From eventually different functions to pandemic numberings
Achilles A. Beros, Mushfeq Khan, Bjørn Kjos-Hanssen, André Nies
aa r X i v : . [ m a t h . L O ] F e b From eventually different functions to pandemicnumberings ⋆ Achilles A. Beros , Mushfeq Khan , Bjørn Kjos-Hanssen − − − ,and Andr´e Nies University of Hawai‘i at M¯anoa, Honolulu HI 96822, USA { beros,mushfeq,bjoernkh } @hawaii.edu University of Auckland, New Zealand [email protected]
Abstract.
A function is strongly non-recursive (SNR) if it is eventuallydifferent from each recursive function. We obtain hierarchy results for themass problems associated with computing such functions with varyinggrowth bounds. In particular, there is no least and no greatest Much-nik degree among those of the form SNR f consisting of SNR functionsbounded by varying recursive bounds f .We show that the connection between SNR functions and canonicallyimmune sets is, in a sense, as strong as that between DNR (diagonallynon-recursive) functions and effectively immune sets. Finally, we intro-duce pandemic numberings, a set-theoretic dual to immunity. It has been known for over a decade that bounding diagonally non-recursivefunctions by various computable functions leads to a hierarchy of computationalstrength [1,16] and this hierarchy interacts with Martin-L¨of random reals andcompletions of Peano Arithmetic [10,11]. The strongly non-recursive functionsform an arguably at least as natural class, and here we start developing analogoushierarchy results for it.
Definition 1.
A function f : ω → ω is strongly nonrecursive (or SNR ) if forevery recursive function g , for all but finitely many n ∈ ω , f ( n ) = g ( n ) . It is strongly non-partial-recursive (or SNPR ) if for every partial recursive function g , for all but finitely many n , if g ( n ) is defined, f ( n ) = g ( n ) . Note that every SNPR function f is SNR, as well as almost DNR: for allbut finitely many n , if ϕ n ( n ) is defined, then f ( n ) = ϕ n ( n ). Also, a function isstrongly nonrecursive iff it is eventually different from each recursive function.Thus it is eventually different in the sense of set theory with the recursive setsas ground model [2]. ⋆ This work was partially supported by a grant from the Simons Foundation ( efinition 2. An order function is a recursive, nondecreasing, and unboundedfunction h : ω → ω such that h (0) ≥ . For a class C of functions from ω to ω ,let C h denote the subclass consisting of those members of C that are bounded by h . Theorem 1.
For each order function h , there exists an order function g suchthat every DNR g function computes an SNPR h function. In order to prove this theorem, we will need a result due to Cenzer andHinman [8], in a form presented in Greenberg and Miller [10].
Definition 3 (Greenberg and Miller [10]).
Let a ≥ and let c > . Let P ca denote the class of functions f bounded by a such that for all e and for all x < c ,if ϕ e ( x ) ↓ , then f ( e ) = ϕ e ( x ) . Theorem 2 (Cenzer and Hinman [8]).
Let a ≥ and c > . Then any DNR a function computes a function in P cca . Moreover, the reduction is uniformin a and c .Proof (Proof of Theorem 1). Let r be the recursive function such that ϕ r ( x ) ( e ) = ϕ e ( x ). For each n ≥
2, let x n ∈ ω be the least such that h ( x n ) ≥ n .We construct g to ensure that any DNR g function computes a function f that is bounded by h and such that for all x > x n and for all e < n , if ϕ e ( x ) ↓ ,then f ( x ) = ϕ e ( x ).In order to compute f on the interval [ x n , x n +1 ), a function in P nn suffices andsuch a function can be uniformly obtained from a DNR n function, by Theorem 2.However, we only need a finite part of this function, and we can recursivelydetermine how much. Note that the reductions in Theorem 2 can be assumedto be total. For n ≥
2, let γ n denote the use of reduction that, given a DNR n function, computes a P nn function.Then, letting m n = max( { γ n ( r ( x )) : x ∈ [ x n , x n +1 )] } ) , it suffices for g to be any recursive function such that for all x ≤ m n , g ( x ) ≤ n .We also have the following counterpart to Theorem 1: Theorem 3.
For each order function g , there exists an order function h suchthat every DNR g function computes an SNPR h function.Proof. For n ≥
1, let τ n : ω → ω n be a uniformly recursive sequence of bijec-tions, and for i < n , π ni : ω n → ω denote the projection function onto the i -thcoordinate.Let r be the recursive function such that for n ≥ e < n , if ϕ e ( n )converges, then ϕ r ( e,n ) , on any input, outputs π ne ( τ n ( ϕ e ( n )).Now, given a DNR g function f , let j (0) = 0 and for n ≥
1, let j ( n ) = τ − n ( h f ( r (0 , n )) , ..., f ( r ( n − , n )) i ) . hen j ( n ) = ϕ e ( n ) for any e < n : If it were, then we would have f ( r ( e, n )) = π ne ( τ n ( j ( n ))) = π ne ( τ n ( ϕ e ( n ))) = ϕ r ( e,n ) ( r ( e, n )) , which contradicts the fact that f is DNR. Thus, j is SNPR h where for n ≥ h ( n ) = max( { τ − n ( h i , ..., i n − i ) : i k < g ( r ( k, n )) for all k < n } ) . Theorem 4 (Kjos-Hansen, Merkle, and Stephan [13]).
Every non-high
SNR is SNPR .Proof.
Supose that f : ω → ω is not high, and that ψ is a partial recursivefunction that is infinitely often equal to it. For each n ∈ ω , let g ( n ) be the leaststage such that |{ x ∈ ω : ψ ( x )[ g ( n )] ↓ = f ( x ) }| ≥ n + 1. Then g is recursive in f .Since f is not high, there is a recursive function h that escapes g infinitelyoften. We define a recursive function j that is infinitely often equal to f . Let j = ∅ . Given j n , let A = {h x, ψ ( x ) i : x / ∈ dom( j n ) , ψ ( x )[ h ( n )] ↓} . Let y be the least such that it is not in the domain of j n ∪ A . Finally, let j n +1 = j n ∪ A ∪ h y, i .It is easily seen that j = S n j n is recursive and infinitely often equal to f . Corollary 1.
Given any order function h , every non-high SNR h function com-putes a DNR h function. The following definitions can also be found in [10] and [12].
Definition 4.
Given σ ∈ ω <ω , we say that a tree T ⊆ ω <ω is n -bushy above σ if every element of T is comparable with σ , and for every τ ∈ T that extends σ and is not a leaf of T , τ has at least n immediate extensions in T . We refer to σ as the stem of T . Definition 5.
Given σ ∈ ω <ω , we say that a set B ⊆ ω <ω is n -big above σ ifthere is a finite n -bushy tree T above σ such that all its leaves are in B . If B isnot n -big above σ then we say that B is n -small above σ . Proofs of the following lemmas can be found in [10] and [12].
Lemma 1 (Smallness preservation property).
Suppose that B and C aresubsets of ω <ω and that σ ∈ ω <ω . If B and C are respectively n - and m -smallabove σ , then B ∪ C is ( n + m − -small above σ . emma 2 (Small set closure property). Suppose that B ⊂ ω <ω is n -smallabove σ . Let C = { τ ∈ ω <ω : B is n -big above τ } . Then C is n -small above σ .Moreover C is n -closed , meaning that if C is n -big above a string ρ , then ρ ∈ C . Definition 6.
Given an order function h , let h <ω denote the set of finite stringsin ω <ω whose entries are bounded by h , and let h n denote the set of such stringsof length n . Theorem 5.
Let h be any order function. Then, uniformly in h , we can find arecursive function π such that if g is any order function such that h ( n ) /g ( π ( n )) is unbounded, then there is a low f ∈ DNR h that computes no DNR g function.Proof. Given σ ∈ h <ω , let q ( σ, e, k ) be an index for the partial recursive functionthat searches for a k -big set A ⊂ h <ω above σ such that Φ τe ( q ( σ, e, k )) convergesand is constant as τ ranges over A , and which then outputs this constant value.Let π ( n ) = max { q ( σ, e, k ) : σ ∈ h n , e, k ≤ n } . Next, we describe a 0 ′ -recursive construction of f . We define a sequence f (cid:22) f (cid:22) f (cid:22) ... of finite strings in h <ω , and B ⊆ B ⊆ B ⊆ ... of r.e. subsets of h <ω such that for each s ∈ ω , B s is h ( | f s | )-small and h ( | f s | )-closed above f s .Let f = hi , and let B be the set of non-DNR strings. Next, we describehow to construct f s +1 and B s +1 given f s and B s .If s = 2 e is even: We ensure that Φ fe is not DNR g . Let k = h ( | f s | ) and let n ≥ k, e be the least such that h ( n ) ≥ k ( g ( π ( n )) + 1). We begin by extending f s to astring σ / ∈ B s of length n . Note that B s is k -small and k -closed above σ . Let x = q ( σ, e, k ), and note that x ≤ π ( n ).Now, if ϕ x ( x ) ↓ to some value i less than g ( x ), then the set A i = { τ (cid:23) σ : Φ τe ( x ) ↓ = i } is k -big above σ , so there is an extension τ of σ such that τ ∈ A i \ B s . Let f s +1 = τ and B s +1 = B s . This forces Φ fe to not be DNR.Otherwise, for each i < g ( x ), A i is k -small above σ , and so C = [ i For every order function h there is an order function g such thatthere is a low DNR h that computes no DNR g . Additionally, we have: Corollary 3. For every order function g there is an order function h such thatthere is a low DNR h that computes no DNR g .Proof. Using the uniformity in Theorem 5 along with the recursion theorem,we construct h knowing its index in advance, thereby obtaining π , and ensuringthat h ( n ) /g ( π ( n )) is unbounded.By combining the strategies for the two corollaries above, we get: Corollary 4. For every order function h there is an order function g such thatthere is a low f ∈ DNR g that computes no DNR h as well as a low f ∈ DNR h that computes no DNR g . Corollary 5. Given any order function h there is an order function g such thatthere is an SNR h that computes no SNR g .Proof. By Theorem 1, there is an order function h ′ such that any DNR h ′ com-putes an SNR h . By Corollary 2, there is an order function g such that there isa low DNR h ′ function f ′ that computes no DNR g function. Then f ′ computesan SNR h function f that computes no SNR g function: if j is recursive in f and is an SNR g function and since it is low, it is itself DNR g by Corollary 1, acontradiction. Corollary 6. Given any order function g there is an order function h such thatthere is an SNR h that computes no SNR g .Proof. By Corollary 3, there is an h ′ and a low DNR h ′ function f ′ that computesno DNR g function. By Theorem 3 there is an h such that f ′ computes an SNR h function f . Then f cannot compute an SNR g function since the latter would beDNR g by Corollary 1.Let O denote the set of all order functions. Recall the Muchnik and Medvedevreducibilities of mass problems: efinition 7. A mass problem A is Muchnik reducible to a mass problem B ,written A ≤ w B and sometimes read “weakly reducible”, if for each B ∈ B , thereis an A ∈ A such that A ≤ T B , where ≤ T is Turing reducibility. If there is asingle Turing reduction Φ such that for all B ∈ B , Φ A ∈ A then A is Medvedevreducible to B , written A ≤ S B and sometimes read “strongly reducible”. We can phrase Corollaries 6 and 5 as follows: ∀ h ∈ O ∃ g ∈ O SNR g w SNR h ; ∀ g ∈ O ∃ h ∈ O SNR g w SNR h . Thus, the Muchnik degrees of various mass problems SNR f have no least orgreatest element. Canonical immunity was introduced by three of the present authors in [6] andshown there to be equivalent, as a mass problem, to SNR, and studied furtherin [5]. Here we give a new Theorem 10 below, analogous to the case of DNR,that was not obtained in [6].Considering lowness notions associated with Schnorr randomness was whatlead those authors to this new notion of immunity. Definition 8. A canonical numbering of the finite sets is a surjective function D : ω → { A : A ⊆ ω and A is finite } such that { ( e, x ) : x ∈ D ( e ) } is recursiveand the cardinality function e 7→ | D ( e ) | , or equivalently, e max D ( e ) , is alsorecursive. We write D e = D ( e ). Definition 9. R is canonically immune ( CI ) if R is infinite and there is arecursive function h such that for each canonical numbering of the finite sets D e , e ∈ ω , we have that for all but finitely many e , if D e ⊆ R then | D e | ≤ h ( e ) . We include proofs of some of the results from [6]. Theorem 6 (Beros, Khan, and Kjos-Hanssen [6]). Schnorr randoms arecanonically immune.Proof. Fix a canonical numbering of the finite sets, { D e } e ∈ ω . Define U c = { X :( ∃ e > c ) (cid:0) | D e | ≥ e ∧ D e ⊂ X (cid:1) } . Since e 7→ | D e | is recursive, µ ( U c ) is recursiveand bounded by 2 − c . Thus, the sequence { U c } c ∈ ω is a Schnorr test. If A is aSchnorr random, then A ∈ U c for only finitely many c ∈ ω . We conclude that A is canonically immune. Theorem 7 (Beros, Khan, and Kjos-Hanssen [6]). Each canonically im-mune set is immune.roof. Suppose A has an infinite recursive subset R . Let h be any recursivefunction. Let R n denote the set of the first n elements of R , and let { D e : e ∈ ω } be a canonical numbering of the finite sets such that D n = R h (2 n )+1 for all n ∈ ω . For all n , D n ⊆ R ⊆ A and | D n | = h (2 n ) + 1 > h (2 n ), and so h doesnot witness the canonical immunity of A .We now show that canonically immune is the “correct” analogue of effectivelyimmune. Let W , W , W , ... be an effective enumeration of the recursivelyenumerable (or r.e.) sets of natural numbers. An infinite set A of natural numbersis said to be immune if it contains no infinite r.e. subset. It is said to be effectivelyimmune when there is a recursive function f such that for all e , if W e is a subset of A , then | W e | ≤ f ( e ). The interest in sets whose immunity is effectively witnessedin this manner originally arose in the search for a solution to Post’s problem; formore on this the reader may see [6]. Theorem 8 (Beros, Khan, and Kjos-Hanssen [6]). Each canonically im-mune ( CI ) set computes a strongly nonrecursive function, i.e., SNR ≤ w CI . Incidentally, Beros and Beros [4] showed that the index set of Medvedev reduc-tions from CI to SNR is Π -complete. Theorem 9 (Kjos-Hanssen [13]). Each SNR function is either of high orDNR Turing degree. Corollary 7 (Beros, Khan, and Kjos-Hanssen [6]). The following are equiv-alent for an oracle A :1. A computes a canonically immune set,2. A computes an SNR function,3. A computes an infinite subset of a Schnorr random. If a canonically immune set A is of non-high Turing degree then by Theorem 8, A computes an SNR function, which by Theorem 9 means that A computes aDNR function, hence A computes an effectively immune set. Our new result isto make this more direct: A is itself that effectively immune set. Theorem 10. If A is non-high and canonically immune, then A is effectivelyimmune.Proof. Let us introduce the notation W e,s u to mean the set of k ∈ W e,s such that when k enters W e,s , at most u othernumbers have already entered W e,s . Roughly speaking, D u consists of the first u + 1 elements of D .et A be non-high and not effectively immune. We need to show that A isnot canonically immune.Since A is not effectively immune, there are infinitely many e for which thereis an s with h e, s i ∈ F where F = {h e, s i : | W e,s | > h ( e ) , and W e,s h ( e ) ⊆ A } . Here we assume h is nondecreasing. So ∀ d ∃ s ∃ e d > d h e d , s i ∈ F Let f ( d ) = s . Then there is a recursive function g ( d ) which is not dominated by f . We may assume s > e d since increasing s will keep h e, s i ∈ F . Let D h e,d i = W e,g ( d ) h ( d )Let ˜ h ( h e, d i ) = h ( d ) for d ≤ e ≤ g ( d ). Using a recursive bijection between thedomain of ˜ h , {h e, d i : d ≤ e ≤ g ( d ) } and ω , we may assume the domain of ˜ h is ω .Then for infinitely many e (namely, there are infinitely many d with f ( d ) ≤ g ( d ), and for each such d there is an e with d ≤ e ≤ g ( d ) that works) we have1. D h e,d i = W e,g ( d ) h ( d ) ⊆ W e,g ( d ) h ( e ) ⊆ A , and2. | D h e,d i | > ˜ h ( h e, d i ) = h ( d ).We have lim n →∞ ˜ h ( n ) = ∞ since for each d we only include e up to the e d above.Let the sets D k +1 be a canonical list of all the finite sets, just in case wemissed some of them using the sets D k . Brendle et al. [7] explored an analogy between the theory of cardinal character-istics in set theory and highness properties in computability theory, following upon work of Rupprecht [15].Their main results concerned a version of Cichon’s diagram [3] in computabil-ity theory. They expressed the cardinal characteristics in Cichon’s diagram aseither d ( R ) = min {| F | : F ⊆ Y and ∀ x ∈ X ∃ y ∈ F ( x R y ) } , d ( R ) = min {| F | : F ⊆ Y and ∀ x ∈ X ∃ y ∈ F ( x R y ) } or b ( R ) = min {| G | : G ⊆ X and ∀ y ∈ Y ∃ x ∈ G ¬ ( x R y ) } , b ( R ) = min {| G | : G ⊆ X and ∀ y ∈ Y ∃ x ∈ G ¬ ( x R y ) } , here X and Y are two spaces and R is a relation on X × Y . As the spacesconsidered admit a notion of relative computability, it is natural to say that anelement x ∈ X is computable (in A ⊂ ω ).They defined two computability-theoretic notions corresponding to d ( R ) and b ( R ) as follows: B ( R ) = { A : ∃ y ≤ T A ∀ x ( x is computable → x R y ) } , B ( R ) = { A : ∃ y ≤ T A ∀ x ( x is computable → x R y ) } , and D ( R ) = { A : ∃ x ≤ T A ∀ y ( y is computable → ¬ ( x R y )) } , D ( R ) = { A : ∃ x ≤ T A ∀ y ( y is computable → ¬ ( x R y )) } . They found that B ( R ) and D ( R ) tend to be highness properties in computabilitytheory, often equivalent to well-known notions.They finally mapped a cardinal characteristic d ( R ) or b ( R ) to D ( R ) or B ( R )respectively, and showed that if b ( R ) ≤ d ( S ), then B ( R ) ⊆ D ( S ) and so on.The resulting analog of Cichon’s diagram is not isomorphic to the originaldiagram, in that some strict inequalities of cardinal characteristics are consistentwith ZFC but the corresponding computability-theoretic notions coincide.Using their point of view we find a new dual notion to canonical immunity:that of a numbering such that no recursive set is large with respect to it . Definition 10. Let D = ( e D e ) be a numbering of the finite subsets of ω and let R ⊆ ω . We say that D is h -endemic to R if there are infinitely many e with | D e | ≥ h ( e ) and D e ⊆ R . D is a pandemic numbering if there is an orderfunction h such that for all infinite recursive sets R , D is h -endemic to R . Recall also that an escaping function is a function f : ω → ω such that f is notdominated by any recursive function. Theorem 11. The Muchnik degrees of pandemic numberings and of hyperim-mune sets coincide.Proof. In one direction, if e max D e is recursively bounded then we show that D is not a pandemic numbering. Namely, we construct a recursive set R whichwaits for h to get large ( h ( e ) > d say) and only then lets its d th element r d enter R , and lets r d be large enough (larger than max D k , k ≤ e ) to prevent | D e | ≥ h ( e ), D e ⊆ R .In the other direction, given a hyperimmune set, we (as is well known) alsohave an escaping function f . At those inputs e where f is greater than a functionassociated with a potential infinite recursive set R k (given as the graph of apartial recursive { , } -valued function), we define the next D e so as to ensure D e ⊆ R and | D e | ≥ h ( e ). Namely, the function to escape is the time e t k ( e ) = t ( h e, k i ) it takes for R k to get h ( e ) many elements. If R k is really infinite then t k is total and so f ( e ) ≥ t k ( e ) for some (infinitely many) e . So we define D h e,k i to consist of the first h ( h e, k i ) elements of R k , if any, as found during a searchtime of f ( h e, k i ) or even just f ( e ).y Corollary 7 and Theorem 11, the dualism between immunity and pandemicsis the same, Muchnik-degree-wise, as that between, in the notation of Brendleet al. [7], – eventually different functions, b ( = ∗ ), and – functions that are infinitely often equal to each recursive function, d ( = ∗ ). Remark 1. As the recursive sets are closed under complement, a notion of bi-pandemic would be the same as pandemic . Thus, while we do not know whethercanonical bi-immunity is Muchnik equivalent to canonical immunity, there is nocorresponding open problem on the dual side. References 1. 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