F then the corresponding mapping γ constructed in the same way as there turns out to be a ( (cid:37) ∗ , (cid:37) ∗ ) -realization of Σ ( ϕ , ϕ , ϕ ) . Acknowledgment
The author thanks an anonymous referee for certain helpful suggestions and corrections.
References [Br03] Vasco Brattka. Computability over topological structures. In:
Computability and Models , eds. S. B.Cooper and S. S. Goncharov, Kluwer Academic Publishers, 2003, 93-136.[Es98] Mart´ın H¨otzel Escard´o. Effective and sequential definition by cases on the reals via infinite signed-digitnumerals.
Electron. Notes Theor. Comput. Sci. , 13:53–68, 1998.[Mo69] Yiannis Moschovakis. Abstract first order computability. I.
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J. Univers. Comput. Sci. , 14:801–844, 2008. We may also follow Alex Simpson’s proof (mentioned in [Es98, Subsection 1.1]) of the needed instanceof the lemma in question and derive the computability of ¯ α directly from the equality¯ α ( x ) = α (max( x, c )) + α (min( x, c )) − α ( c ) . This work is licensed under the Creative Commons Attribution License. To view a copy of thislicense, visit https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/