aa r X i v : . [ m a t h . L O ] J un COMPUTABILITY AND NON-MONOTONE INDUCTION
DAG NORMANN
Abstract.
Non-monotone inductive definitions were studied in the late 1960’sand early 1970’s with the aim of understanding connections between the com-plexity of the formulas defining the induction steps and the size of the ordinalsmeasuring the duration of the inductions. In general, any type 2 functional willgenerate an inductive process, and in this paper we will view non-monotoneinduction as a functional of type 3. We investigate the associated computa-tion theory inherited from the Kleene schemes and we investigate the natureof the associated companion of sets with codes computable in non-monotoneinduction. The interest in this functional is motivated from observing that con-structions via non-monotone induction appear as natural in classical analysisin its original form.There are two groups of results: We establish strong closure properties ofthe least ordinal without a code computable in non-monotone induction, andwe provide a characterisation of the class of functionals of type 3 computablefrom non-monotone induction, a characterisation in terms of sequential op-erators working in transfinite time. We will also see that the full power ofnon-monotone induction is required when this principle is used to constructfunctionals witnessing the compactness of the Cantor space and of closed,bounded intervals. Introduction
Motivation and history.
With the introduction of set theory in the secondhalf of the 19th century, mathematicians had more tools in their toolbox thanbefore, they had a richer language in which to express mathematical properties,but they also had tools like transfinite recursion and the use of the axiom of choice.One of these tools, inspired from the new ordinal numbers introduced by Cantor, isnon-monotone induction over the set of integers, seen as an operator of order four,or of type 3 in the terminology of type theory.It is worth noticing that the set-theoretical language mostly used at the time isof third order, while coding is needed to capture the same concepts in second orderarithmetic (SOA). In a series of papers[14–20], Sam Sanders and the author haveinvestigated the logical and computability strength of some of the results using suchtools, when expressed in a language close to how it was originally done.Non-monotone inductive definitions were studied in the late 1960’s and early1970’s, but the general interest has been low since then. Examples of papers on thesubject are [1, 3, 24, 29]. The inductive definitions were classified according to thecomplexity of the formulas defining them, and the key property of interest was thecomplexity of the corresponding closure ordinals. This could be expressed in terms
Department of Mathematics, The University of Oslo, P.O. Box 1053, Blindern N-0316Oslo, Norway
E-mail address : [email protected] . of reflection properties as in [24] or by comparing classes of closure ordinals as in[29].In this paper we will view non-monotone inductive definability over N via afunctional I of type 3 (Definition 2.1), and investigate the strength of Kleene com-putability (Definition 2.3) relative to I . As there is no justifiable Church-Turingthesis for the computability theory of higher order functionals, Kleene computabil-ity is just one possible model, but since this model has proved to be fruitful for theanalysis of discontinuous functionals of type 2, and for computability relative to theSuperjump as defined by Gandy [5], see Harrington [8], Kleene computability is anatural model for the investigation of the computational strength of non-monotoneinduction.The motivation for bringing up non-monotone induction once again is the obser-vation that this functional represents a natural upper complexity-bound for otherfunctionals appearing as realisers for classical theorems such as the Heine-Boreltheorem and the Baire Category theorem, when these theorems are formalised ina set-theoretic language and not within the restricted language of second orderarithmetic.The first application of non-monotone inductive definitions known to the authoris due to E. Borel [2]. The motivation of Borel was to give a direct proof of thetheorem now known as the Heine-Borel theorem. The assumption was that we aregiven a way to associate an open neighbourhood O x to each x in a closed interval[ a, b ] and the claim was that we can then explicitly find a finite sub-covering. Inthe terminology of today, Borel constructed a functional taking the map x O x as the argument and yielding a finite subcovering as the value. The definition ofthis functional is by transfinite recursion, building up finite subcoverings of largerand larger closed subintervals, a construction that can be viewed as a simultaneousnon-monotone inductive definition of Dedekind cuts for numbers c ≤ b and finitesubcoverings of each closed interval [ a, d ] for d < c . In [14], a realiser Θ of theuncountable Heine-Borel theorem ( HBU ) is defined. This realiser selects a finite set x , . . . , x n such that the corresponding open neighbourhoods form a subcovering.It is proved in [14] that Θ , in conjunction with E , computes the Suslin functional(see below), and in Normann [13, Theorem 1(c)] it is shown that any realiser Θ of HBU as above , in conjunction with the Suslin operator, computes the functional I to be defined below. We will slightly improve this theorem, see Section 6.2.Realisers Ξ for the Lindel¨of lemma for Baire space N N (homeomorphic to theirrationals) is one class of functionals discussed in [13], where it is proved in The-orem 1 that any such realiser will compute I and that there is at least one suchrealiser computable in I . Thus non-monotone induction reflects the complexity ofwitnessing the Lindel¨of lemma in this special case.In [19] the aim is to investigate real line topology with the purpose of classify-ing the complexity of theorems and concepts in terms of their reverse mathematics and computational complexity . Representations of open sets, such as being count-able unions of rational neighbourhoods, are based on mathematical insight, andanalysing the logical and computational strength of such insight is part of the aimof [19]. Given representations of open sets as in classical reverse mathematics, us-ing second order arithmetic, the Baire Category Theorem is effective in the sensethat given ( a representation of) a sequence of dense open sets we can compute a OMPUTABILITY AND NON-MONOTONE INDUCTION 3 fast-converging Cauchy-sequence for a point in the intersection. In [19, Theorem6.5] it is proved that, using non-monotone induction, we can find a functional ξ taking a sequence { X k } k ∈ N of subsets of R as arguments and yielding an x ∈ R asvalue, such that whenever each X k is dense and open then ξ ( { X k } k ∈ N ) ∈ \ k ∈ N X k . In [19, Theorem 6.6] it is proved that no such functional ξ can be computable in anyfunctional of type 2, but it remains open to decide if the full power of non-monotoneinduction is needed for obtaining a functional ξ like this.1.2. Overview and results.
In Section 2 we will define the functional I that is ourmain subject of investigation, and we will define the Kleene-computations via theschemes S1-S9 with I as the one argument of type 3. We observe two interpretationsof these schemes, one where we follow Kleene and restrict the application schemeS8 to total inputs and one where we relax on this requirement. We show that thetwo interpretations lead to the same class of functions of type 1 computable in I .We use this to prove what is known as stage comparison and Gandy selection forthe interpretation using partial inputs.In Section 3 we investigate the least ordinal π not computable in I , and theassociated companion L π . We prove that the set of codes f ∈ WO of π is notcomputable in I , and thus in particular not a Π -set (Corollary 3.8). We alsoestablish a number of reflection properties for π .Section 4 is a preparation for Section 5. In Section 4 we introduce what wecall hyper-sequential procedures and in Section 5 we narrow down this concept to inductive procedures . These procedures model nested systems of non-monotone in-ductions, using our new concept of blockings to organise the nesting. The inductiveprocedures can be used to characterise the class of functionals of type 3 computablein I .In Section 6 we look at some of the functionals serving as realisers for classicaltheorems in analysis, primarily theorems where the proof in some way dependson the compactness of the unit line or Cantor space. We will see that when suchrealisers are constructed in a natural way, they implicitly have the full power of non-monotone induction. In conjunction with the Suslin functional S , all realisers ofthe theorems in question will compute I . We will illustrate how to use compactnessfor computing I in the proof of Lemma 6.9, a lemma that is a slight improvementof [13, Theorem 1 (c)].In Section 7 we briefly discuss what it means to relativise these results to func-tionals of type 2 and in Section 8 we summarise the paper and discuss a few openproblems. 2. Non-monotone induction and computability
Inductive definitions.
Mathematically we can identify the Cantor set C = { , } N with the powerset P ( N ) of the integers, where we identify a set with itscharacteristic function. In this paper, we will use both notations, as it sometimes isessential that we consider the set as C , the compact set, and sometimes consider theset P ( N ) where the inclusion ordering is essential. This view will be relevant when COMPUTABILITY AND NON-MONOTONE INDUCTION we define non-monotone induction, but mathematically we use C as the formal def-inition of the set under consideration, and treat it as P ( N ) when this is convenient.When elements of C are viewed as characteristic functions, the point-wise ordering ≤ coincides with the inclusion ordering ⊆ . Definition 2.1.
Let F : C → C be a functional of type 2.a) We view F as an inductive definition, defining the increasing sequence f β in C where β runs over the countable ordinals, by transfinite recursion asfollows:(1) f is the constant zero(2) f β +1 = max { f β , F ( f β ) } (3) If β is a limit ordinal, f β = sup γ<β f γ .b) There will, for cardinality reasons, be a least countable ordinal α F suchthat f α F = f α F +1 . Then α F is the least ordinal α such that F ( f α ) ≤ f α .We let I be defined by I ( F ) = f α F , with the notation introduced above.If we need to point to the functional F , we write f Fβ . Remark 2.2.
We are not fully in the realm of Kleene-computability, since this isdeveloped for total functionals of pure type only. However, if G is of pure type 2,we may consider G as a code for F G ( f )( n ) = min { G ( n ˆ f ) , } , where f ∈ C and with the standard concatenation-understanding of n ˆ f ∈ N N .Using standard coding, we my also consider I as a functional of type 3. For thesake of readability, we prefer to use a customised version of Kleene’s definition, asdefined in Section 2.2, when we investigate the computational strength of I . Example 1.
We view C as the powerset of N and let G : C → N . For purecardinality reasons, there must be A = B ⊆ N such that G ( A ) = G ( B ), and, by theaxiom of choice, there will be a functional Φ such that for every G , Φ( G ) is such apair. Now, the axiom of choice is not needed for this, as will be seen from an easyapplication of I :Given G : C → N , let F G be defined by F G ( A ) = A ∪ { G ( A ) } . We then see thatthe transfinite iteration of F G will generate a strictly increasing sequence of sets { A β } β ≤ α exactly until we have an α , and a β < α , such that G ( A β ) = G ( A α ).In [20] the complexity of such functionals Φ witnessing that there is no injectionfrom C to N is studied in more detail, and it is proved that no such functional canbe computed from an object of type two.2.2. Kleene computability.
Kleene [9] defined a relation { e } ( ~ Φ) = a , in the formof a positive inductive definition with nine cases, where e is an index , a naturalnumber that serves as a G¨odel number for a generalised algorithm, and ~ Φ is asequence of functionals of pure types in the type-structure of total functionals. Thenine cases in the definition are called schemes and are numbered as S1 - S9. For arecent introduction to Kleene computability, see Longley and Normann [10, Chapter5]. In this section we will mainly be concerned with computations of the form { e } ( I , ~F , ~f , ~a ) OMPUTABILITY AND NON-MONOTONE INDUCTION 5 where ~F is a sequence of functionals of type 2, ~f is a sequence of functions of type 1and ~a is a sequence from N . In Definition 2.3 we will restrict S1 - S9 to this case. InSection 4.2 we will give a more general version of S8, accommodated to the contentof that section. Our version of S8 here, when restricted to the use of I as the onlyobject of type 3, will be equivalent to using the version of S8 in Section 4.2 to thefunctional of pure type 3 that will represent I . Definition 2.3.
Using transfinite recursion, we define the relation { e } ( I , ~F , ~f , ~a ) = c, where I is as defined, ~F = ( F , . . . , F m ) is a sequence from N N → N , ~f =( f , . . . , f n ) is a sequence from N N , ~a = ( a , . . . , a k ) is a sequence from N and c ∈ N , as follows.S1 If e = h i , then { e } ( I , ~F , ~f , ~a ) = a + 1.S2 If e = h , q i , then { e } ( I , ~F , ~f , ~a ) = q .S3 If e = h i , then { e } ( I , ~F , ~f , ~a ) = a .S4 If e = h , e , e i , { e } ( I , ~F , ~f , ~a ) = b and { e } ( I , ~F , ~f , b, ~a ) = c , then { e } ( I , ~F , ~f , ~a ) = c .S6 If e = h e , τ , τ , τ , i , where τ , τ and τ are permutations of (the indexsets for) the input sequences ~F , ~f and ~a , then { e } ( I , ~F , ~f , ~a ) = { e } ( I , ~F τ , ~f τ , ~a τ ).S7 If e = h i , then { e } ( I , ~F , ~f , ~a ) = f ( a ).S8 For this scheme there will be subcases, one for each type >
1. For us,there will be two subcases, where the case for type 3 is where we adjust thedefinition to application of I :2. If e = h , , d i then { e } ( I , ~F , ~f , b, ~a ) = F ( g ) when g ( a ) = { d } ( I , ~F , ~f , a, ~a )is a total function. We write { e } ( I , ~F , ~f , ~a ) = F ( λa. { d } ( I , ~F , ~f , a, ~a )) .
3. If e = h , , d i we let { e } ( I , ~F , ~f , b, ~a ) = I ( F G )( b ) where G ( f ) = { d } ( I , ~F , f, ~f , ~a ) . S9 If e = h i then { e } ( I , ~F , ~f , d, ~a ) = c if { d } ( I , ~F , ~f , ~a ) = c . Remark 2.4.
We have excluded S5, the scheme of primitive recursion, from ourdefinition. There are two reasons for this. The main reason is that one mayprove the recursion theorem on the basis of the other schemes, and thus S5 willbe redundant. The other reason is that, since recursion is iterated composition, allarguments involving S5 that we need will be covered by how we deal with S4.Kleene computability inherits several of the key properties of classical com-putability, such as the S n,m -theorem and the recursion theorem. The existenceof universal algorithms is axiomatised in the form of S9. In the sequel, we willassume familiarity with these basic properties.2.3. The computability theory of I . We first prove that the prototype of dis-continuity is computable in I . Definition 2.5.
We define the functional E of type 2 by E ( f ) = (cid:26) ∀ k ( f ( k ) = 0)1 if ∃ k ( f ( k ) > Lemma 2.6.
The functional E is computable in I . COMPUTABILITY AND NON-MONOTONE INDUCTION
Proof.
Given f ∈ N N , we want to decide if ∃ k ( f ( k ) > F f ( A ) = { k : f ( k ) > } ∪ { k : k + 1 ∈ A } . Then ∃ k ( f ( k ) >
0) if and only if 0 ∈ I ( F f ). (cid:3) Remark 2.7. E is sometimes denoted as ∃ , and is equivalent, within S1 - S9, toFeferman’s µ .The Suslin functional S is defined by S ( f ) = (cid:26) ∀ g ∃ n ( f (¯ g ( n )) = 0)1 if ∃ g ∀ n ( f (¯ g ( n )) > Lemma 2.8.
The Suslin functional S is computable in I .Proof. We use that E is computable in I . Given f , we let T f be the treeof finite sequences s = ( s , . . . , s n − ) such that we for all m ≤ n have that f ( h s , . . . , s m − i ) = 0. Then S ( f ) = 0 if and only if T f is well founded. Forall f , the subset of T f consisting of all sequences that cannot be extended to aninfinite branch in T f can be defined using an arithmetical inductive definition, andwe then use E to decide if this subset is the whole tree T f . (cid:3) Totality vs. partiality.
In the original definition of higher order computabil-ity via Kleene’s S1 - S9, all objects were assumed to be total. This can be consideredto be a defect of S8, where the input λξ. { d } ( ξ, − − −− ) has to be defined for all ξ of the type in question in order to accept the termination of Ψ( λξ. { d } ( ξ, − − −− )),even if Ψ is defined in such a way that it only requires some values of the inputfunctional. In the case of I , we only need F to be total on the set of functions f β for β ≤ α F in order to identify I ( F ). Remark 2.9.
A similar phenomenon takes place for Gandy’s Superjump S , intro-duced in [5]. The superjump is defined by S ( F, e ) = (cid:26) { e } ( F, e ) ↓ { e } ( F, e ) ↑ where ↓ means that there is a value of the computation, while ↑ means the converse.In order to find the value of S ( F, e ) we only need to know F restricted to the set of f computable in F , the so called of F . This was used by Harrington [8] inan essential way when he classified the computational strength of S , and was alsoimportant in Hartley’s [7] analysis of the countably based functionals ( See Section4 for a further discussion). We will show that loosening up the requirement oftotality of the input functional to I does not add to the computational strength of I . This is as it is for S , but not, for instance, as for computations with continuousinputs in general. Then we add considerable strength by relaxing on S8, see e.g.[10, Sections 6.4 and 8.5] for results and further references. Definition 2.10.
We write { e } t ( I , ~F , ~f , ~a ) = b if { e } ( I , ~F , ~f , ~a ) = b accordingto the original definition, while we write { e } p ( I , ~F , ~f , ~a ) = b if we interpret S8according to the following extension of I to partial F : C → C . We will not acceptnon-total inputs to F , and for each f ∈ C we either have that F ( f ) ∈ C or totallyundefined. We stick to the notation from Section 2.1:i) By recursion on β , f Fβ is defined if β = 0 or β > f Fγ and F ( f Fγ )are defined for all γ < β . OMPUTABILITY AND NON-MONOTONE INDUCTION 7 ii) I ( F ) is defined if there is an ordinal α such that f Fα +1 is defined and f Fα = f Fα +1 iii) If I ( F ) is defined, and α is as in ii), I ( F )( n ) = f Fα ( n ) for each n ∈ N .When the context is clear, we will talk about t -computations and p -computations. Theorem 2.11.
There is a computable (in the sense of Turing) function ρ suchthat if { e } p ( I , ~F , ~f , ~a ) = b , then { ρ ( e ) } t ( I , ~F , ~f , ~a ) = b .Proof. We use the recursion theorem to define ρ , and define it by cases accordingto S1 - S9. It is obvious what to do in all cases except application of I . Thefinal correctness proof will, of course, be by induction on the complexity of the { e } p -computation (we will define the rank or norm of a terminating computationformally below, definitions that do not rely on the correctness of this theorem),but as is common for this kind of argument, we assume that ρ does the job on allsubcomputations, and we define ρ by self-reference.So assume that { e } p ( I , ~F , ~f , b, ~a ) = I ( F G p )( b )where G p ( f ) = { d } p ( I , ~F , f, ~f , ~a ), and that the recursion terminates as defined inDefinition 2.10. Assume further, as an induction hypothesis, that we can replace G p with the, still partial, G t ( f ) = { ρ ( d ) } t ( I , ~F , f, ~f , ~a ) . We assume familiarity with the concept of a prewellordering R on a domain D ⊆ N . Since E and S are t -computable in I , we also have that the set ofprewellorderings will be t -computable in I .If R is a prewellordering on D , each element in the domain D will have an ordinalrank, and we let R β be the elements in D with ordinal rank below β . We willconstruct a total functional H mapping prewellorderings to prewellorderings suchthat we can decide b ∈ I ( F G t ) from I ( H ). The definition of H ( R ) is as follows,observing that we only need E when we know that R is a prewellordering. We let f β be as in the definition of I ( F G t ), and we identify f β with A β = { b ∈ N : f β ( b ) = 1 } .- By R -recursion, compare R α with A α until we either have disagreement or that R α = A α with F ( A α ) ⊆ A α .- In the first case, α must be a successor ordinal β + 1. We let H ( R ) be R restricted to R β = A β , and then end-extended with F ( A β ) \ A β . In the other casewe let H ( R ) be R restricted to R α .Since we in the computation of H ( R ) only will ask for values F ( A β ) , our as-sumption shows that H is total. I ( H ) will be a prewellordering R , and we willhave that it matches the prewellordering induced by F G t . We then have that b ∈ I ( F ) ⇔ b ∈ dom ( I ( H )) . It is now a matter of routine to define a suitable candidate for ρ ( h , d i ) in a com-putable way from d and an alleged index for ρ , so we may define a working ρ bythe classical recursion theorem. (cid:3) From now on, if we write { e } , then we mean { e } p . COMPUTABILITY AND NON-MONOTONE INDUCTION
The norm of a computation and Gandy Selection.
The advantage ofusing p -computations is that now all computation trees will be countable, and allcomputations will have a countable ordinal as rank. We give a direct definitionof this rank. In order to simplify the readability we introduce the following asa convention: With the expression λ ( g, c ) . { d } ( I , ~F , g, ~f , c, ~a ) we really mean thefunction ( ∗ ) F ( g )( c ) = min { , { d } ( I , ~F , c ˆ g, ~f , ~a ) } . Definition 2.12.
Let C I be the set of finite sequences h e, ~F , ~f , ~a i such that forsome b we have { e } ( I , ~F , ~f , ~a ) = b. If h e, ~F , ~f , ~a i ∈ C I we define the norm ||h e, ~F , ~f , ~a i|| by transfinite recursion asfollows:i) If e corresponds to S1 - S3 or S7, we let the norm be zero.ii) If { e } ( I , ~F , ~f , ~a ) = { e } ( I , ~F , ~f , { e } ( I , ~F , ~f , ~a ) , ~a ), where { e } ( I , ~F , ~f , ~a ) = c , we let ||h e, ~F , ~f , ~a i|| = max {||h e , ~F , ~f , ~a i|| , ||h e , ~F , ~f , c, ~a i||} + 1 . The cases S6 and S9 are handled in a similar way, and are left for the reader.iii) If { e } ( I , ~F , ~f , ~a ) = F ( g ) where g ( b ) = { d } ( I , ~F , ~f , b, ~a ) we let ||h e, ~F , ~f , ~a i|| = sup {||h d, ~F , ~f , b, ~a i|| + 1 : b ∈ N } iv) If { e } ( I , ~F , ~f , b, ~a ) = I ( λ ( g, c ) . { d } ( I , ~F , g, ~f , c, ~a ))( b ), we let F be as in( ∗ ), and we let α and f β for β ≤ α be as in the definition of I . By theassumption, f β is well defined and total for all β ≤ α , where F ( f α ) ≤ f α .We let ||h e, ~F , ~f , b, ~a i|| = sup {||h d, ~F , c ˆ f β , ~f , ~a i|| + 1 : β ≤ α ∧ c ∈ N } . If h e, ~F , ~f , ~a i 6∈ C I we let ||h e, ~F , ~f , ~a i|| = ℵ , the first uncountable ordinal. Lemma 2.13 (Stage Comparison) . There is a a partial functional P in two variables, p -computable in I , such that i) P ( h e, ~F , ~f , ~a i , h d, ~G, ~g, ~c i ) terminates if at least one of h e, ~F , ~f , ~a i and h d, ~G, ~g, ~c i is in C I and then ii) P ( h e, ~F , ~f , ~a i , h d, ~G, ~g, ~c i ) = 1 if ||h e, ~F , ~f , ~a i|| ≤ ||h d, ~G, ~g, ~c i|| iii) P ( h e, ~F , ~f , ~a i , h d, ~G, ~g, ~c i ) = 0 if ||h d, ~G, ~g, ~c i|| < ||h e, ~F , ~f , ~a i|| .Proof. We use the recursion theorem to construct P , and the definition is split into81 cases, according to the schemes corresponding to e and d . S8 splits into twocases, S8.2 and S8.3 for applications of F and I , while S5 is redundant and leftout. This is why we have 9 × P does in cases where one or both indices do not correspond toKleene-indices at all, but we leave these trivial cases for the reader. Fortunately,many other cases are trivial as well, in particular those where one of the indices e or d represents a basic computation S1 - S3 or S6. Moreover, all cases not involvingS8.3 are covered by the literature, see e.g. [5].We will give the details for three cases (S4 , S8.3), (S8.2 , S8.3) and (S8.3 ,S8.3). The remaining cases follow by similar, or even simpler, arguments. As isnormal practise for this kind of construction/proof we define P by self reference, OMPUTABILITY AND NON-MONOTONE INDUCTION 9 assuming for each case, as an induction hypothesis, that P works for the immediatesubcomputations.Case (S4 , S8.3): Let { e } ( I , ~F , ~f , ~a ) = { e } ( I , ~F , ~f , { e } ( I , ~F , ~f , ~a ) , ~a )and let { d } ( I , ~G, ~g, b, ~c ) = I (( G )( b ) , where G = λ ( g, c ) . { d } ( I , ~G, c ˆ g, ~g, ~c )).Let g α be element α in the sequence inductively defined from G . We now considerthe following induction, that can easily be formalised via an inductive definition:Use P to compare ||h e , ~F , ~f , ~a i|| with the ranks needed to compute g , g , . . . until the first is bounded in norm by one of the latter computations or until thelatter induction terminates.In the first case, let c = { e } ( I , ~F , ~f , ~a ) and start over again, now comparing thecomputations involved in computing the g α ’s with ||h e , ~F , ~f , c, ~c i|| .If ||h e, ~F , ~f , ~a i|| ≤ ||h d, ~G, ~g, ~c i|| , this will be verified through the two induc-tions, and the composition will terminate at least as fast as the induction. If ||h d, ~G, ~g, ~c i|| < ||h e, ~F , ~f , ~a i|| , at least one of the two inductions will result in thefull induction induced by G , and we can deduce that this terminates faster thanthe composition.Case (S8.2 , S8.3): Let { e } ( I , ~F , ~f , ~a ) = F ( f )where f ( a ) = { e } ( I , ~F , ~f , a, ~a )and let { d } ( I , ~G, ~g, b, ~c ) = I ( G )( b )where G is as in the previous case.As in the previous case, we simulate the induction in the second part while, ateach step, comparing the length of the computations needed with those of each { e } ( I , ~F , ~f , a, ~a ). We use E in doing this. If we for each a reach a step inthe induction where we need a computation that dominates the computation of { e } ( I , ~F , ~f , a, ~a ), we know that the left hand side will terminate at most with thesame rank as the right hand side. If we are able to complete the induction on theright hand side before termination of all sub-computations on the left hand side,we know that the right hand side terminates first. This stepwise comparison untilthe value of P is settled can be expressed as an inductive definition.Case (S8.3 , S8.3): Let { e } ( I , ~F , ~f , a, ~a ) = I ( λ ( a ′ , f ′ ) . { e } ( I , ~F , a ′ ˆ f ′ , ~f , ~a ))( a )and let { d } ( I , ~G, ~g, b,~b ) = I ( λ ( b ′ , g ′ ) . { d } ( I , ~G, b ′ ˆ g ′ , ~g,~b ))( b ) . Notice that the norms of these computations will be independent of the choices of a and b . Let F and G be the partial functionals involved in these inductions, whereat least one is sufficiently total for the induction to terminate. We now describe asimultaneous inductive definition of two increasing sequences f α and g β of elementsof C , where we use E and P to make all the comparisons involved: * Let f = g be the constant zero.* Assume that f , . . . , f α and g , . . . , g β are constructed.* Consider all computations involved in computing all f δ ( a ) for δ ≤ α and incomputing F ( f α )( a ) for all a , and then consider all computations involvedin computing all g γ ( b ) for γ ≤ β and in computing G ( g β )( b ) for all b .* If the norm of each computation in the first set is bounded by the norm ofsome computation in the second set, we add f α +1 = max { f α , F ( f α ) } andkeep g , . . . g β .* On the other hand, if there is one computation in the first set whose normstrictly bounds all norms of the computations in the other set, we add g β +1 = max { g β , G ( g β ) } and keep f , . . . , f α .* At least one of these two inductions will terminate through this process,and when it does, we know which one will terminate with lowest ordinalnorm.We leave the formal definition of this inductive definition for the reader. (cid:3) Theorem 2.14 (Gandy Selection) . There is a p -computable selection operator ν such that for all e , ~F , ~f and ~a we have ∃ n { e } ( I , ~F , ~f , n, ~a ) ↓ ⇒ { e } ( I , ~F , ~f , ν ( e, ~F , ~f , ~a ) , ~a ) ↓ . Proof.
This is a soft consequence of Lemma 2.13, with an argument well known inthe literature, see e.g. [4, Theorem 3.1.6], [11, Theorem 3], [25, Theorem X.4.1] orthe original [5]. (cid:3)
For many of the inductive definitions used, we add at most one new element to theinductively defined set at each stage. Such definitions can be defined by functionals F of pure type 2, identifying 2 N with the power set of N via characteristic functions: Definition 2.15.
Let G : 2 N → N and let H G : 2 N → N be defined by H G ( A ) = A ∪ { G ( A ) } . An inductive definition F is single valued if it is in the form H G . We let I ( G ) = I ( H G ) Lemma 2.16.
The functionals I and I are computationally equivalent modulo E .Proof. I is trivially, and outright, computable in I . In order to prove the otherdirection, we let F : 2 N → N be given, and we will construct a single valued G that simulates F . We assume that F is nontrivial, i.e. that F ( ∅ ) = ∅ . We let G operate on sets B of finite binary sequences, and we totally order these sequencesusing the standard lexicographical ordering by first comparing the first place wheretwo sequences are different, and if this does not help, by length. This is not a wellordering, but G , as we define it, will only generate well ordered sets of sequences.There will be three cases in the definition of G ( B ):(1) B has no maximal element s . Let A be the set of n such that s ( n ) = 1 forat least one s ∈ B . If F ( A ) ⊆ A , let G ( B ) be the (sequence number of)the empty sequence. If not, let n be the least number in F ( A ) \ A , and let OMPUTABILITY AND NON-MONOTONE INDUCTION 11 G ( B ) = s where s is the binary sequence of length n + 1 approximating thecharacteristic function of A ∪ F ( A ).(2) If there are elements s < · · · < s k in B so that B = { s ∈ B : s < s } hasno maximal element and such that B = B ∪ { s , . . . , s k } , let A be the set of n such that s ( n ) = 1 for at least one s in B . Ifthe sequences s , . . . , s k do not approximate the characteristic function of A ∪ F ( A ), let G ( B ) = 0 (the value does not matter), while otherwise, welet G ( B ) = s where s is the least proper extension of s k that approximates A ∪ F ( A ).(3) Otherwise, let G ( B ) = 0.The induction induced by G will, one step at the time, build up approximations tothe characteristic functions of the sets appearing in the induction induced by F . If I ( F ) uses α many steps, I ( G ) will use ω · α many steps. Clearly G is computablein F and E , and the closure set of F is arithmetical in the closure set of G . Thus I is computable in I and E . (cid:3) The companion of I In this section we will analyse the computational power of I in terms of settheory. Recall that a set X is hereditarily countable if the transitive closure trcl( X )is countable. Hereditarily countable sets X will have codes , essentially structures( D, R, A ) where D ⊆ N , R is a binary relation on D , A ⊆ D and ( D, R, A ) isisomorphic to (trcl( X ) , ∈ trcl( X ) , X ). Such codes can further be coded as functionsin N N in a natural way. Definition 3.1.
The companion M of I is defined as the set of sets X with codesthat are computable in I . Remark 3.2.
The companion of other functionals are defined in analogy with this.For instance, the companion of E will be L ω CK1 , the companion of S (the Suslinfunctional) is L β for the first recursively inaccessible ordinal β while the companionof S (the Superjump) is L ρ where ρ is the first recursively Mahlo ordinal . Lemma 3.3.
There is a countable ordinal π such that M = L π .Proof. Since { e } p is absolute for L , we have that M is a transitive subset of L . L will be closed under a certain map sending a code for an ordinal α to a code for L α (this map is computable in E ), so M will be an initial segment of L . (cid:3) Lemma 3.4.
Let F : C → C be a partial functional computable in I such that I ( F ) is defined. Let α be the corresponding closure ordinal for F . Then α < π .Proof. For each F there is an F ′ computable in F and E such that F ′ generates aprewellordering R where R β +1 = F ( R β ) \ R β for each ordinal β . Then α will be theordinal rank of the inductively definable prewellordering R so α will be computablein I whenever F is computable in I . (cid:3) The aim of this section is to find closure- and reflection-properties of L π . Since S is computable in I we have that the set of codes for hereditarily countable setsis computable in I . Given codes f i for sets X i , i ∈ N , , we only need E to unifythe codes in the form of a code for { X i : i ∈ N } . Further, given codes f , . . . , f n for sets X , . . . , X n , and a ∆ -formula Φ( x . . . , x n ), E can decide the truth valueof Φ( X , . . . , X n ). Finally, if Φ( x , . . . , x n , y ) is a ∆ -formula, f , . . . , f n are codescomputable in I for X , . . . , X n ∈ L π and L π | = ∃ Y Φ( X , . . . , X n , Y )then we can use Gandy selection for I to compute (an index for) a code g for a set Y such that Φ( X , . . . , X n , Y ). This leads to a proof of Lemma 3.5. L π is an admissible structure. Let WO be the set of codes for countable ordinals. This is a Π - set, and it iseasy to prove that the following sets are Π as well:(1) The set of f ∈ WO that codes ω CK1 .(2) The set of f ∈ WO that codes the first recursively inaccessible ordinal.(3) The set of f ∈ WO that codes the first recursively Mahlo ordinal.We say that these ordinals are Π -characterisable . Many ordinals of distinction areΠ -characterisable, for instance all clockable ordinals in the sense of infinite timeTuring machines ([6]), see Welch [27] for a survey and further references on suchmachines. Definition 3.6.
Let P be a class of ordinals. We say that P is I -decidable if thereis an I -computable function ∆ : N N → N such that ∆( f ) = 0 if and only if f codesan ordinal α and P ( α ) holds.That π is not Π -characterisable follows from the following much stronger: Theorem 3.7.
Let P be a property on ordinals that is I decidable and such that P ( π ) . Let X ⊂ π be closed, unbounded and Σ over L π . Then there is an α ∈ X such that P ( α ) .Proof. We can code a partially enumerated set { f d : d ∈ D } of functions as theset of pairs h d, f d ( n ) i where d ∈ D and n ∈ N . The idea is to construct aninductive definition Γ that is computable in I and such that Γ generates a code foran ordinal both in X and satisfying P . Γ will not be total, but sufficiently totalfor the induction to terminate. In defining Γ as computable in I , we use that theSuslin functional S is computable in I . We define Γ( R ) as follows: • If R does not code an enumerated set { f d : d ∈ D } , we let Γ( R ) = R . Notethat the empty set codes the empty set of functions. • Assume that R codes { f d : d ∈ D } . If f d WO for some d ∈ D , letΓ( R ) = R . • Assume now that f d ∈ WO codes α d or all d ∈ D , and use E to computea code g for the least upper bound α of { α d : d ∈ D } . If each α d are in X ,then α ∈ X since X is closed. If P ( α ) we let Γ( R ) = R . This is where wewant the induction to close. • Otherwise, we apply Gandy selection for I and search for an index e for acode g of an ordinal β > α such that β ∈ X . We then letΓ( R ) = R ∪ {h e, g ( n ) i : n ∈ N } . If α < π , we can use the recursion theorem for I to see that L π is closed under the α -iteration of Γ, and that Γ generates codes for an increasing sequence of ordinals γ β for β < α . Since we always use an index e for an ordinal larger than those appearingat earlier stages, we do not risk to mix up codes for different ordinals. Since X is OMPUTABILITY AND NON-MONOTONE INDUCTION 13 closed, all ordinals obtained during this iteration will be codes for ordinals γ β ∈ X .Since P ( π ) and this induction will stop when we hit a γ β with P ( γ β ), and since byLemma 3.4 no such induction will stop at π , there must be an ordinal γ β < π suchthat P ( γ β ). (cid:3) Corollary 3.8.
The closure ordinal π of I is not Π -characterisable. We also have
Corollary 3.9.
The closure-ordinal π of I is recursively Mahlo.Proof. We have to prove that if X ⊆ π is π -computable, closed and unbounded,then X contains an admissible ordinal. Since the class of countable, admissibleordinals is I -decidable, this is a direct consequence of Theorem 3.7. (cid:3) Since being recursively Mahlo and other even stronger closure properties are also I -decidable, we may extend this argument in order to prove that π satisfy thesestronger properties, and that every closed unbounded subset of π that are Σ over L π also contain elements satisfying these stronger properties. We will not pursuethis further here.We will now consider an alternative way of expressing that π must be a “large”countable ordinal. What is “large” is of course subject to the perspective one istaking. Definition 3.10.
An ordinal γ is reflecting if for all formulas Φ( x , . . . , x n ) andelements X , . . . , X n in L ( γ ), L γ | = Φ( X , . . . , X n ) ⇒ ∃ β < γ [ L β | = Φ( X , . . . , X n ) . ]Note that if γ is reflecting, then γ is admissible and recursively inaccessible. Corollary 3.11.
The closure ordinal π of I is reflecting.Proof. If α < β , X , . . . , X n are in L α and L π | = Φ( X , . . . , X n ), then the setof γ > α such that L γ | = Φ( X , . . . , X n ) is I -decidable, contains π and thus, byTheorem 3.7, contains an ordinal β with α < β < π . (cid:3) π will not be the least reflecting ordinal: Corollary 3.12.
Let π be the closure ordinal of I . If α < π , then there is areflecting ordinal γ with α < γ < π .Proof. This is also a consequence of Theorem 3.7, since being reflecting is I -decidable. Indeed, if X ⊆ π is closed, unbounded and Σ over L π , then X containsarbitrarily large reflecting ordinals. (cid:3) Remark 3.13.
These results do of course not imply that L π has an elementarysubstructure, or even a substructure satisfying the same first order sentences. The theory of L π is not I -decidable, so there is no way to unify these arguments to allformulas simultaneously. Classes of functionals of type 3
Motivation.
We introduced the functional I in Section 2 and illustrated itscomputational strength through an analysis of the companion in Section 3. Inthis section we will give an analysis of computations relative to I resembling an operational semantics . Our approach is inspired by the success-story of using nestedsequential procedures for modelling functionals definable in LCF (Scott, [26]) orequivalently in PCF (Plotkin, [23]) from objects of type 1. For an introduction tonested sequential procedures, see [10, Chapter 6].Since we are only concerned with functionals of type ≤ hyper-sequential procedures until we find a characterisation of the functionals of type 3that are computable in I and some functional of type 2. The gain will mainlybe that we obtain a more civilised, and less ad hoc, way of expressing relativecomputability for certain functionals of type 3 than when we refer to the Kleeneschemes directly. We will use this to give mathematical support to the informalclaim that if realisers of classical theorems based on compactness arguments arecomputable in I , then the full power of I is required.A functional Φ of type 3 is normal if E is computable in Φ, where E ( F ) = (cid:26) ∀ f ∈ N N ( F ( f ) = 0)1 if ∃ f ∈ N N ( F ( f ) > , and where F is assumed to be total.The set of functionals of type 3 that are neither normal nor computable in type 2objects is mainly unexplored with respect to computability-theoretical properties.The classical object of this kind is Gandy’s Superjump S . S is of course a naturalfunctional in the context of higher order computability theory. Recently, examplesthat are natural from other perspectives have emerged. In [14, §
3] we introducedclasses of realisers Θ for the general Heine-Borel theorem and a weaker class offunctionals Λ that compute realisers for the Vitali Covering theorem. In [13] wealso considered functionals Ξ that serve as realisers for the Lindel¨of Lemma forBaire Space. In this paper we introduced I , which, under the name IND, wasproved in [13] to compute Lindel¨of realisers Ξ. This plethora of elements in a sofar unexplored class of functionals justifies a more coherent study of this class. Wewill return to some of these functionals in Section 6.2.Hartley [7] investigated the fully typed hierarchy of hereditarily countably basedfunctionals, based on a definition due to Stan Wainer, and obtained some generalresults. For instance, he proved that if we assume the Continuum hypothesis to-gether with ZFC, Φ is countably based if and only if E is not computable in Φand any functional of type 2.The original definition of the countably based functionals is by a generalisationof the definition of the continuous functionals e.g. as based on domain theory , see[10, Chapter 10] for a recent introduction. In this paper, we will only be interestedin objects of types 0, 1, 2 and 3, and we define the countably based functionals forthese cases, suiting our own purposes: Definition 4.1.
All integers are countably based. Moreover(1) All total functions f : N → N are countably based. OMPUTABILITY AND NON-MONOTONE INDUCTION 15 (2) All partial functionals F mapping a subset of N N to N are countably based.(3) Let Φ be a partial functional taking countably based functionals of type 2as arguments and yielding integers as values. Φ is countably based if wefor each F and n such that Φ( F ) = n find a countable set A ⊆ N N suchthat F is total on A and such that for all G of type 2, if G is total on A and agrees with F on A , then Φ( G ) = n .In (3), a base element for Φ will be a countable set A together with the restrictionof an F to A with the property described. E will not be countably based, since in order to know that E ( O ) = 0 we needto know that O ( f ) = 0 for all f ∈ N N .One problem with the countably based functionals Φ is that the base elementsof Φ are not well structured as individual sets, and a suitable class of base elementsfor Φ may not be well structured as a class. Much of the way of thinking inheritedfrom the computability theory of the continuous functionals is useless. The aim ofthis section is to introduce a more restricted class, the hyper-sequential functionals,where we have added some further structure. Examples of hyper-sequential func-tionals will be the Superjump and I . However, the first concept we introduce willbe too general for our purpose, for instance, all functionals of type 3 computableusing an infinite time Turing machine the way suggested by Welch [28] will behyper-sequential.4.2. Hyper-sequential functionals.
The definitions.
In this section we will define what we mean with a hyper-sequential procedure. A transfinite calculation using a functional F as an oraclecan be viewed as a sequence of queries of the form “what is F ( f )?”, where thenext query will depend on the answer to the previous ones. We will capture suchdeterministic procedures with our concepts defined in 4.2. Our aim will be to isolatethe procedures that will correspond to computations relative to I , and in order tofully capture those , our calculations also must contain some documentation . In acomputation { e } ( I , F, ~f , ~a ), there may be subcomputations with extra arguments g or b of type 1 or 0. Our abstract calculations will contain a LOG of functions g ,and the use of this LOG will be to show that procedures corresponding to Kleene-computations in I are definable at the level of Π . This will be made precise later.In order to formally describe this LOG we take the liberty to add an extra element ∗ to N , and to claim that objects involving this ∗ will be of a certain complexity,for instance Π , without going to the trouble of coding. Definition 4.2. a) A string is a sequence { ( f β , a β ) } β<α where α is a countable ordinal, each f β ∈ N N and each a β ∈ N ∪ {∗} . We call f β a query , and sometimes writesit as F ( f β ) =?.b) A hyper-sequential procedure is a set Ω of strings where each string will begiven an integer value, and such that whenever { ( f β , a β ) } β<α and { ( f ′ β , a ′ β ) } β<α ′ are in Ω they are either equal or there is a β < min { α, α ′ } such that f β = f ′ β , a β = a ′ β , a β = ∗ , a ′ β = ∗ and ( f γ , a γ ) = ( f ′ γ , a ′ γ ) for all γ < β . Formally,will let Ω be a set of pairs ( t, b ) where t is a string and b is the associatedvalue. c) If { ( f β , a β ) } β<α is a string and F ∈ T p (2), we say that the string matches F if F ( f β ) = a β for all β < α with a β ∈ N .d) If a string t is in a a hyper-sequential procedure Ω, has a value a andmatches F , we call t a calculation , calculating Ω( F ) = a .e) If Ω is a hyper-sequential procedure, then Ω defines (or computes ) thepartial functional Φ( F ) = Ω( F ) of type 3. When Φ( F ) is defined, thecalculation of Ω( F ) will be unique.f) A total functional of type 3 is hyper-sequential if there is a hyper-sequentialprocedure that defines it.g) If t = { ( f β , a β ) } β<α is a calculation, and a β = ∗ we say that β is in theLOG of t .We will from now on use the words procedure and sequential in the meaning ofhyper-sequential procedure and hyper-sequential. Remark 4.3.
A procedure can be viewed as a strategy for a transfinite game wherePlayer I, the computing device, plays queries and Player 2, the input, answers eachquery using F . In some matches of the games, corresponding to the calculations,Player 1 wins in the sense of providing an output, while in other matches, PlayerII wins because it either stops after countably many steps without a value, or itgoes on through ℵ many steps. We will discuss this further when we considerprocedures with more structure. If we then still use the picture of games withrules, the LOG will represent places where Player 1 will enter a sub-game followingdifferent rules, and the LOG will help the referee to verify that the whole match isplayed according to the general, nested, rules of the game.Note that if t is a string that is an initial segment of several calculations, thenthe next f β will be the same for all such extensions, and if β is in the LOG of oneof them, it will be in the LOG of all extensions. Definition 4.4.
Let { ( f β , a β ) } β<α be a string.A sub-string is a sequence { ( f γ , a γ ) } γ<β for some β ≤ α .We can concatenate strings in the usual way: If we for each ordinal γ < γ havea string { ( f γ,β , a γ,β ) } β<α γ we let the concatenation { ( f β , a β ) } β<α be defined by- α = P γ<γ α γ - If β = P γ<γ α γ + β where γ < γ and β < α γ , then ( f β , a β ) =( f γ ,β , a γ ,β ).We will prove that the class of sequential functionals of type 3 is closed underKleene-computability as defined through the schemes S1 - S9. To be more precise,we will prove that if ~ Φ = (Φ , . . . , Φ n ) consists of sequential functionals and λF. { e } ( ~ Φ , F, ~f , ~a )is total, then it is itself sequential. To make this precise, we need to extend S8 todeal with general inputs of type 3. For the sake of notational simplicity, we assumethat the arguments of our computations will be of the form as above, that we dropthe scheme S6 of permutation and that we use an alternative indexing for scheme S8so that we can read out from the index for which of the arguments in the list ~ Φ theoracle call is made. (Alternatively we could modify S6 to cater for permutations ofthe list of inputs of all four types.) We still leave out S5, primitive recursion, partly
OMPUTABILITY AND NON-MONOTONE INDUCTION 17 because it is redundant in the presence of S9, and partly because it can be handledin analogy to composition S4. Thus we add the following scheme to Definition 2.3,while replacing the one occurrence of I with a sequence ~ Φ of functionals of type 3:S8 If e = h , , i, d i then { e } ( ~ Φ , ~F , ~f , ~a ) = Φ i ( λf. { d } ( ~ Φ , ~F , f, ~f , ~a ))In the original definition by Kleene, this is only supposed to make sense when { d } ( ~ Φ , ~F , f, ~f , ~a ) terminates for all f ∈ N N , but when we are working with count-ably based Φ i we normally only require that a base element is a sub-function of λf. { d } ( ~ Φ , ~F , ~f , ~a ).As we will see in the sequel, being sequential the way we define it here is quitegeneral, and thus the fact that this class is closed under Kleene computabilitymay be of restricted interest. However, we will later refer to the construction ofprocedures imbedded in the proof of Lemma 4.7 in situations where we will showthat much more restricted classes of functionals still are Kleene closed.Since we, in this section, are primarily interested in functionals of type 3 com-putable in a given sequence of sequential functionals of the same type, we restrictthe number of arguments of type 2 to one. We can do this because the number oftype 2 arguments will not increase as we move down the paths of the computationtree. The number of arguments of type 0 and of type 1 may increase, so we needto consider arbitrarily long finite lists of such input arguments. Definition 4.5.
Let ~ Φ = (Φ , . . . , Φ n ) be a sequence of sequential functionalsdefined from the procedures Ω , . . . , Ω n . Let F be of type 2 and let ~f , ~a be finitesequences of objects of type 1 and 0 resp. Assume that { e } ( ~ Φ , F, ~f , ~a ) = b . Byrecursion on the length of this computation we define the calculation t e,~ Φ ,F, ~f,~a withvalue b as follows, where we use ˆ to denote concatenation of strings (recall that calculations are strings that, in the given context, have values) :- If e is an index for an initial computation, i.e. for S1, S2, S3 or S7, we let t e,~ Φ ,F, ~f,~a be the empty string, i.e. with α = 0.- If { e } ( ~ Φ , F, ~f , ~a ) = { e } ( ~ Φ , F, ~f , { e } ( ~ Φ , F, ~f , ~a ) , ~a ) , let c = { e } ( ~ Φ , F ~f , ~a ). Let t e,~ Φ ,F, ~f,~a = t e ,~ Φ ,F, ~f,~a ˆ t e ,~ Φ ,F, ~f,c,~a . - In the case of S9, we just use the calculation for the immediate subcompu-tation.- Let { e } ( ~ Φ , F, ~f , ~a ) = F ( λc. { e } ( ~ Φ , F, ~f , c, ~a )). Let f ( c ) = { e } ( ~ Φ , F, ~f , c, ~a ).Then t e,~ Φ ,F, ~f,~a = t e ,~ Φ ,F, ~f, ,~a ˆ t e ,~ Φ ,F, ~f, ,~a ˆ · · · · · · | {z } ω ˆ( f, F ( f )) . - Let { e } ( ~ Φ , F, ~f , ~a ) = Φ i ( λg. { e } ( ~ Φ , F, g, ~f , ~a )). Let H ( g ) = { e } ( ~ Φ , F, g, ~f , ~a )and let { ( g β , b β ) } β<α be the calculation in Ω i that is matching H .We let t e,~ Φ ,F, ~f,~a be the concatenation of { ( g β , ∗ )ˆ t e ,~ Φ ,F,g β , ~f,~a } β<α . This ends the definition.
Remark 4.6.
We inserted the pairs ( g β , ∗ ) in the LOG in order to remind us of thefact that we at that stage are simulating a subcomputation with an extra argument g β . We need the information about this extra argument in order to say that a stringis ‘correct’, in a sense made precise later. Lemma 4.7.
Let e , ~ Φ , ~f and ~a be fixed as in Defintion 4.5. Then the set { ( t e,~ Φ ,F, ~f,~a , b ) : { e } ( ~ Φ , F, ~f , ~a ) = b } will be a procedure.Proof. Assume that both { e } ( ~ Φ , F, ~f , ~a ) and { e } ( ~ Φ , G, ~f , ~a ) terminate. We proveby induction on the ordinal ranks of the computations that the corresponding cal-culations satisfy Definition 4.2 b). The proof is split into cases corresponding tothe Kleene schemes.If e is an index for an initial computation, the claim is trivial, and the inductionstep is trivial in the case of application of S9.Let e be an index for composition, and let e and e be as in the construc-tion. If the calculations for { e } ( ~ Φ , F, ~f , ~a ) and { e } ( ~ Φ , G, ~f , ~a ) are different, thenby the induction hypothesis they split at a first point, and there the f -parts arethe same while the a -parts differ. Since these calculations are initial segments ofthe calculations under consideration, the concatenated calculations also satisfy thedefinition.If the calculations for the e -computation are equal, then, by the indiction hy-pothesis, the values are the same, c , and then our conclusion follows from theinduction hypothesis for { e } ( ~ Φ , F/G, ~f , ~a ).Application of F/G : In this case, we construct calculations as the concatenationof ω + 1 items, first the corresponding calculations for each c ∈ N , and at the end,pairs ( f, F ( f )) and ( f ′ , G ( f ′ )) respectively. If there is a least c where the corre-sponding two calculations differ, the f -parts will agree while the a -parts will differat a minimal location in these calculations, by the induction hypothesis. Then the f -parts will agree and the a -parts will differ at the corresponding minimal loca-tion in the concatenated calculation. If the two concatenations of the calculationsinherited for each c are equal, it follows from the induction hypothesis that thearguments f and f ′ for F and G resp. are equal, so at the top pair ( f, F ( f )) and( f ′ , G ( f ′ )) we will have that the query parts are equal.Application of Φ i : Let { ( g β , b β ) } β<α and { ( g ′ β , b ′ β } β<α ′ be the two calculationsin Ω i matching the corresponding functionals H and H ′ as in the definition in thiscase. First we see that if the two concatenated calculations agree as far as theyboth go, we can use the induction hypothesis, sub-induction on β < min { α, α ′ } andthe fact that Ω i is a procedure to show that g β = g ′ β and that H ( g β ) = H ′ ( g ′ β ) forall β . Since Ω i is a procedure, it follows that α = α ′ , that the two concatenatedcalculations are equal and that the values are the same.If the two concatenated calculations differ, there will be a least β < min { α, α ′ } such that they differ in the sections computing H ( g β ) and H ′ ( g ′ β ). Then g β = g ′ β ,so by the induction hypothesis there is a least location in those sections where theydiffer, and there the f parts are equal while the a -parts differ. So, the calculationsconstructed will satisfy the definition. (cid:3) OMPUTABILITY AND NON-MONOTONE INDUCTION 19
Theorem 4.8.
The class of hyper-sequential functionals of type 3 is closed underrelative Kleene-computability.Proof.
Immediate from Lemma 4.7. (cid:3)
Remark 4.9.
We can deduce, from the proof of Theorem 4.8, that all functionalsof type 3 computable in functionals of lower types will be hyper-sequential.4.2.2.
Mixed types.
Some of the objects we are interested in are of types at level ≤ I is one prominent example. There are two natural waysto extend the concept of sequential functionals to objects of such types. One is toidentify such types as the fixed points of computable retracts on the correspondingpure types, the retracts being explicitly definable as Kleene-computable where theschemes S5 and S9 are not needed. Then an object will be, by definition, sequentialif the representation in the pure type is so.The other alternative is to extend the intuition of sequentiality to objects of thesegeneral types. A type like this will be of the form σ , . . . , σ n → N , where each σ i has level ≤
2. Thus a calculation will be a well-ordered set of querieswith answers where each query is of the form F i ( ~f ) =? for some i , varying with thequery. Each ~f will consist of functions and/or integers, and the functions may be ofone or several number variables. To keep track of all this in its full generality willrequire some heavy notation, but there will be no genuine mathematical problems.Given this, we can define what we mean with a procedure adjusted to each type,and then the sequential objects of that type. It is obvious that the two approachesare equivalent, but not being pressed, we prefer to omit all details. In some of ourexamples, we will use the latter, intuitive approach.4.2.3. Examples.
Our first example is what motivated us to isolate the concept ofhyper-sequential functionals:
Theorem 4.10.
The functional I is hyper-sequential.Proof. Let F : 2 N → N , F ′ ( a ˆ f ) = F ( f )( a ) and let f F be the constant 0. We find f F = F ( f F ) through the ω -series of queries F ′ ( a ˆ f F ) =?, then f F = F ( f F ) ∪ f F (identifying a characteristic function with the corresponding set) through the ω -sequence of queries F ′ ( a ˆ f F ) =? and so on. This is clearly a hyper-sequentialprocedure. (cid:3) In his CiE-2019-paper [28], Philip Welch introduced infinite time Turing ma-chines that can take functionals F of type 2 as oracles. The idea is to have aspecial oracle tape, and whenever the oracle F is called upon, we consider the ora-cle tape as the input information, and what the consequence of the oracle call willbe will depend on the precise ITTM-model we are using. We have Theorem 4.11.
Every ITTM-computable functional is sequential.
We leave this theorem without a proof, since the proof is easy, but requiresfamiliarity with the ITTM-model.Clearly, all sequential functionals are countably based. To what extent the con-verse is true is unknown, but we do have:
Theorem 4.12.
If the continuum hypothesis CH holds, all countably based totalfunctionals will have extensions to partial functionals that are are sequential.Proof.
We work within ZFC + CH. Let { f α } α< ℵ be an enumeration of N N . LetΦ be countably based and let X be set of base elements for Φ. The elements of X will be triples ( A, φ, a ) where A ⊆ N N is countable, φ : A → N and a ∈ N . Thesignificance is that whenever F extends φ to all of N N , then Φ( F ) = a , and thatfor each F there will be at least one ( A, φ, a ) ∈ X where F is an extension of φ .The sequential procedure will then be to compute F ( f α ) up to the first α wherethere is some ( A, φ, a ) ∈ X such that • A ⊆ { f α : α < α } . • For f α ∈ A we have that F ( f α ) = φ ( f α ).We will have that Φ( F ) = a independent of which ( A, φ, a ) we chose with thisproperty. (cid:3)
Denotation procedures.
There is no reason to believe that the continuumhypothesis can be avoided in Theorem 4.12, but the theorem still suggests thatthe concept of hyper-sequential functional is too general to be of interest, and theintention is to investigate possible refinements of the concept. Now we will considerprocedures that will include some extra information, a number or denotation d β for each β in the index ordinal of a calculation. In its full generality, this does notrestrict the class of functionals definable from procedures, but it gives us a toolfor discussing the complexity of them. Thus, in the theorems of this section, theconstructions of the procedures with denotations used to prove them will be asimportant as the theorems themselves. Definition 4.13. A denotation procedure Ω, d-procedure for short, will be a set Ωof calculations with denotations ( { ( f β , a β , d β ) } β<α , c )where each a β ∈ N ∪ {∗} and(1) The denotations d β are in N .(2) The corresponding set of calculations without the denotations is a proce-dure.(3) For each ( { ( f β , a β , d β ) } β<α , c ) ∈ Ω, if β < γ < α , then d β = d γ .By abuse of terminology, we will use Ω both for a d-procedure and for thecorresponding procedure, making it clear in each case if we consider the denotationsor not. Clearly, all d-procedures will define sequential, partial functionals as well.In fact we have Observation 1.
By the axiom of choice, all procedures Ω can be extended to d-procedures. We simply use the axiom of choice to select one enumeration of α for eachcalculation ( { ( f β , a β ) } β<α , c ) ∈ Ω and use this to define the additional d β s foreach calculation. There is of course no extra knowledge to be harvested from thisargument, but it illustrates a possibility that we have to bring under control in thed-procedures that we construct: Definition 4.14.
Let Ω be a d-procedure, let { ( f β , a β , d β ) } β<α be a calculationin Ω and let β < α . The delay of the denotation of the calculation at point β OMPUTABILITY AND NON-MONOTONE INDUCTION 21 is the least ordinal γ such that for all other calculations { ( f ′ δ , a ′ δ , d ′ δ ) } δ<α ′ in Ω, if( f δ , a δ ) = ( f ′ δ , a ′ δ ) for all δ < β + γ , then d δ = d ′ δ for all δ ≤ β .The delay tells us for how much longer we must run a calculation before we cantell what the denotation will be.A key property of a d-procedure is that we can use the denotations to code theprocedure in a manageable way as a subset of the continuum. Definition 4.15. a) Let Ω be a d-procedure. The representation of Ω will be the set of quadru-ples ( D, ≺ , { ( f d , a d ) } d ∈ D , c ) derived from calculations ( { ( f β , a β , d β ) } β<α , c )in Ω as follows:i) D is the set of d β for β < α and ≺ is the corresponding ordering on D .ii) When d = d β , f d is the f β and a d is the a β of the calculation.iii) c is the value of the calculation.We code these items as elements of N N in some standard way.b) We say that a d-procedure Ω is Π if the representation of Ω is a Π -set.c) If ( D, ≺ , { ( f d , a d ) } d ∈ D , c ) is a calculation in a d-procedure and ( D ′ , ≺ ′ , { ( g d , b d ) } d ∈ D ′ ) satisfies that ≺ ′ is a well ordering of D ′ , each g d is oftype 1 and each b d is of type 0, we say that ( D ′ , ≺ ′ , { ( g d , b d , } d ∈ D ′ ) is iso-morphic to an initial segment of ( D, ≺ , { ( f d , a d ) } d ∈ D ) if there is an orderisomorphism ρ from D ′ to an initial segment of D such that f ρ ( d ) = g d and a ρ ( d ) = b d for all d ∈ D ′ . Lemma 4.16.
The functional I is definable from a d-procedure that is Π .Proof. For each b , we will construct a procedure for the 0 - 1-valued function λG. I ( F G )( b ) , where F G ( f )( a ) = min { , G ( a ˆ f ) } is as in Remark 2.2. The only difference betweenthese procedures will be in the value part, the c in each string.Let G be given. We will describe the calculation with denotation that will match G and conclude with the value I ( F G )( b ). Let { f β } β ≤ α be the sequence constructedwhile defining I ( F G ).For each β ≤ α , let g a,β = a ˆ f β . We see that in order to “compute” I ( F G )( b )we have to evaluate G on all functions g a,β for all β ≤ α , a sequence of queries oforder type ω ( α + 1). So, we define the calculation matching G as( { ( h γ , b γ , d γ ) } γ<ω ( α +1) , c )where- h ω · β + a = g a,β for β ≤ α and a < ω .- b ω · β + a = G ( g a,β ) for β and a as above.- d ω · α + a = h , a i for a ∈ ω .- d ω · β + a = h x + 1 , a i , where x is minimal such that f β +1 ( x ) = 1 while f β ( x ) = 0, if β < α and a ∈ ω .- c = f α ( b ).It remains to prove that the representation is Π . We do this through the followingsteps: (1) Since the set of pairs ( D, ≺ ) where D ⊆ N and ≺ is a well ordering of D isΠ , the set Ω of quadruples ( D, ≺ , { ( h d , b d ) } d ∈ D , c ) where ( D, ≺ ) is a wellordering as above is Π . We call the elements in Ω strings .(2) If Ω is is the set of strings in Ω where the order type of ( D, ≺ ) equals ω · ( α + 1) for some α , we still have a Π -set.(3) Let Ω be the strings in Ω that corresponds to a possible evaluation of I on some G . This requires that the calculation is locally correct , i.e. thateach ( h d , a d ) is in relation to its ( D, ≺ )-predecessors as prescribed by therecursion step. This can be decided arithmetically, so Ω is also Π .(4) For a string in Ω , we can arithmetically decide if the enumeration ( D, ≺ )is as in the construction above, so the representation Ω of the calculationswith denotations in the procedure for I will also be Π . (cid:3) Remark 4.17.
We introduce delays in this construction. Whenever we simulateone step in the induction, we must wait until we know if we are at the final step ornot before we can decide what the denotation will be, and this involves a delay oflength ω . Definition 4.18.
Let Φ be a total functional of type 3. We say that Φ is Π -definable if Φ is definable from a Π d-procedure. Lemma 4.19.
The class of Π -definable total functionals of type 3 is closed underrelative Kleene computability.Proof. We build on the proof of Theorem 4.8 and the construction in Definition 4.5.We just have to show how to add the denotations d β to each item in the calculation,and then show that the complexity of the representation is preserved. We definethe d-procedure as follows:In the cases of initial computations there are no ordinals to be denoted, and inthe case of S9 we can keep the denotations as they are.In the case of composition, we can use d
7→ h , d i to denote the items in the firstpart and the map d
7→ h , d i to denote the items in the second part.When we compute g and then apply F to g , we use the map d
7→ h c + 1 , d i to denote the items coming from the calculation computing g ( c ) and end the fullsubcalculation with ( g, F ( g ) , h , i ).In the case where we apply the procedure Ω i for Φ i to a partial functional H oftype 2 for which we have an index, our calculation will be the concatenation of thecalculatioins related to the computations of H ( g β ) = a β , where we also inserted( g β , ∗ ) in front of each such local calculation. If the Ω i -denotation for the pair( g β , a β ) in the calculation evaluating Φ i ( H ) is d , we use h d , i to denote ( g β , ∗ )in the calculation we construct, and if an item in the calculation defined from thecomputation of H ( g β ) = a β is d , we let h d , d + 1 i be the denotation of thecorresponding item in the concatenated calculation.It is clear that if two calculations, as in Definition 4.5 are equal, the denotationswill be the same as well. This defines a d-procedure.It remains to show that the representation of this d-procedure will be Π whenthe representations of the d-procedures for Φ , . . . , Φ n are Π . This will be the OMPUTABILITY AND NON-MONOTONE INDUCTION 23 hard, technical part of our proof, and we first give a brief explanation of what weaim to do:We let the Π -representations for ~ Φ be given. Using the recursion theorem forcomputing relative to E , we will design an algorithm that, given e , ~f , ~a and arepresentation ( D, ≺ , { ( f d , a d ) } d ∈ D )of a string with denotations (a d-string for short) will semi-check, in the sense ofproviding an algorithm relative to E that terminates when the property holds, ifthere is some F such that this string matches F and that the d-string gives us therepresentation of the calculation we constructed for the computation { e } ( ~ Φ , F, ~f , ~a ).In addition, if our algorithm finds the representation ( D, ≺ , { ( f d , a d ) } d ∈ D ) adequateas the representation of a d-calculation, it will produce the value of the computation { e } ( ~ Φ , F, ~f , ~a ), which then will be the same for any F matching the given d-string(which by now is confirmed as a d-calculation). Since termination of E -algorithmsis of complexity Π , this will prove the lemma. Without stressing this point every-where needed, we assume that the given d-string matches itself, in the sense thatif both ( f, a ) and ( f, a ′ ) occur, maybe at different places, then a = a ′ .As usual, our E -procedure will be defined by cases following S1 - S9, where weonly focus on the nontrivial cases.If e is an index for an initial computation, we check if the given string is empty.If so, it is fine as a calculation, and we can read off the value from the index, thegiven ~f and ~a .Composition: { e } ( ~ Φ , F, ~f , ~a ) = { e } ( ~ Φ , F, ~f , { e } ( ~ Φ , F, ~f , ~a ) , ~a ) . First we check if ( D, ≺ ) is of the form ( D , ≺ ) + ( D , ≺ ) where each d ∈ D is ofthe form h , d ′ i and each d ∈ D is of the form h , d ′ i .Let D ′ = { d ′ : h , d ′ i ∈ D } and consider the corresponding string inheritedfrom the given one. If this is ok for the computation { e } ( ~ Φ , F, ~f , ~a ), we computethe value c ′ , and now ask if the ( D , ≺ ) is ok for { e } ( ~ Φ , F, ~f , c ′ , ~a ) in the samesense.Application of F : { e } ( ~ Φ , F, ~f , ~a ) = F ( λc. { e } ( ~ Φ , F, ~f , c, ~a )) . First we check if the given string has a last element ( g, a, h , i ) and if what comesbefore can be seen as an ω -sum of intervals I c where the denotations are of the form h c + 1 , d i .If this is the case, the given string generates, in analogy with the case for compo-sition, strings t c , and we check for each of them if they are ok for the computation { e } ( ~ Φ , F, ~f , c, ~a ), and with value g ( c ).If they are all ok we accept the given string as a calculation, and see that thevalue of F ( λc. { e } ( ~ Φ , F, ~f , c, ~a )) must be a .Application of Φ i : { e } ( ~ Φ , F, ~f , ~a ) = Φ i ( λg. { e } ( ~ Φ , F, g, ~f , ~a )) . This is where we need the extra information stored in the LOG. We proceed asfollows:
In the given string, first check if ( D, ≺ ) is the union of intervals where the firstelement of the interval is of the form h d , i and the rest are of the form h d , d + 1 i .Then check for each of these intervals , where g = f h d , i , if the rest of thisinterval, after replacing h d , d + 1 i with d , is ok for { e } ( ~ Φ , F, g, ~f , ~a ), and if so,compute the value a .Finally, we collect these pairs ( g, a ) with denotation d into a string, and checkif this is a calculation in Ω i with some value c . For this, we use Gandy selection,and we then find the correct value as well.In order to complete the argument we must prove that if this process works,then the computation in question, relative to any F matching the given string,will terminate with the chosen value, and prove that if the computation terminatesfor a total F , then our process terminates on the corresponding representation ofthe d-calculation, and again, that it gives the right value. Both arguments are byinduction on the length of computations, the first for E -computations and thelatter for the computation of { e } ( ~ Φ , F, ~f , ~a ). The details are trivial. (cid:3) Inductive procedures
As a consequence of Lemmas 4.16 and 4.19 we see that all total functionals oftype 3 computable in I will be definable from a d-procedure that is Π , but theconverse is not true, see Theorem 6.1.The aim of this section is to narrow down a subclass of the d-procedures further inorder to approach a characterisation of the class we are primarily interested in, thefunctionals computable in I .5.1. Computability in E . Matters are trivial if the d-procedure is hyperarith-metical:
Theorem 5.1.
Let Φ be of type 3. Then Φ is computable in E if and only if Φ isdefinable from a d-procedure with a ∆ -representation.Proof. First let Φ be definable from the d-procedure Ω, and assume that the rep-resentation is ∆ . By the boundedness theorem for Σ -sets of codes for ordinals,see e.g. [25, Exercise II 5.9], there will be a computable ordinal λ such that allcalculations in Ω have order-types bounded by λ . Let ( X, ⊳ ) be a computablewell-ordering of length λ , and for each x ∈ X , let X x = { y ∈ X : y ⊳ x } . For each F , and by recursion on the ⊳ -rank of x ∈ X , we will use F and E to compute a string indexed by X x that matches F and is, modulo the choice of denotations,isomorphic to the calculation in Ω matching F , until Ω tells us what the valueΦ( F ) must be. We use the recursion theorem, and explain the step from x to its ⊳ -successor x ′ . So, as an induction hypothesis, we assume that we have constructedthe string t = { ( f y , a y , y ) } y ⊳ x . This string is isomorphic to an initial segment of(( D, ≺ , { ( f ′ d , a ′ d ) } d ∈ D , c ) if there is a d ∈ D with the same rank as x , and the corre-sponding isomorphism r from X x to D d will satisfy that f y = f ′ r ( y ) and a y = a ′ r ( y ) for all y ∈ D x . Now, the set Ω t of calculations in Ω such that the string t isisomorphic to an initial segment will be ∆ relative to t .In order to know what to do next, we first have to split between the two cases:are we able to give out a value for Φ( F ) or must we continue the evaluation, that is,identifying, up to isomorphism, a larger part of the calculation matching F ? SinceΩ has a calculation for Φ( F ), we know that there is at least one calculation in Ω OMPUTABILITY AND NON-MONOTONE INDUCTION 25 of which t is isomorphic to an initial segment. Moreover, if t is actually isomorphicto a calculation in Ω, this is unique, and Ω provides us with the value. So, in orderto decide between the two cases, we ask a “∆ ( t )”-question, i.e. a Σ ( t )-questionand a Π ( t )-question that are equivalent. Those are:‘is t isomorphic to a proper initial segment of some element of Ω t ?’and ‘is t isomorphic to proper initial segments of all elements in Ω t ?’ .In the case the answer is ‘no’, we have constructed a copy of the calculation matching F . Then we can compute the unique value Φ( F ) = c from the data. On the otherhand, if the answer is ‘yes’ , there will be a next query f x that will be unique for allcalculations in Ω t . { f x } is a ∆ -singleton relative to t , and we can compute each f x ( n ) from t and E . In both cases, we can rely on Gandy selection. This provesthe theorem one way.Now assume that Φ( F ) = { e } ( E, F ). We can construct a d-procedure Ω forΦ in analogy with the one we constructed in the proof of Lemma 4.19, withoutrelativizing the construction to a set of ~ Φ with Π -procedures. That the result nowwill be ∆ can be seen from the following consideration:i) Whenever { e } ( E, F ) ↓ we can compute the corresponding calculation( D, ≺ , { ( f d , a d ) } d ∈ D , c )uniformly in e , F and ∃ , by use of the recursion theorem. It is worthnoticing that there will be no delay here, given { f d , a d ′ } d ′ ≺ d , we do notonly have a unique value for the next f , but also for its denotation d , evenif the calculations are matching different F s.ii) Next we observe that when checking if a representation ( D, ≺ , { f d , a d } d ∈ D )of a d-string is a real representation of a real calculation of a value, wecan relax the requirement that ( D, ≺ ) is a well ordering. The checking theway we did it in the proof of Lemma 4.19 is partially computable by therecursion theorem, and can be proved to terminate for a given e under theassumption that there is at least one total F matching the given d-stringsuch that { e } ( E, F ) ↓ .This shows that the d-procedure will be ∆ in this case. (cid:3) Remark 5.2.
We have essentially used that the element of a ∆ -singleton is itselfhyperarithmetical, and this implicitly provides us with a next -function in this case.There is no delay in the ∆ -d-procedure constructed in the above argument, thedenotations in an initial segment of a calculation is uniquely determined by thesegment itself.5.2. The class W ( I ) . In this section we will give a closer analysis of the classof functionals of type 3 that are computable in I . We will do so by investigatingadditional properties of the elements in the following class: Definition 5.3.
Let W ( I ) be the set of d-procedures for functionals Φ computablein I as constructed in the proofs of Lemma 4.16 and Lemma 4.19. Tame d-procedures.
In this sub-section we will introduce two properties sharedby all d-procedures in W ( I ). Definition 5.4.
Let Ω be a procedure defining a total functional.a) Let Ω pre be the set of of ( D, ≺ , { ( f d , a d ) } d ∈ D ) that are isomorphic to aninitial segment of a calculation in Ω.b) Let next Ω be the function mapping t ∈ Ω pre to the disjoint union of N N and N such that- If t is isomorphic to a calculation t ′ in Ω, then next Ω ( t ) is the valueof this calculation- If t is isomorphic to a proper initial segment t ′ of a calculation t ′′ inΩ, then next Ω ( t ) = ( f, c ) where F ( f ) =? is the next query after t ′ in t ′′ (independent of the choice of t ′′ ) and c ∈ { , } . Moreover, if c = 0,then the next pair in t ′′ after t ′ will be of the form ( f, ∗ ) while if c = 1,we continue t ′ with a pair ( f, a ) in t ′′ for some a ∈ N . Definition 5.5.
Let Ω be a d-procedure for a total functional. We say that Ω is tame if Ω is Π , Ω pre is Π and next Ω is partially computable in E . Lemma 5.6.
Let Ω be the d-procedure for I . Then Ω is tame.Proof. We need the full complexity of Π to formulate that we are dealing withwell orderings ( D, ≺ ), the rest is actually arithmetical. Each step in the underlyingrecursion takes ω many steps when we evaluate according to Ω. It is clearly arith-metical to decide if any ordered set of pairs ( f, a ) indexed over N locally satisfiesthe recursion in I , so checking if a d-string is in Ω pre is arithmetical when we knowthat the representation is well ordered. If a string is locally correct, the next queryis arithmetically defined if there is one, and the value is arithmetically express-ible from the list of queries answers if the d-string corresponds to a calculation, so next Ω is computable in E as requested. (cid:3) Lemma 5.7.
The class of functionals definable from tame d-procedures Ω is closedunder Kleene computability.Proof. Let ~ Φ = Φ , . . . , Φ n be defined from the tame d-procedures Ω , . . . , Ω n . Wealready know that the d-procedure for any Φ computable in ~ Φ is Π .We use the recursion theorem to define, for each index e and extra inputs ~f and ~a , a set X e, ~f,~a of d-strings (( D, ≺ , { ( f d , a d ) } d ∈ D )that is Π uniformly in e, ~f , ~a , together with the function next e, ~f,~a defined on X e, ~f,~a , and show thati) If F is of type 2 and { e } (Φ , . . . , Φ n , F, ~f , ~a ) = c and ( D, ≺ , { ( f d , a d ) } d ∈ D , c )is the associated d-calculation obtained from the proof of Lemma 4.19, thenany string isomorphic to an initial segment of ( D, ≺ , { ( f d , a d ) } d ∈ D ) is in X e, ~f,~a .ii) If a string is in X e, ~f,~a and matches some F for which { e } (Φ , . . . , Φ n , F, ~f , ~a )terminates, then the string is isomorphic to an initial segment of the d-calculation obtained through the proof of Lemma 4.19. OMPUTABILITY AND NON-MONOTONE INDUCTION 27 iii) next e, ~f,~a is computable in E uniformly in the parameters and acts asspecified.We define X e, ~f,~a and next e, ~f,~a by cases according to the scheme e represents.Note that since we are dealing with semi-decidable sets, we cannot take NO for ananswer, and search-procedures have to use Gandy selection:- e is an index for an initial computation given by S1-S3, S7: X e, ~f,~a will consistof the empty string only. next e, ~f,~a is trivially given in all these cases.- Composition: { e } ( ~ Φ , F, ~f , ~a ) = { e } ( ~ Φ , F, ~f , { e } ( ~ Φ , F, ~f , ~a ) , ~a ) . Given ( D, ≺ , { ( f d , a d ) } d ∈ D ), where ( D, ≺ ) is a well ordering, we use ( D, ≺ ) recur-sion to test if the initial segments of ( D, ≺ , { ( f d , a d ) } d ∈ D ) are in X e , ~f,~a until weeither found an initial segment that is in X e , ~f,~a and with a value c or we find thatthe given string is in X e , ~f,~a , and thus in X e, ~f,~a . If this search fails, the given stringis not in X e, ~f,~a , and non-termination is not a problem. If this search ends with aproper substring that is in X e , ~f,~a and with a value c , we compare the rest of thestring with X e , ~f,c,~a in the same way. The next -function for e will be inheritedfrom the next -functions of e and e , c , and these can be used to check that thegiven string does not go too far, beyond where we should have a valued string.- We leave the cases for permutation and S9 for the reader, as those cases areeven simpler.- Application of F : { e } ( ~ Φ , F, ~f , ~a ) = F ( λa. { e } ( ~ Φ , F, ~f , a, ~a )) . In analogy with how we dealt with composition, we can compare a given stringpoin-by-point with elements in X e , ~f, ,~a , in X e , ~f, ,~a and so forth until we eitherfind that the given string is a concatenation of finitely many strings in these sets, allexcept the last one maximal, that it is the concatenation of one maximal string fromeach X e , ~f,a,~a in increasing order or that it even contains a final ( g, b ) at the end.In the last case, we also must check if g ( a ) is the value of the string from X e , ~f,a,~a used in the concatenation before accepting the given string. When accepted, weinherit the next -function in the obvious way.- Application of Φ i : { e } ( ~ Φ , F, ~f , ~a ) = Φ i ( λg. { e } ( ~ Φ , F, g, ~f , ~a )) . Here it may be useful to look back on Definition 4.5. We explain informallyhow we, point by point, compare the initial segments of the given string ( D, ≺ , { ( f d , a d ) } d ∈ D ) with the requirements for Ω i and the various sets X e ,g, ~f,~a where we may assume that we have defined these sets as a part of the inductionhypothesis: • If d is the ( D, ≺ )-least element, f d has to be the first query g in Ω i , givento us by next Ω i , with a d = ∗ . • We check the next segment of ( D, ≺ , { ( f d , a d ) } d ∈ D ) for membership in X e ,g, ~f,~a until we have reached a value or until the given string is exhausted. • In the latter case, the string is in X e, ~f,~a and in the first case, we let b bethe value, check that the the pair ( g , b ) is in in Ω i pre and use the next -function of Ω i to verify that the next query in Ω i will be the next query in( D, ≺ , { ( f d , a d ) } d ∈ D ). • By transfinite recursion on ( D, ≺ ) we can continue this comparison untilthe given string is exhausted or until it does not compare with strings in Ω i (whenever we have found a new value there, and can use its next -functionto find the new g β ), or with the strings in the sets X e ,g β , ~f,~a .It is now routine to verify the properties i) - iii). (cid:3) Blocking.
It is not the case that all functionals definable by a tame d-procedures will be computable in I , see Theorem 6.1. The point with the de-notations is that they may make it easier to design non-monotone inductions thatare copying evaluations in a procedure, but the obstacle will be that we will not,in general, be able to tell from a part of a calculation what the correct denotationof the next query will be, there may be a delay as defined in 4.14. We find thisphenomenon in the procedure for I , where we first must establish, within each in aseries of ω -length sub-calculations, if we reached the final fixed point or not beforereading off the correct denotation. That the situation would be simpler withoutthis obstacle, is seen from the following lemma: Lemma 5.8.
Let Ω be a d-procedure for a total functional such that for any sub-string { ( f β , a β , d β ) } β<α of a calculation in Ω , the denotations d β are uniquely givenby { ( f β , a β ) } β<α .If Ω in addition is tame, and the unique choice of d β is computable from { ( f γ , a γ } γ ≤ β and E uniformly at each stage, then the functional defined by Ω is computable in I .Proof. We code an entry ( f β , a β , d β ) as the set {h ¯ f β ( n ) , a β , d β i : n ∈ N } . These sets will be disjoint, so we may code each initial segment of a string as apair of sets, where one is the union X β of such single codes and the other is thecorresponding ordering of the denotations. We will use that the Suslin functional S is computable in I .Given F , we design a non-monotone inductive definition Γ F computable in S thatsimulates the evaluation of the calculation matching F :(1) Given X , we use S to check if X codes an initial segment of a calculationin Ω as coded above. If not, let Γ F ( X ) = X , and if it does, continue.(2) Use next Ω to find the next query f α , use F to find a α = F ( f α ) and finallythe E algorithm that gives us the unique denotation d α .(3) Let Γ F ( X ) be X extended with the code for ( f α , a α , d α ) in the set to theleft and all pairs h d β , d α i for β < α in the set to the right.It is clear that the set I (Γ F ) will code the calculation in Ω matching F , togetherwith the ordering of all the denotations used in that calculation, and we can use next Ω to compute Ω( F ).Further details are left for the reader. (cid:3) OMPUTABILITY AND NON-MONOTONE INDUCTION 29
We will now add further structure to d-procedures, blocks . It will be like insertingcommands of the form \ begin { block } and \ end { block } bracketing blocks and sub-blocks. These imaginary commands must satisfy, for each calculation, the standardrules of bracketing, allowing for infinite branchings in the length, but only finitenesting in depth. Where to put these commands will determined by the initialsegment of the string up to where the command is, and the use will be to markthat there is now an uncertainty to what the denotations will be at the end, andthat we have to carry out a sub-procedure, or evaluate the calculation for F a bitfurther, in order to find the true denotations of the calculation. We will consider twoexamples before giving the abstract definition of a tame d-procedure with blockings: Example 2.
Let Ω be the d-procedure for I . Recall that, given G , Ω( G ) williterate the induction given by F G , generating the sequence { f β } β ≤ α by evaluating G on 0ˆ f , f , . . . ω f , f . . . ω ( α +1) .Each ω -sequence will be a block in this case, and after each block we know whatthe denotation for the query G ( n ˆ f β ) = ? will be, but not while we are inside ablock. However, in order to view the calculation within a block as a sub-procedure,we only need denotations that are unique for queries within this block, and ignorethe larger picture. It is not hard to modify the proof of Lemma 5.8 to see that wecan simulate the procedure for I using I . Example 3.
In the case of composition we constructed the calculations as con-catenations of strings for the two parts, and when defining the new denotations, wepaired the denotations from the first part with 0 and from the second part with 1.We may consider the first part as one block and the other part as another one, butif we from the larger picture know that we are entering a composition, there is noneed for this. There is no delay in deciding what the denotations are inherited inthe construction of denotations for compositions, as there is for the construction ofdenotations for transfinite recursions with an unknown end.We will need the blocking structure on calculations to characterise functionalsof type 3 computable in I in terms of procedures.The blocks will be organised in a nested way, with some blocks being sub-blocksof others. The point is that, within each block, we will define denotations alongthe way, and when the need of a delay is observed, we enter a sub-block where weform temporary denotations that at the end of the block will be rewritten to thetrue ones. The nesting of the blocks will reflect that there may be delays within aperiod of delay, so the rewriting of denotations may go through several levels.We will now give the full definition: Definition 5.9.
Let Ω be a tame d-procedure.a) A block in Ω is an interval t in a calculation t ˆ t ˆ t in Ω. Blocks t in t ˆ t ˆ t and t ′ in t ′ ˆ t ′ ˆ t ′ are considered to be equal if t ˆ t = t ′ ˆ t ′ .b) A blocking of Ω is a set of blocks for each calculation t in Ω satisfying:i) Given two blocks in t , they are either disjoint or one is included in theother.ii) For each calculation, all chains of blocks totally ordered by inclusionwill be finite.iii) t is one of the blocks in t .iv) The level of a block t in t is the number of other blocks in t properlycontaining t as a substring. v) If t = t ˆ t and t ′ = t ˆ t ′ are two calculations in Ω with a commoninitial substring t , and if a block of level m in t starts at the beginningof t , then a block of level m starts in t ′ at the beginning of t ′ .Moreover, if the two blocks coincide until one of them ends, they areequal.iii) above just makes the rest easier to express.c) The blocking is tame if we in addition havei) There is a partial function block Ω computable in E such that foreach string t in Ω pre , block Ω ( t ) decides for each m if there is a blockof level m starting at the next query next Ω ( t ) and decides the levelsof the blocks, if any, ending before the next query.ii) For each block s = { ( f δ , a δ ) } γ ≤ δ<β in a calculation t there is a uniqueinjective denotation { d sδ } γ ≤ δ<α , where these denotations are computedusing the two functions denote and redenote (with subscript Ω ifneeded) both computable in E and where ∗ if t ˆ( f β , a β ) is an initial substring of the calculation t and s isthe block of highest level containing ( f β , a β ) then denote ( t , a β )will be the denotation d sβ . ∗ If s is a block of level m > t = t ˆ s ˆ s ˆ s ˆ t , where t = s ˆ s ˆ s is the block of level m −
1, then redenote with input t ˆ s ˆ s and m will give us d t restricted to s ˆ s . Comment 1.
It is c), ii) that captures the essence of blocking, a block representsthe local delay of deciding what the denotation one level up will be like, and willmake it possible to simulate the evaluation of a procedure as a nested applicationof I . Definition 5.10. An Inductive Procedure is a tame Π -procedure with a tameblocking. Theorem 5.11.
A functional Φ of type 3 is definable from an inductive procedureif and only if Φ is computable in I .Proof. We show first that if Φ is definable from an inductive procedure, then Φis computable from I . We use a nested version of the argument in Lemma 5.8,using everywhere that the functions next , block , denote and redenote arecomputable in E , and thus in I . We use the recursion theorem for Kleene com-putations to make the following precise:Given F , we construct the inductive definition Γ F as in Observation 5.8, usingthe denotation as it comes, until we hit the beginning of a block s . Then we start theexecution of a sub-procedure simulating the evaluation of F along s as an inductivedefinition Γ sF in the same way. This sub-procedure will come to an end when thefull evaluation along s is simulated. At this stage we can describe the next step forΓ F : From the output of Γ sF , E and the assumption on blocks, we can computethe correct denotations along the string up to the end of s , and Γ F just ads the fullsimulation of the evaluation of F in Ω using this denotation.If s has sub-blocks, then Γ sF will have sub-procedures in a similar way, this is whywe need the recursion theorem to formally describe this procedure.In order to prove the other direction we elaborate on the proofs that the class offunctionals satisfying our requirements is closed under Kleene computability, and OMPUTABILITY AND NON-MONOTONE INDUCTION 31 assume that Φ , . . . , Φ n now are defined from inductive procedures.The only case we need to consider is that of application of Φ i , { e } ( ~ Φ , F, ~f , ~a ) = Φ i ( λg. { e } ( ~ Φ , F, g, ~f , ~a )) . Let G ( g ) = { e } ( ~ Φ , F, g, ~f , ~a ). Then there is a calculation { ( g β , b β ) } β<α in Ω i matching G . For each β < α , the pair ( g β , b β ) will be replaced by a substring of thecomposed calculation as follows: ( g β , ∗ ) will just be preserved as it is, while ( g β , b β )is replaced by a string starting with ( g β , ∗ ) and continued with the calculation of F ( g β ).When we defined the denotations for these composed calculations, we gave themdirectly from the denotations in the Ω i -calculation and from the denotations in thecalculations of G ( g β ) without adding any further delay. Thus we can inherit theblocking structure of the Ω i -calculation, and whenever ( g β , a β ) is in one of theseblocks before we compose all the substrings, we let all blocks in t , where ( g β , ∗ )ˆ t is inserted for ( g β , a β ) and t is the calculation of G ( g β ), be new blocks of a higherlevel.When we constructed the d-procedure in this case, we gave the rules for trans-forming the denotations in this simulating block to denotations of the correspondingitems in the full calculation, and this clearly can be relativised to the blocks in Ω i .In order to tie the whole thing up showing that the definability and computabil-ity requirements of what we construct are satisfied, we need to use the recursiontheorem for E , induction on the ordinal lengths of ~ Φ , F -computations and in-duction/recursion on the level of blocks in a string. The details are tedious, butsimple. (cid:3) The I -computable functions revisited. In this section we will considerpure computations { e } ( I , ~a ) , without function and functional parameters.In our definition of a procedure, we used the parameter F to give values toqueries, but at certain points we also inserted elements of the LOG, functionsappearing as arguments in sub-computations but not necessarily as arguments inthe main computable function or functional we design the procedure for.When transforming a computation in I without functional arguments to a pro-cedure this LOG will now be the backbone of the calculations. Since there will beno genuine queries anymore, we can even drop the ∗ for marking element-hood inthe LOG. Thus a pure string will be a triple ( D, ≺ , { g d } d ∈ D ) where ( D, ≺ ) is a wellordering and each g d ∈ N N . We will consider such strings that are Π -singletons,where the set of other strings isomorphic to initial segments of the given one is Π ,where we have a next -function computable in E and where we have functions block , denote and redenote as before, making this one-point set an induc-tive procedure. We call this a pure inductive calculation , and these pure inductivecalculations will reflect I -computations with integer inputs. Remark 5.12.
When transforming a computation { e } ( I , ~a ) = b to a pure inductivecalculation we first of all linearised the computation. Then we hid some of theindexing in the function next , and also in how we designed the denotations, we actually “hid” all intermediate “Kleene-calculations” that do not involve the schemeS8. However, for transfinite computations, this hiding will not significantly reducethe length of a computation. On the other hand, when we translate a pure inductivecalculation to a Kleene-computation, we may add to the length of the computation,partly because it takes time to compute next , block , denote and redenote whenever needed, and partly because we have to add time to the time-span of aninduction in order to verify that the induction comes to a halt when it does.Recall the definition of π as the first ordinal with no code computable in I . Wehave the following characterisation. Lemma 5.13. π is the supremum π ∗ of the ordinal lengths of the pure inductivecalculations.Proof. First, we will prove that π ≤ π ∗ . Let α < π , and let ( X, ≺ ) be I -computableand a well ordering of ordinal length α . We prove this direction by constructinga pure inductive calculation with at least α many steps. We will use the pureinductive calculations deciding x ∈ X and x ≺ y as building blocks, and with thehelp of those we simulate, in the form of a grand pure inductive calculation, theinduction building up X one point at each step, a process that needs exactly α many steps.Then we prove that π ∗ ≤ π . Let t be a pure inductive calculation. Using thesame strategy we used when showing that a functional definable from an inductiveprocedure is computable in I , a strategy involving the recursion theorem for I , wecan show that there is a nested computation relative to I that computes a code forthe ordinal length of t , whenever we enter a block, we compute the length of thatblock as a subcomputation, and then at the end of the block, ads a copy of thiscode to the well ordering we are building up. Actually, it will be the denotationswith their ordering we compute, and in a block, the local denotations within thatblock. (cid:3) Theorem 5.14.
Let α < π . If α is the rank of a pure inductive calculation t , then α is Π -characterisable.Proof. If f ∈ WO has rank β , we can decide if β = α as follows: By recursion onthe wellordering coded by f we can use next t and the E -algorithm organisingthe blockings to compute the corresponding stages in t with full information aboutwhere in the blocking structure we are at each step. If this simulation terminatesexactly at the end of t , we accept f , otherwise we refute it. The set of f s acceptedwill be Π . (cid:3) Tracing computations in I Partial procedures.
In order to make constructions of procedures smoother,we have not insisted on the natural requirement that for each f and calculation,there is at most one occurrence of the query F ( f ) =?. However, when it doesappear several times, the calculation will be based on the same answer everywhere.When we refer to a query F ( f ) =? we will always mean the first occurrence. When { e } ( I , F, ~f , ~a ) ↓ for all F , it is clear that the associated procedure will lead toterminating calculations for all F . This means that when f appears in a query F ( f ) = ? there will be an extension into a calculation for all a ∈ N . Conversely, wecan prove that for every calculation ( { ( f β , a β ) } β<α , c ) constructed in the procedure OMPUTABILITY AND NON-MONOTONE INDUCTION 33 for λF. { e } ( I , F, ~f , ~a ), if F matches this string, then c will be the value of thiscomputation.We will now discuss what happens if we consider computations { e } ( I , F ) thatdo not terminate for all inputs F . In this case, we can still define a set Ω pre ofstrings with denotations, in these strings we may enter blocks and sub-blocks, andthey behave as required for inductive procedures, since we have established theseproperties for each e , F , ~f and ~a such that { e } ( I , F, ~f , ~a ) ↓ .We need to consider parameters ~f and ~a , since such parameters occur in sub-computations, so we reason within this generality.If we consider the construction of the inductive procedure more carefully, we canobserve what we construct in the case of non-termination more closely, again bycases according to what the index e is like:If e is an index for an initial computation, we constructed a trivial procedureyielding the correct output without any queries made.In the case of composition, we first constructed the procedure for the innercomponent, and for the calculations in this procedure (the strings that give us ananswer), we concatenated with calculations from the procedure of the correspondingouter component. In the case the composed computation does not terminate, wedo construct a string modelling the leftmost non-terminating subcomputation.In the cases where there will be exactly one subcomputation, what we do is usingthe string of that one.In the cases of application of F or application of I , we are doing exactly asabove, in case of non-termination we build up the string until we reach the leftmostnon-terminating subcomputation, and ending the string in Ω with a copy of a stringfor this leftmost one.In the case that e is not an index at all, the procedure will be trivial, but withnon-termination as the conclusion.If { e } ( I , F, ~f , ~a ) ↑ , there will be a leftmost Moschovakis witness, a descendingsequence of unsettled computations such that every computation to the left willterminate, and it will be exactly the string corresponding to evaluate { e } ( I , F, ~f , ~a )along this descending sequence of subcomputations that will be constructed in thiscase. If Ω is constructed like this, we will simply have some strings where theconclusion must be ⊥ instead a proper value. However, since being a Moschovakiswitness is semi-decidable, this was the original point with them, we see that Ω pre will still be Π . We will also have functions next , block , denote and redenote computable in E . What may be lacking is that blocks may be entered withoutever being left, that we may have an infinite descending sequence of blocks (thatwill then not have end points) and that we will not have a E -computable way todefine the denotations for the blocks unless they have an end. So, the procedurewill not be an inductive procedure. This is as it has to be, since we can definethe characteristic function of a set of functions of type 2 that is complete semi-computable in I using a procedure like this, replace the value ⊥ with 0 and thevalue a ∈ N with 1 as the values of calculations.These considerations contain the proof of Theorem 6.1.
There is a total procedure Ω such that Ω pre is Π , and such thatthere is a partial function next Ω that is computable in E , but where the functionaldefined by Ω is not computable in I .Proof. The only property left is that we must be able to decide, using a E -algorithm, if a string t in Ω pre is maximal or not, and in case it is maximal, ifit has a value or not. By recursion on the indexing of t we can follow the pointsin the computation tree corresponding to the points in t . If this point in the com-putation tree has an index that is none of the indices of S1 - S9, we can concludethat there is no value. If the blocking depth is infinite, we can conclude that thestring represents non-termination. This can be checked by E , using block andcalculating the lim sup of the block level of the items of the string. In all othercases, there will be a next query or there will be a value given to us by the original next -function (cid:3) We also have
Theorem 6.2.
There is a non-terminating computation { e } ( I , ~a ) such that thelength of the string simulating the leftmost Moschovakis witness will have length atleast π (= ω I ) .Proof. If this were not the case we can use Gandy selection for I -computations andmake I -semidecidable equivalent to I -decidable, obtaining the standard contradic-tion by diagonalisation. (cid:3) Remark 6.3.
Moschovakis witnesses were introduced in [12], and they were signifi-cant for the understanding of higher order computing relative to normal functionalsand in set recursion. They are also introduced in [10, Section 5.2.2], and they wereapplied in the proof of [19, Theorem 6.6].6.2.
Functionals computable in I . In a series of papers [14–21], written jointlywith Sam Sanders , we investigate classes of functionals of type 3 that serve asrealisers for classical theorems in analysis. There are unsettled question concerningthe relative computability of the elements of these classes. In this section we willsee that in the case that elements of these classes are computable in I , we can“almost” compute the Suslin functional from them, and consequently, they will“almost” be computationally equivalent to I itself. We will make the “almost” precise by replacing a functional Φ computable in I with one that traces the historyof the computation, not just gives the value. We call this the honest version of ΦThere is an analogue with what we do in complexity theory where the complexityof a set is often measured by the resources required to decide membership in theset and not by what we can decide using small resources combined with the set asan oracle. In a mathematically precise way we will see that if we compute realisersfor some classical theorems of analysis from I , we need the full power of I in doingso. Definition 6.4.
Let Ω ∈ W ( I ) be the inductive procedure constructed from Φ ascomputable in I . We define the honest version H (Φ) as the functional (of mixedtype) that sends a functional F of type 2 to the fixed point of the inductive definitionΓ F as constructed from Ω in the proof of Theorem 5.11, i.e. as the history of theevaluation of Φ( F ) from I . OMPUTABILITY AND NON-MONOTONE INDUCTION 35 An open covering of a set X in a topological space T is normally defined as a set C of open sets in T whose union is a superset of X . However, if we make use of theconcept of realisers, a realiser of the open covering will be a map sending x ∈ X toan open set O x ∈ T such that x ∈ O x . When Borel [2] gave an attempt of a directproof of the Heine-Borel theorem, he actually, without knowing the concept, usedthis idea of a realiser; with free translation he expressed his assumption as follows:(*) Assume that we have a way of attaching an open neighbourhood O x of x to each x ∈ [ a, b ].In the papers with Sanders, we have considered coverings and related conceptsprimarily over the Cantor space C = { , } N and the Baire space B = N N givenas functionals F of type 2, where F ( f ) defines the neighbourhoods C f ( F ( f )) and B f ( F ( f )) of extensions of f ( F ( f )), depending on which space we consider f to bean element of.We have been looking at the following three classes: Definition 6.5. a) A strong realiser for the Heine-Borel theorem will be a functional Θ suchthat whenever F : C → N , then Θ( F ) = { f , . . . , f n } such that C ⊆ C f ( F ( f )) ∪ · · · ∪ C f n ( F ( f n )) . b) A weak realiser for the Heine-Borel theorem will be a functional θ suchthat whenever F : C → N then θ ( F ) = { s , . . . , s n } where each s i is a finitebinary sequence, { C s , . . . , C s n } is a covering of C and for each i = 1 , . . . , n there is an f i ∈ C such that f i ( F ( f i )) = s i .c) A Pincherle realiser will be a functional M such that whenever F : C → N ,then M ( F ) = N ∈ N and N satisfies:(-) If G : C → N satisfies that G ( g ) ≤ F ( f ) whenever g ( F ( f )) = f ( F ( f ))( F is considered as a realiser for local boundedness) then G is boundedby N on C .The following lemma is trivial: Lemma 6.6.
Every strong realiser for the Heine-Borel theorem computes a weakone, and every weak realiser for the Heine-Borel theorem computes a Pincherlerealiser.
The proof is left for the reader.
Lemma 6.7.
Let M be a Pincherle realiser that is countably based. Let F : C → N be arbitrary, and let X ⊂ C be countable such that M ( G ) = M ( F ) for all G suchthat F and G are equal when restricted to X . Then { C f ( F ( f )) : f ∈ X } is a coveringof C .Proof. Assume not, let M ( F ) = N and define F ( f ) = (cid:26) F ( g ) if g ∈ S f ∈ X C f ( F ( f )) N + 1 if g S f ∈ X C f ( F ( f )) Then M ( F ) = M ( F ) because the two functions agree on X . However, if we define G ( g ) = (cid:26) g ∈ S f ∈ X C f ( F ( f )) N + 1 if g S f ∈ X C f ( F ( f ))6 COMPUTABILITY AND NON-MONOTONE INDUCTION then G obviously satisfies the boundedness condition induced by F , but is notbounded by N , contradicting that N = M ( F ). (cid:3) Theorem 6.8.
Let M be a Pincherle realiser that is computable in I . Then theSuslin functional S is computable in H ( M ) and E .Proof. In [14, Theorem 5.1] it is proved that there is a functional F computable in E such that the collection of neighbourhoods defined from F and the hyperarith-metical binary sequences is not a covering of C . The construction easily relativizesto an arbitrary f ∈ N N so it suffices to show how we can compute a complete Π -setfrom H ( M ), F and E .For a general procedure Ω and an arbitrary G , the calculation of Ω( G ) will form acountable basis for Ω( G ). If F and M are as given, Lemma 6.7 then shows that thecalculation of Ω M ( F ) must contain queries that are not hyperarithmetical. How-ever, in an inductive procedure, if the input functional is computable in E , thenall queries appearing at the level of a computable ordinal must also be computablein E . This follows from the assumption that the next -function is computable in E . So, the calculation of Ω M ( F ) must have a rank that goes beyond ω CK . The setof (indices for the) computable well-orderings will then be both Σ and Π in thiscalculation, and thus computable from this calculation using E . The calculationitself is computable from F and H ( M ), so we are through. (cid:3) In [13] it is proved that I (under the name of IND) is computable in the Suslinfunctional S and any strong realiser for the Heine-Borel theorem. We can improvethis as Lemma 6.9.
Let M be a Pincherle realiser. Then I is computable in M and S .Proof. Let F : C → C be given, and consider F as an inductive definition, definingthe sequence { f β } β<α . This set is coded as a prewellordering ( A, (cid:22) ) where x (cid:22) y if f α ( x ) = f α ( y ) = 1 and we for all β ≤ α have that f β ( y ) = 1 → f β ( x ) = 1. Identify (cid:22) with {h x, y i : x (cid:22) y } . We will see how to compute (cid:22) from F , S and M . We let x, y, z, w, n, m etc. range over N .The idea is, for each n, x, y to construct a functional G n,x,y such that if x (cid:22) y then M ( G n,x,y ) ≥ n and such that G n,x,y is independent of n otherwise. We willthen have that x (cid:22) y ↔ ∀ m ∃ n ( M ( G n,x,y ) > m ) . We will now define G n,x,y ( X ) in cases, where we in all cases except in the lastone have defined G n,x,y ( X ) independently of n , x and y , and so large that (cid:22) willbe different from X for at least one argument z < G n,x,y ( X ). We rename X to (cid:22) X = {h z, w i : h z, w i ∈ X } . Let h z, w i ∈ X ≺ if h z, w i ∈(cid:22) X and h w, z i 6∈(cid:22) X .For all cases below, we assume that none of the earlier cases apply.(1) If (cid:22) X is not a preordering, there is a finite initial binary subsequence s of(the characteristic function of) X such that no extension of s is a preorder-ing. In this case, let G n,x,y ( X ) be the length of the least such s .(2) Let W X be the domain of the well founded part of ≺ X (computable inthe data using S ), and for each z ∈ W X let f Xz be the characteristicfunction of { w ∈ W X : w ≺ X z } and g Xz be the characteristic functionof { w ∈ W X : w (cid:22) X z } .If there is a ≺ X -least w such that g Xw = max { f Xw , F ( f Xw ) } , we knowthat (cid:22) X differs from (cid:22) , and we need to find a finite approximation to OMPUTABILITY AND NON-MONOTONE INDUCTION 37 (the characteristic function of) X where this is manifested. Choose thenumerically least such w . There will be two sub-cases:- There is a z such that f Xw ( z ) = 0, F ( f Xw )( z ) = 0 but g Xw ( z ) = 1. Selectthe numerically least such z . Then we cannot have both z (cid:22) w and w (cid:22) z ,while we have z (cid:22) X w and w (cid:22) X z , so we let G n,x,y ( X ) = max {h z, w i , h w, z i} + 1 . - There is no such z . Then there is a z such that f Xw ( z ) = 0, F ( f Xw )( z ) = 1,but g Xw ( z ) = 0. Then we do have z (cid:22) w and not z (cid:22) X w , so we let G n,x,y ( X ) = h z, w i + 1 for the numerically least such z .(3) If we get to this point, the well-founded part of (cid:22) X is an initial segment of (cid:22) , and we want to decide if this initial segment is proper or not. This istested by letting g be the characteristic function of W X : the initial segmentis proper unless F ( g ) ≤ g . If the initial segment is proper, we can find z such that z (cid:22) z but z W X , recognised by g ( z ) < F ( g )( z ) . If h z, z i 6∈ X ,we let G n,x,y ( X ) = h z, z, i + 1. If h z, z i ∈ X , z is still not in the wellfounded part of ≺ X so there will be a w such that w ≺ z and w is not inthe well founded part of ≺ X . Since z is of minimal rank in ≺ outside X W ,we cannot have that w ≺ z when w X W . We can find such z and w using E and search over N , and we let G n,x,y ( X ) = max {h z, w i , h w, z i} + 1.(4) So far, we have defined G n,x,y ( X ) independently of n , x and y . If we havereached this far, we know that I ( F ) = W X , and we let • G n,x,y ( X ) = n if x ∈ W X , y ∈ W X and x (cid:22) X y • G n,x,y ( X ) = 0 otherwise.Through items (1) - (3) above, we have constructed G n,x,y such that unless X is apreordering with (cid:22) as its well founded part, G n,x,y ( X ) is such that (cid:22) is not in theneighbourhood of X induced by G n,x,y ( X ). Moreover G n,x,y ( X ) is independent of n (and of x and y ) in this case. We further have that G n,x,y is independent of n ifwe do not have x (cid:22) y , while the function F n,x,y ( X ) = (cid:26) n if X = (cid:22) G e,x,y if x (cid:22) y . In this case we musthave that M ( F e,x,y ) ≥ n . This is what we aimed to obtain. (cid:3) Corollary 6.10.
Let M be a Pincherle realiser that is computable in I . Then I and ( H ( M ) , E ) are computationally equivalent. Relativisations
It is a matter of routine to relativise concepts of computability to functions f : N → N or to subsets of N . Our characterisation of the functionals of type 3computable in I using inductive procedures do relativise directly to objects of type1. We even had to do so in order to cope with inductive procedures themselves,since there will be function parameters in subcomputations of the form { e } ( I , ~a ).In this section we will briefly discuss how the concept of an inductive procedurerelativises to parameters ~F of type 2, without going into any technical details. Thekey observation is that we can easily extend the definition of procedures to copewith multiple inputs ~F , or, if we are interested in functionals of type 3 computable in a fixed functional G of type 2, to input pairs F, G . We only have to add thecoordinate of each query when asked during a calculation.Given a partial functional λ ( F, H ) { e } ( I , F, H ) there will be a procedure Ω asbefore, where Ω and Ω pre are Π , there are functions next , block , denote and redenote computable in E . The calculations/strings corresponding to terminat-ing computations will have a blocking accepting the axioms we gave, respecting therules of bracketing and with no infinitely deep chains of blocks. Our main problemin describing what we mean with an inductive G -procedure for a total functional λF. { e } ( F, G ) is to find the right relativisation of Π to G . Another, minor problemis that we must allow information about G in the LOG of a calculation, or in someother way, see Remark 7.4. Definition 7.1.
Let G be a functional of type 2.a) For g ∈ N N , let { g pi } i ∈ N be an enumeration of the set of functions primitiverecursive in g , where the enumeration is uniformly computable in g .A weakly arithmetical formula Φ( g, G ) is a formula that is arithmetical in g and λi.G ( g pi ).b) X ⊆ N N is weakly arithmetical in G if it is defined by g ∈ X ↔ Φ( g, G )for some weakly arithmetical formula Φ.c) X ⊆ N N is Π [ G ] if X is the intersection of a Π -set and a set that is weaklyarithmetical in G . Definition 7.2.
Let Ω be a procedure for a total functional Φ( F ). Let G be oftype 2. Ω is an inductive G -procedure if the following are satisfied:(1) Ω and Ω pre are Π [ G ].(2) There is a function next Ω partially computable in E and terminating onΩ pre .(3) The calculations in Ω have blockings, and the blocking structure is guidedby the partial functions bloc , denote and redenote , computable in E and with the standard properties. Theorem 7.3.
Let G be of type 2 and Φ of type 3. Then Φ is computable in I and G if and only if Φ is definable by an inductive G -procedure.Proof. One way is proved exactly in the same way that we proved that if Φ isdefinable by an inductive procedure, then Φ is computable in I .For the other direction, let Φ( F ) = { e } ( I , F, G ). We consider the procedure Ω + for the partial functional λF, H. { e } ( I , F, H ), where both Ω + and Ω +pre are Π andwhere the functions next , block , denote and redenote are computable in E .Intersecting with a set that is weakly arithmetical in G we get the procedures inΩ + and the strings in Ω +pre that are matching G . Those will be our Ω and Ω pre . (cid:3) Remark 7.4.
Pairs ( g, G ( g )) will still be present in the calculations. They may beconsidered to be elements of the LOG of the G -calculations. This would actuallyrequire a re-definition of our concepts of procedure and calculation , but we leavehow to do it to the reader in case of interest.It may be possible to avoid appearances of pairs ( g, G ( g )) in the procedure for λF. { e } ( F, G ), but then at the cost of the complexity of the next -function and the
OMPUTABILITY AND NON-MONOTONE INDUCTION 39 other functions guiding the blockings. These will then have to be computed by I and G , and not just by E .8. Summary and Open problems
In this paper we have investigated non-monotone induction as given by a func-tional I of type 3 from the perspective of higher order computability theory. Wehave established strong closure properties for the companion L π of the set of func-tions computable in I , and we have represented computations relative to I and pa-rameters of type 2 in the form of inductive procedures and sequential calculations.Computations relative to I can be linearised in a natural way, since application of I can be seen as the result of a linear process indexed by some ordinal, and theordinal rank of a calculation reflects the length of a computation seen as a linearprocess. We have shown that the length of any terminating computation, withinteger inputs in addition to I , is Π -characterisable. There are two open problemsrelated to this: Problem 1.
Are all ordinals α < π Π -characterisable? Problem 2.
Are there ordinals computable in I that are not the length of anycomputation { e } ( I , ~a )?A positive solution to Problem 1 would give us a nice characterisation of the clo-sure ordinal π , but we conjecture that the answer is negative. We also believe thatwhen the two problems are solved, the solution will show that they are connected.Problem 2 asks if there is a gap-structure for computing with I as it is for infinitetime Turing machines, see Hamkins and Lewis [6] or Welch [28], and for recursion in E , and not as for computing relative to the Superjump S . In case there are gaps,it will be of interest to see how the gap structure coincides with the gap structureof infinite time Turing machines computing in time bounded by π .This problem also suggests that there is a distinction between various functionalsof type 3 similar to the one between predicative and non-predicative arguments inmathematics: in order to compute S ( F ) we need to generate the 1-section of F , theset of functions computable in F , and we need that F is total on its own 1-section,and then we have enough information to deduce what S ( F ) will be. This will alsowork when F is partial, as long as it is not so partial that it is undefined for aninput it is able to compute. In order to compute I ( F ) for a partial F we needthat F is total on functions computable from F and I , including the final productof the ‘computation’. In our main theorem for establishing closure properties of π , it was essential for the argument that we construct an induction where π tellsus to stop, and that we thus must stop before π . We can only consider partialinductive definitions computable in I when they also make sense in the case whenthe recursion lasts π steps in order to deduce that they must stop at an earlierstage. This is a kind of non-predicativity.We will end this paper with an example of a partial functional F : C → C thatis computable in I and total on the set of f ∈ C that are computable in I , butwhere the closure set X of the associated inductive definition is not computable in I because F ( X ) is undefined. Example 4.
We define F as a partial function from P ( N × N ) to P ( N × N ): - If X is a well ordering, use Gandy selection for I to find an index e for awell ordering Y ⊆ N × N with domain B and of length extending that of X , and let F ( X ) = X ∪ { ( a, h e, b i ) : a ∈ X ∧ b ∈ B } ∪ { ( h e, b i , h e, c i ) : ( b, c ) ∈ Y } . - If X is not a well ordering, we let F ( X ) = X .During the induction, a new index e must be found each time, so F ( X β ) will bean end-extension of X β with a well ordering of the order-type of some Y β withindex e β for all β < π . The recursion will stop after π steps because then F ( X π ) isundefined. Clearly, X π is not computable in I . Acknowledgements.
I thank Sam Sanders for involving me in the project thispaper is a spin-off of, for reading a preliminary version of this paper, and for givingvaluable feedback on the exposition. Our joint project started with him asking meif I could say anything about the computational properties of some weird-lookingfunctionals of type 3. The rest is history.I am grateful to John Hartley for his comments on the exposition.I am grateful to editors and anonymous referees of other papers from our jointproject, their sharp comments often helped me think more clearly about how topresent higher order computability in the context of those papers, and then of thisone.I also thank the participants of the seminar on mathematical logic at the Uni-versity of Oslo for attending my informal talks on the subjects of this paper, andgiving valuable feedback.
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