Anti-complex sets and reducibilities with tiny use
Johanna N.Y. Franklin, Noam Greenberg, Frank Stephan, Guohua Wu
aa r X i v : . [ m a t h . L O ] O c t ANTI-COMPLEX SETS AND REDUCIBILITIESWITH TINY USE
JOHANNA N.Y. FRANKLIN, NOAM GREENBERG, FRANK STEPHAN,AND GUOHUA WU
Abstract.
In contrast with the notion of complexity, a set A is called anti-complex if the Kolmogorov complexity of the initial segments of A chosen bya recursive function is always bounded by the identity function. We showthat, as for complexity, the natural arena for examining anti-complexity is theweak-truth table degrees. In this context, we show the equivalence of anti-complexity and other lowness notions such as r.e. traceability or being weaktruth-table reducible to a Schnorr trivial set. A set A is anti-complex if andonly if it is reducible to another set B with tiny use , whereby we mean thatthe use function for reducing A to B can be made to grow arbitrarily slowly,as gauged by unbounded nondecreasing recursive functions. This notion ofreducibility is then studied in its own right, and we also investigate its rangeand the range of its uniform counterpart. Introduction
In a recent talk [24], Nies gave a general framework for relating lowness notionsand their dual highness notions with what he names “weak reducibilities” (with aprominent example being LR , the low-for-randomness partial ordering). Even be-fore their extensive investigation in the context of effective randomness, in classicalrecursion theory, both strong and weak reducibilities gave rise to lowness and high-ness classes. For example, truth-table (or weak truth-table) reducibility inducedthe classes of superlow and superhigh Turing degrees; in the other extreme, hy-perarithmetic reducibility (and the hyperjump) allowed the definition of the usefulclass of hyperlow hyperdegrees (see [29]). In this paper we give a new notion ofrelative strength which, surprisingly, leads to a lowness notion which is analogousto familiar ones in the context of the weak truth-table degrees.The motivation for our notion, which we call “Turing reducibility with tinyuse”, comes from recent investigations into strengthenings of weak truth-table re-ducibility in a direction which is incomparable with truth-table reducibility, namely computable Lipshitz reducibility cl (also known as sw , strong weak truth-table reducibility) and identity-bounded Turing reducibility ibT , and also, to a smallerextent, related reducibilities such as C and H . These reducibilities were intro-duced in order to combine the traditional Turing reduction, that is, computationof the membership relation using an oracle, and calibration of relative randomness,usually on the domain of left-r.e. reals (see [2]). The second author was supported in part by the Marsden grant of New Zealand and by NTUgrant RG58/06, M52110023. The third author is supported in part by the NUS grants R252-000-308-112 and R252-000-420-112; he worked on this paper while on sabbatical leave to VictoriaUniversity of Wellington in October and November 2010.
Recall that computable Lipschitz reductions are weak truth-table reductions inwhich the use function is bounded by n + c for some constant c . The idea of a Turingreduction with tiny use is to further restrict the use function of the reduction torecursive functions which grow more and more slowly. A set A is reducible to aset B with tiny use if one can use arbitrarily little of the oracle B to computearbitrarily much of A , so not only does B contain all the information that A has,it compresses that information arbitrarily well. To make the definition precise, weinvoke the following definition first made by Schnorr [30]: an order function (orsimply an order ) is a recursive function which is nondecreasing and unbounded. Definition 1.1.
Let
A, B ∈ { , } ω . We say that A is reducible to B with tiny use and write A T ( tu ) B if for every order function h , there is a Turing reduction of A to B whose use function is bounded by h .Let us agree on some notation and terminology. If Φ is a Turing reduction (a Turingmachine with an oracle) and Φ B = A , then we let, for every n < ω , the use of thisreduction, ϕ ( n ) = ϕ B ( n ), be m + 1, where m is the largest number which is queriedby Φ while computing A ↾ n . Here A ↾ n is the string A (0) A (1) . . . A ( n − A ( n ) = Φ B ( n ), Φ first computes Φ B ( m ) for all m < n . Thus B ↾ ϕ ( n ) is the shortest initial segment of B which, serving as anoracle for Φ, outputs A ↾ n .The motivation for considering reducibility with tiny use comes from a result ofGreenberg and Nies [12], who showed that if A is a recursively enumerable, stronglyjump-traceable set and B is an ω -r.e. random set, then A is reducible to B withtiny use. In fact, in [13] it is shown that this is a characterisation of the stronglyjump-traceable r.e. sets.The relation T ( tu ) yields a lowness notion in a very simple way: we consider thecollection of sets A for which there is some B such that A T ( tu ) B . An immediateanalysis of T ( tu ) shows that this collection is invariant in the weak truth-tabledegrees and induces an ideal in these degrees. This ideal can be characterised inthree other ways, for which we make a sequence of definitions.Recall that for their work characterising lowness for Schnorr randomness as re-cursive traceability, extending a fundamental result of Terwijn and Zambella [32],Kjos-Hanssen, Merkle and Stephan [16] defined a set A to be complex if there isan order function f such that C ( A ↾ f ( n )) > n for all n (here C denotes plainKolmogorov complexity ). They showed that a set A is complex if and only if thereis some diagonally nonrecursive function f wtt A . As an analogue, we make thefollowing definition: Definition 1.2.
A set A ∈ { , } ω is anti-complex if for every order function f ,for almost all n , C ( A ↾ f ( n )) n .Thus anti-complexity is a mirror image of complexity: complexity indicates incom-pressibility in that one can effectively find locations of high complexity, whereasanti-complexity denotes a high level of compressibility and hence low informationcontent. Recall that a machine is a partial recursive function M : { , } ∗ → { , } ∗ . If M is a machine,then the M -complexity of a string σ in the range of M , denoted by C M ( σ ), is the length of theshortest string τ ∈ M − { σ } . If σ is not in the range of M , then we write C M ( σ ) = ∞ . A machine U is optimal if for every machine M there is some constant c such that for all σ ∈ { , } ∗ , C U ( σ ) C M ( σ ) + c . Then C denotes C U for some fixed optimal machine U . EDUCIBILITIES WITH TINY USE 3
Traceability, in both its recursive and r.e. versions, is a notion which has turnedout to be extremely useful in algorithmic randomness and classical recursion theory.Recent work of Franklin and Stephan [11] has indicated that it is also useful in thecontext of strong reducibilities. They have shown that the class of Schnorr trivialsets is invariant in the truth table degrees and that a set is Schnorr trivial if andonly if its truth-table degree is recursively traceable (this means that only thefunctions which are truth-table reducible to the set receive a recursive trace, allwith a uniform bound of some order). Since the natural environment for T ( tu ) isthe weak truth-table degrees, we find that traceability in these degrees plays a rolehere. The characterisation theorem is as follows. Theorem 1.3.
The following are equivalent for a set A . (1) There is a set B such that A T ( tu ) B . (2) A is anti-complex. (3) deg wtt ( A ) is r.e. traceable. (4) A is weak truth-table reducible to a Schnorr trivial set. We note that the equivalence of (3) and (4), together with Franklin and Stephan’sresult, yields a theorem which has no explicit connection to effective randomness,and yet we currently do not know of any direct proof that does not involve T ( tu ) and Kolmogorov complexity: a weak truth-table degree a is r.e. traceable if andonly if there is some weak truth-table degree b > a which contains a set B whosetruth-table degree is recursively traceable.As notions of traceability of total functions are equivalent to their strong versions(see Lemma 3.3), it follows that a Turing degree a is r.e. traceable if and only ifevery weak truth-table degree contained in a is r.e. traceable. Theorem 1.3 thenimplies the following characterisation of r.e. traceability in the Turing degrees. Theorem 1.4.
The following are equivalent for a Turing degree a . (1) a is r.e. traceable. (2) Every set A ∈ a is anti-complex. (3) Every set A ∈ a is weak truth-table reducible to a Schnorr trivial set. Among r.e. degrees, we note that the equivalence between array recursiveness andr.e. traceability holds in the weak truth-table degrees. Recall that a very strongarray ¯ F = h F n i n<ω consists of a recursive sequence of pairwise disjoint finite setssuch that for all n , | F n | > n , and that an r.e. set A is ¯ F -ANR if for every r.e. set B there are infinitely many n such that A and B coincide on F n . Theorem 1.5.
The following are equivalent for a weak truth-table degree a con-taining an r.e. set. (1) For no very strong array ¯ F does a contains an ¯ F -ANR set. (2) For some very strong array ¯ F , a contains no ¯ F -ANR set. (3) There is an ω -r.e. function that dominates all functions recursive in a . (4) a is r.e. traceable. (5) For all A ∈ a , A T ( tu ) K ( here K = { e : ϕ e ( e ) ↓} is the halting set ) . This result implies the result from [7] that the array recursive r.e. wtt-degrees forman ideal.Together with T ( tu ) , we also investigate a uniform version uT ( tu ) , where a singlereduction witnesses the relation T ( tu ) . This relation is, in general, much stronger FRANKLIN, GREENBERG, STEPHAN, AND WU than T ( tu ) (for example, if A is nonrecursive and A uT ( tu ) B , then B is high),but their domains are the same, and so the condition “there is a set B such that A uT ( tu ) B ” can be added as a fifth equivalent condition in Theorem 1.3. Aneven stronger version of this theorem which bounds the complexity of such B isTheorem 3.8. We prove Theorem 1.3 in Section 3. In Section 4 we investigatethe distribution of the anti-complex sets in the Turing degrees, discuss high andrandom degrees, prove Theorems 1.4 and 1.5, and investigate anti-complexity andtiny use in the r.e. degrees. One corollary of our investigations is an answer toQuestion 7.5.13 from Nies’s book [22]. Theorem 1.6.
Not every high Turing degree contains a partial-recursively randomset.
The motivation behind this question is to find an exact boundary between weakernotions of randomness, such as Schnorr randomness and recursive randomness,which occur in every high Turing degree, and stronger notions of randomness, suchas Martin-L¨of randomness, which do not. We provide a proof of Theorem 1.6 inSection 4.In Section 5, we investigate the dual highness notions: the sets B for which thereis a nonrecursive set A such that A T ( tu ) B (or the more stringent A uT ( tu ) B ).We investigate the situation in both the hyperimmune-free ( -dominated) degreesand in the r.e. and ∆ degrees. For example, we show that every high Turing degreecontains sets A and B such that A uT ( tu ) B and that for every nonrecursive r.e.set B there is some nonrecursive r.e. set A such that A T ( tu ) B .Throughout the paper, we also mention strong reducibilities (such as truth-tableand many-one) with tiny use. In particular, in Theorem 3.11 we use truth-tablereducibility with tiny use to obtain a new characterisation of Schnorr triviality:a set A is Schnorr trivial if and only if it is truth-table reducible to some set B with tiny use. This result strengthens the intuition, arising from Franklin andStephan’s characterisation of Schnorr triviality in terms of recursive traceability inthe truth-table degrees, that strong reducibilities have deep connections with weakrandomness notions. Along this vein, Day [1] has recently given characterisationsof both Schnorr randomness and computable randomness as the complements ofthe domains of relations weaker than truth-table reducibility with tiny use. Forexample, he showed that a set A is not Schnorr random if and only if there is someset B such that A tt B with use function which does not dominate n − h ( n ) forsome order function h .In the following section we supply the rest of the basic definitions and make somebasic observations. 2. Basics
We first define the uniform reducibility.
Definition 2.1.
Let
A, B ∈ { , } ω . We say that A is uniformly reducible to B with tiny use (and write A uT ( tu ) B ) if there is a Turing reduction Φ B = A whoseuse function is dominated by every order function. Observation 2.2. (1) If A uT ( tu ) B , then A T ( tu ) B . (2) If A T ( tu ) B , then A wtt B . EDUCIBILITIES WITH TINY USE 5
Remark 2.3.
Despite the fact that our reductions imply weak truth-table reduc-tions, we prefer the notation T ( tu ) to wtt ( tu ) . This is because a weak truth-tablereduction first marks the use, then queries the oracle and finally computes the value,whereas Turing reductions with tiny use would — at least in the uniform case —not do the operations in this order, as otherwise the use is automatically boundedfrom below by an order function. Next, we see that our relations are invariant in the wtt-degrees.
Observation 2.4. If A wtt E and E T ( tu ) B , then A T ( tu ) B ; if A T ( tu ) E and E wtt B , then A T ( tu ) B . Thus the relation T ( tu ) is invariant onweak truth-table degrees and is preserved by increasing the degree on the range anddecreasing the degree on the domain. The same holds for uT ( tu ) . Observation 2.5.
For a fixed B ∈ { , } ω , the classes { A : A T ( tu ) B } and { A : A uT ( tu ) B } are wtt-ideals. Another formulation for our notions uses not the use functions but their discreteinverses. If Φ B = A is a Turing reduction, then for every n < ω we let Φ( B ↾ n )be the longest initial segment of A which is calculated by Φ querying the oracle B only on numbers smaller than n .In general, if f : ω → ω is a nondecreasing and unbounded function but notnecessarily recursive, we let f − , the discrete inverse of f , be defined by letting f − ( k ) be the greatest n such that f ( n ) k (let us assume that f (0) = 0, as it is forevery use function, so f − is total; otherwise f − is defined for almost all numbers).That is, if f ( n + 1) > f ( n ), then the interval [ f ( n ) , f ( n + 1)) gets mapped by f − to n . We note that if f is recursive (and is thus an order function), then so is f − .According to this definition, if Φ B = A with use ϕ , then for all n , Φ( B ↾ n ) = A ↾ ϕ − ( n ). Observation 2.6.
Let f and g be nondecreasing and unbounded. (1) If f bounds g , then g − bounds f − . (2) (cid:0) f − (cid:1) − bounds f . If f is growing slower than the identity function, thatis, if for all n , f ( n + 1) f ( n ) + 1 , then (cid:0) f − (cid:1) − = f . This observation suffices for the following corollary, noting that when investigatingslow-growing recursive orders, we may assume that the orders grow slower than theidentity function.
Corollary 2.7.
Let
A, B ∈ { , } ω . (1) A T ( tu ) B if and only if for every order function g , there is a Turingreduction Φ B = A such that the map n
7→ | Φ( B ↾ n ) | bounds g . (2) A uT ( tu ) B if and only if there is a Turing reduction Φ B = A such thatthe map n
7→ | Φ( B ↾ n ) | dominates every recursive function. (A functionwhich dominates every recursive function is called dominant .) Some other basic results follow.
Proposition 2.8.
Let
A, B ∈ { , } ω . (1) If A T ( tu ) A , then A is recursive. (2) If A is recursive, then A uT ( tu ) B . FRANKLIN, GREENBERG, STEPHAN, AND WU
Proof.
Let f ( n ) = n + 1. If Φ A = A and for all n we have Φ( A ↾ n ) ⊇ A ↾ n + 1,then we can recursively compute A ( n ) by applying Φ to A ↾ n , which we alreadycomputed. For (2), use a reduction Φ B = A whose use function is a constant 0. (cid:3) Corollary 2.9. If A T ( tu ) B and A is nonrecursive, then deg wtt ( A ) < deg wtt ( B ) . As a result, if deg wtt ( B ) is minimal, then every A T ( tu ) B is recursive. Proposition 2.10.
Let B ∈ { , } ω . If there is some nonrecursive A such that A uT ( tu ) B , then B is high. Recall that a set B is high if B ′ > T ∅ ′′ . Proof.
For any Turing reduction, if Φ B is total, then the map n
7→ | Φ( B ↾ n ) | iscomputable in B (indeed, weak truth-table reducible to B ). The map Φ whichwitnesses A uT ( tu ) B dominates every recursive function. By Martin [20], thismap has high Turing degree. (cid:3) We will review the situation in Proposition 2.10 in greater detail in Section 5.3.
Sets bounded by other sets with tiny use
In this section we prove Theorem 1.3. It will follow from Theorem 3.8 and Propo-sitions 3.4, 3.9 and 3.10.3.1.
Anti-complexity and traceability.
For functions f, g : ω → ω , we write f + g if there is some constant c such that g + c bounds f . Lemma 3.1.
A set A is anti-complex if and only if for every f wtt A , C ( f ( n )) + n. This lemma shows that the notion of anti-complexity (like its analogue notion,complexity) is wtt-degree invariant.
Proof.
We first note that A is anti-complex if and only if for every order function f , C ( A ↾ f ( n )) + n . One direction is immediate from Definition 1.2. For theother direction, suppose that for every order function f , C ( A ↾ f ( n )) + n . Let f be an order function. Applying the hypothesis twice to the functions n n and n n + 1, there is a constant c such that for all n , C ( A ↾ f (2 n )) n + c and C ( A ↾ f (2 n + 1)) n + c . If n > c , then C ( A ↾ f (2 n )) and C ( A ↾ f (2 n + 1)) areless than or equal to 2 n , so Definition 1.2 holds.Assume that for every g wtt A , C ( g ( n )) + n . Let f be an order function andlet g ( n ) be a natural number code for A ↾ f ( n ). Then g wtt A , so as we justobserved, A is anti-complex.Now assume that A is anti-complex and let h be an order function. Let f wtt A and let g be a recursive bound for the use function for the reduction of f to A .Using this reduction, we see that C ( f ( n )) + C ( A ↾ g ( n )). Again, as we justobserved, C ( A ↾ g ( n )) + n . (cid:3) We show that anti-complexity can also be characterised as a weak truth-table ana-logue of a very useful concept in the Turing degrees, that of r.e. traceability. Recallthat a r.e. trace for a function f is a uniformly recursively enumerable sequence h T n i of finite sets such that for all n , f ( n ) ∈ T n , and that a trace h T n i is bounded by an order function h if the function n
7→ | T n | is bounded by h . EDUCIBILITIES WITH TINY USE 7
Definition 3.2.
A weak truth-table degree a ∈ D wtt is r.e. traceable if there is anorder function h such that every f wtt a has an r.e. trace which is bounded by h .The standard argument of Terwijn and Zambella [32] shows that the choice of orderdoesn’t matter: Lemma 3.3.
A weak truth-table degree a is r.e. traceable if and only if for everyorder function h , every f wtt a has an r.e. trace which is bounded by h .Proof. Suppose that h is an order function which witnesses that a weak truth-tabledegree a is r.e. traceable. Let ˆ h be any other order function and let f wtt a .Let g ( n ) be the least k such that ˆ h ( k ) > h ( n ). This function is well definedbecause ˆ h is unbounded and is recursive. Hence the map n f ↾ ( g ( n + 1)) isweak truth-table below a , and so it has a trace h T n i which is bounded by h .The function g is unbounded because h is unbounded. Let g − be the discreteinverse of g , so g − ( k ) is the greatest n such that h ( n ) ˆ h ( k ) (note that g − isdefined on almost every number). Then | T g − ( k ) | ˆ h ( k ) and g ( g − ( k ) + 1) > k ,so f ↾ l is an element of T g − ( k ) for some l > k . Hence we can let S k be thecollection of all values σ ( k ) for all strings of length greater than k that contain onlynumbers which are elements of T g − ( k ) . Then h S n i will be an r.e. trace for f whichis bounded by ˆ h . (cid:3) Proposition 3.4.
A set A is anti-complex if and only if deg wtt ( A ) is r.e. traceable.Proof. Suppose that A is anti-complex and let f wtt A . By Lemma 3.1, there issome constant c such that for all n , C ( f ( n )) n + c . Then letting T n = { y : C ( y ) n + c } , h T n i is an r.e. trace for f and for all n , | T n | n + c +1 . Hence (by changing finitelymany entries for every function), deg wtt ( A ) is r.e. traceable, witnessed by the orderfunction h ( n ) = 2 n .The other direction follows an idea of Kummer’s, who showed that every arrayrecursive r.e. Turing degree contains only sets of low complexity [18] (see also [4]).Suppose that deg wtt ( A ) is r.e. traceable and let f wtt A . By Lemma 3.3, let h T n i be an r.e. trace for f which is bounded by the order function h ( n ) = n . Wecan construct a machine M which on input σ , first computes U ( σ ), interprets theresult as a pair ( n, m ) and, if m < n , outputs the m th element enumerated into T n .Then for all n , if f ( n ) is the m th element enumerated into T n , then M shows that C ( f ( n )) + C ( n, m ).Now the standard coding of pairs as numbers is polynomial; so there is someconstant c such that for all n and all m n , h n, m i n c . For all x , the identitymachine witnesses that C ( x ) + log x . Hence for all n and all m n , C ( n, m ) + log ( h n, m i ) log n c = c log n + n. Thus we see that the condition of Lemma 3.1 holds. (cid:3)
Porism 3.5. If A is anti-complex, then there is some c < ω such that for all f wtt A , C ( f ( n )) + c log n . (In fact, by working a bit harder, we can have c = 2 .) FRANKLIN, GREENBERG, STEPHAN, AND WU
Tiny use.
Given A ∈ { , } ω , the function n C ( A ↾ n ) is far from mono-tone. Nevertheless, we are interested in some form of inverse, which is possiblebecause lim n C ( A ↾ n ) = ∞ . We let g A ( k ) be the least n such that for all m > n , C ( A ↾ m ) > k . Observation 3.6.
For all A ∈ { , } ω , g A T A ⊕ K . As before, K = ∅ ′ is thehalting problem. For any string x , we let x ∗ be the least element of U − { x } (where U is the universalmachine we use for plain complexity), so C ( x ) = | x ∗ | . We also let A ∗ = n(cid:0) A ↾ g A ( k ) (cid:1) ∗ : k < ω o . Once again, we get A ∗ T A ⊕ K . Lemma 3.7.
For every A ∈ { , } ω , the map k ( A ↾ g A ( k )) ∗ is bounded by somerecursive function.Proof. There is a constant c such that for all τ ∈ { , } ∗ , C ( τ
0) and C ( τ
1) are bothless than or equal to C ( τ ) + c (consider the machine which on input σi , for i = 0 , U ( σ ) i ).For any k < ω , let τ k be a binary string and i ∈ { , } be such that A ↾ g a ( k ) = τ k i . By the definition of g A ( k ), C ( τ k ) k , and so C ( A ↾ g A ( k )) k + c . Hence( A ↾ g A ( k )) ∗ < k + c +1 .To ensure the last inequality, we need some agreement about the coding of stringsby numbers. This coding is obtained by some ω -ordering of all binary strings; weorder binary strings by length first. We let | x | denote the length of the stringidentified with the number x , so for all x , 2 | x | x < | x | +1 . (cid:3) Theorem 3.8.
The following are equivalent for A ∈ { , } ω . (1) There is some set B such that A T ( tu ) B . (2) A is anti-complex. (3) g A is dominant. (4) A uT ( tu ) A ∗ . We remark that we are not aware of a shorter proof of the equivalence of (2) and (3).This suggests that the study of the relation T ( tu ) is important for the seeminglyindependent study of anti-complexity in the wtt-degrees. Proof. (1) implies (2): Assume that A T ( tu ) B . For any functional Φ such thatΦ B = A , for all n , C (Φ( B ↾ n )) + C ( B ↾ n ). Also, for all x , C ( x ) + | x | , so for all n , C (Φ( B ↾ n )) + n . Suppose that f wtt A , so there is some functional Γ suchthat Γ A = f and the use of this computation is bounded by a recursive function g . We can find some Φ such that for all n , Φ( B ↾ n ) is longer than A ↾ g ( n ), so C ( f ( n )) + n . By Lemma 3.1, A is anti-complex.(2) implies (3): Suppose that A is anti-complex and let f be an increasingrecursive function. By definition, for almost all n , C ( A ↾ f ( n )) n . Hence, foralmost all n , g A ( n ) > f ( n ). It follows that g A dominates every recursive function.(3) implies (4): For every A ∈ { , } ω we have A T A ∗ because A = [ { U ( σ ) : σ ∈ A ∗ } (in other words, A ( x ) = U ( σ )( x ) for any σ ∈ A ∗ such that x < | U ( σ ) | , and forevery x there is indeed some σ ∈ A ∗ such that | U ( σ ) | > x ). EDUCIBILITIES WITH TINY USE 9 If g A is dominant, then this reduction witnesses that A uT ( tu ) A ∗ . To see this,let Φ code the described reduction and let f be an increasing recursive function;we see that n
7→ | Φ( A ∗ ↾ n ) | dominates f .Let g be a recursive function which dominates k ( A ↾ g A ( k )) ∗ (Lemma 3.7).Since g A is dominant, for almost all k , g A ( k ) > f ( g ( k + 1)). Suppose that k islarge enough that ( A ↾ g A ( k )) ∗ < n ( A ↾ g A ( k + 1)) ∗ . Then x g ( k + 1) and so g A ( k ) > f ( n ). Then | Φ( A ∗ ↾ n ) | > g A ( k ) so | Φ( A ∗ ↾ n ) | > f ( n ) as required.(4) implies (1): This is clear from the definitions. (cid:3) Schnorr triviality.
Franklin and Stephan [11] characterise the Schnorr trivialsets (defined by Downey and Griffiths in [5]) as those sets whose truth-table degreeis recursively traceable , that is, there is some order function h which bounds tracesfor all functions f truth-table reducible to the degree a , but where the trace h T n i is required to be given recursively (as a sequence of finite sets) rather than merelyuniformly recursively enumerably. In other words, there is a recursive function g such that for all n , g ( n ) is the canonical index for the finite set T n (in Soare’s[31] notation, T n = D g ( n ) ). Again, the Terwijn-Zambella argument shows that anyorder would do.Schnorr triviality is not invariant in the weak truth-table degrees [11, Theorem4.2]. However, the downward closure of the wtt-degrees containing Schnorr trivialsets is familiar. Proposition 3.9.
Every Schnorr trivial set is anti-complex.Proof.
Let A be Schnorr trivial. Fix an order function h . Let Φ be a weak truth-table functional with a recursive bound g on the use function of Φ. Since the map n A ↾ g ( n ) is truth-table reducible to A , by the characterisation mentionedabove, there is a recursive trace h T n i for this map which is bounded by h . If Φ A istotal, then we can enumerate a trace S n for f with bound h by outputting Φ σ ( n ) forthose σ ∈ T n for which Φ σ converges with domain greater than n . Hence deg wtt ( A )is r.e. traceable; by Proposition 3.4, A is anti-complex. (cid:3) Proposition 3.10.
Let A ∈ { , } ω . If g A is dominant, then A is weak truth-tablereducible to some Schnorr trivial set.Proof. Let f , f , . . . be a sequence of (total) recursive functions such that • each f i is strictly increasing, • the range of f i +1 is contained in the range of f i , and • every recursive function is bounded by some f i .(Note that the halting problem K can compute such a sequence.)By Lemma 3.7, let g be a recursive function which bounds the function k ( A ↾ g A ( k )) ∗ .For each k >
0, let q k = (cid:10) ( A ↾ g A ( k )) ∗ , f i ( k ) (cid:11) , where i is the greatest numbersuch that h g ( k ) , f i ( k ) i g A ( k − k > q k g A ( k − B = { q k : k > } . We claim that B is Schnorr trivial and that A wtt B .To see the latter, let n < ω . Let k = g − A ( n ) (that is, the greatest k such that g A ( k ) n ). Then q k +1 g A ( k ) n and A ↾ g A ( k + 1) can be effectively obtainedfrom q k +1 . This procedure allows us to generate A ↾ n effectively from B ↾ ( n + 1).To see that B is Schnorr trivial, we appeal to the characterisation mentionedabove. Here is where we use the fact that g A is dominant. The point is thatfor every i , all but finitely many elements of B are pairs whose second coordinate is contained in the range of f i . This is because the map k
7→ h g ( k ) , f i ( k ) i isrecursive and thus dominated by g A , so for all but finitely many k we will have q k = (cid:10) ( A ↾ g A ( k )) ∗ , f i ′ ( k ) (cid:11) for some i ′ > i , and the range of f i ′ is contained in therange of f i .Now let Ψ be a truth-table functional; there is some i such that f i bounds theuse function of Ψ. After specifying a fixed initial segment of B (specifying those q k ′ whose second coordinate is not in the range of f i ), there are at most 2 kg ( k ) manypossibilities for B ↾ f i ( k ) because, apart from the finitely many fixed elements,there are only kg ( k ) many numbers below f i ( k ) which can be elements of B , asthey all have the form h p, f i ( m ) i for some p < g ( k ) and m < k . After applyingΨ, we get a recursive trace for Ψ( B ) whose k th element has size at most 2 kg ( k ) .Hence deg tt ( B ) is recursively traceable (in the tt-degrees), so as quoted above, B is Schnorr trivial. (cid:3) Truth-table reductions with tiny use.
Another connection between tinyuse and Schnorr triviality is obtained by examining truth-table reducibility. Recallthat A tt B if and only if there is a Turing reduction Φ for which Φ X is total forall X and Φ B = A . We say that A tt ( tu ) B if for every order function h there issuch a functional whose use function is bounded by h . Equivalently, for every orderfunction h , there is a truth-table reduction of A to B for which the size of the n th truth table is bounded by h ( n ). This notion is invariant in the truth-table degrees.Since the use function for a total Turing functional is recursive (equivalently, thesize of the n th truth-table of a tt-reduction is recursive), there is no uniform notionin this context.The class of all A such that there is a B with A tt ( tu ) B gives us a newcharacterisation of the Schnorr trivial sets. Theorem 3.11.
Let A ∈ { , } ω . There is a set B such that A tt ( tu ) B if andonly if A is Schnorr trivial.Proof. We begin by assuming that A tt ( tu ) B . Let h be an order function. Thereis a total reduction Φ such that Φ B = A whose use function is bounded by n log( h ( n )). Then a recursive trace for n A ↾ n with bound h can be obtained byapplying Φ. Hence deg tt ( A ) is recursively traceable.Now suppose that A is Schnorr trivial. Again the point is that deg tt ( A ) isrecursively traceable, so for any recursive function f , the function A A ↾ f ( n )has a recursive trace bounded by the identity function.Let h f i i be an enumeration of all increasing total recursive functions. For each i < ω , let (cid:10) D in (cid:11) n<ω be a recursive trace for the function A A ↾ f i ( n ) such thatfor all n , | D in | = n .We let B be the collection of triples ( i, n, m ) such that A ↾ f i ( n ) is the m th element of D in .Let i < ω . Let Φ i be the following truth-table functional: given an oracle X andinput x ∈ [ f i ( n − , f i ( n )), find the least m n such that ( i, n, m ) ∈ X ; if the m th element of D in is a string σ of length f i ( n ), output σ ( x ). If not, or if there is no m n such that ( i, n, m ) ∈ X , output 0. It is clear that for all i < ω , Φ Bi = A .The standard coding of triples of natural numbers by natural numbers growspolynomially. Hence, if g is, say, an exponentially growing recursive function, thenfor almost all i , for all n and m n , ( i, n, m ) < g ( n ). Hence for almost all i , EDUCIBILITIES WITH TINY USE 11 | Φ i ( B ↾ g ( n )) | > f i ( n ), whence the function n
7→ | Φ i ( B ↾ n ) | dominates f i ◦ g − . Ofcourse every recursive function is dominated by some f i ◦ g − , so A tt ( tu ) B . (cid:3) The distribution of anti-complex sets
In this section we investigate how the anti-complex sets are distributed in the Turingdegrees and among certain classes of sets. Three question are natural: • Which Turing degrees contain anti-complex sets? • Which Turing degrees contain only anti-complex sets? • What kind of sets can be anti-complex?The answer to the second question was mentioned in the introduction:
Proposition 4.1.
A Turing degree a contains only anti-complex sets if and onlyif a is r.e. traceable.Proof. A Turing degree a is r.e. traceable if and only if for every order function h ,every f ∈ a has an r.e. trace bounded by h . Since a Turing degree a is the unionof the weak truth-table degrees contained in a , by Lemma 3.3, a Turing degree a isr.e. traceable if and only if every weak truth-table contained in a is r.e. traceable.The result now follows from Theorem 1.3. (cid:3) Theorem 1.4 now follows from Theorem 1.3.4.1.
High and random anti-complex sets.
Franklin [10] shows that every highdegree contains a Schnorr trivial set. It follows from Proposition 3.9 that everyhigh degree contains an anti-complex set. We improve this result in Corollary 5.5.Nies [23] constructed a ∆ perfect tree, all of whose branches are jump-traceableand thus have r.e. traceable Turing degree. Every perfect ∆ tree contains a highpath, and so there is a high r.e. traceable Turing degree. It follows from Proposition4.1 that there is a high Turing degree that has only anti-complex elements. Notethat such a high degree cannot be ∆ , as every r.e. traceable Turing degree is GL .Now every high degree contains Schnorr random and recursively random sets [25].Hence there is a recursively random, anti-complex set. On the other hand, sufficientrandomness precludes anti-complexity: Kuˇcera [17] has shown that every Martin-L¨of random set weak truth-table computes a diagonally nonrecursive function, soevery Martin-L¨of random set is complex and thus certainly not anti-complex.This result can be strengthened to show that partial-recursively random sets arenot anti-complex. Proof of Theorem 1.6.
Let A be an anti-complex set. By Porism 3.5, there is someconstant c < ω such that C ( A ↾ n ) + c log n ; so for almost all n , C ( A ↾ n ) ( c +1) log n . Hence Theorem 7 of [21] shows that no Mises-Wald-Church stochasticset is anti-complex. Every partial-recursively random set is Mises-Wald-Churchstochastic (see Section 7.4 of [3]), and so no partial-recursively random set is anti-complex. As we just discussed, there is a high Turing degree all of whose elementsare anti-complex, and so such a degree cannot contain a partial-recursively randomset. (cid:3) Anti-complex-free Turing degrees.
Not every Turing degree contains anti-complex sets. In fact, we can find a counterexample within the r.e. degrees. Notethat this counterexample cannot be very low, as all array recursive (and hencesuperlow) r.e. degrees are r.e. traceable and cannot be high.This result extends the result of Downey, Griffiths and LaForte [6] that there isan r.e. degree that contains no Schnorr trivial sets and utilizes their techniques.These techniques involve prefix-free complexity. Recall that a machine M is prefix-free if its domain is an antichain of { , } ∗ , that is, for all distinct σ, τ ∈ dom M , σ is not an initial segment of τ . There is a prefix-free machine, optimalamong all prefix-free machines, and so prefix-free Kolmogorov complexity , which isoften denoted by K , but which we denote by H (to differentiate from the haltingset K = ∅ ′ ), equals C V for some optimal prefix-free machine V . Lemma 4.2. If A ∈ { , } ω is anti-complex, then for every order function f , H ( A ↾ f ( n )) + n .Proof. We follow the proof of Proposition 3.4. If A is anti-complex and f is an orderfunction, then since deg wtt ( A ) is r.e. traceable, there is an r.e. trace h T n i , boundedby the identity function, for the function n A ↾ f ( n ). The same argument in theproof of Proposition 3.4 shows that for all n there is some m n such that H ( A ↾ f ( n )) + H ( m, n ) . It is no longer true that H ( x ) + log x , but even a crude bound such as H ( n ) + n would do to show that for some constant c we have H ( m, n ) + c log n + n as required. (cid:3) Theorem 4.3.
There is an r.e. Turing degree that contains no anti-complex sets.Proof.
For any prefix-free subset D of { , } ∗ , we let µ ( D ) = X τ ∈ D −| τ | be the measure of the subset of the Cantor space defined by D by taking all infiniteextensions of elements of D .Theorem 9 of [6] states that there is an r.e. set A such that for all B ≡ T A thereis a prefix-free machine M such that µ (dom( M )) is a recursive real and such thatfor infinitely many m , H ( B ↾ m ) > C M ( m ).The r.e. degree we seek is the Turing degree of A . Let B ≡ T A ; we show that B is not anti-complex. Let M be a machine for B as described in the previousparagraph.We first note that dom( M ) is a recursive subset of { , } ∗ : If h M s i is a somerecursive enumeration of M , then dom( M ) ↾ { , } n = dom( M s ) ↾ { , } n forany stage s such that µ (dom( M )) − µ (dom( M s )) < − n ; such a stage s can be foundeffectively from n . Now the range of M may not be recursive, but C M ↾ range( M )is a partial recursive function.We can compute a strictly increasing recursive function f such that for all n , X m>f ( n ) m ∈ range M − C M ( m ) − n EDUCIBILITIES WITH TINY USE 13 by finding some s ( n ) such that µ (dom( M )) − µ (dom( M s ( n ) )) − n and letting f ( n ) be greater than any number in the range of M s ( n ) . Let L = (cid:8) ( C M ( m ) − f − ( m ) , m ) : m ∈ range M (cid:9) . The set L is recursively enumerable. Recall that for any set D ⊆ ω , the weight wt ( D ) of D is P ( n,m ) ∈ D − n . We have wt ( L ) = X m ∈ range M f − ( m ) − C M ( m ) = X n n X m ∈ range Mm ∈ [ f ( n ) ,f ( n +1)) − C M ( m ) X n n X m ∈ range Mm > f ( n ) − C M ( m ) X n n − n = X n − n < ∞ . The Kraft-Chaitin Theorem (see [3, 22]) now ensures that for all m , H ( m ) + C M ( m ) − f − ( m )(recall that for m / ∈ range M , we let C M ( m ) = ∞ ).Suppose that B is anti-complex. Then by Lemma 4.2, H ( B ↾ f ( n )) + n .Let m < ω and let n = f − ( m ). We can uniformly compute A ↾ m if we aregiven both m and B ↾ f ( n + 1). Since H measures prefix-free complexity, we have H ( B ↾ m ) + H ( m ) + H ( B ↾ f ( n + 1)) (a description for B ↾ m is a descriptionfor m concatenated with a description for B ↾ f ( n + 1)). Overall we get, for all m , H ( B ↾ m ) + H ( m ) + f − ( m ) + C M ( m ) − f − ( m ) . Since f is increasing, f − is unbounded, which would make it impossible to haveinfinitely many m ∈ range M such that H ( B ↾ m ) > C M ( m ). Hence B cannot beanti-complex. (cid:3) Anti-complex r.e. and ω -r.e. sets. The results so far show that if A is anti-complex, then there is some set B T A ⊕ K such that A uT ( tu ) B . In general,as we will see shortly, one cannot improve this to B wtt A ⊕ K . However, if A isr.e., then we get an improved bound as follows. Proposition 4.4. If A is an anti-complex r.e. set, then A uT ( tu ) K .Proof. We claim that if A is r.e., then A ∗ wtt K ; the rest follows from Theorem3.8. Fix a recursive enumeration h A s i of A and let, at stage s , g s ( k ) be the leastnumber n such that no initial segment of A s of length at least n has a U -descriptionof length at most k . Then g s converges to g A and is an ω -r.e. approximation of g A .We can have g s +1 ( k ) = g s ( k ) only in three cases: • there is some σ ∈ dom U s +1 \ dom U s of length at most k and U ( σ ) ⊂ A s +1 ; • there is some σ ∈ dom U s of length at most k such that U ( σ ) A s but U ( σ ) ⊂ A s +1 ; or • there is some σ ∈ dom U s of length at most k such that U ( σ ) ⊂ A s but U ( σ ) A s +1 .For each σ , each case can happen at most once, and the first two cannot bothhappen at different stages. Hence our approximation for g s ( k ) changes at most2 · k +1 many times.Hence g A wtt K , and it is straightforward to see that A ∗ wtt g A ⊕ K ⊕ A for any set A because once we know g A ( k ), we only need to query K about strings below g ( k ) (where g ( k ) > ( A ↾ g A ( k )) ∗ is recursive) to find ( A ↾ g A ( k )) ∗ and hence A ∗ . (cid:3) Theorem 1.5 now follows from Proposition 4.4 and the techniques of Downey,Jockusch and Stob [7, 8] and Ishmukhametov [14]. The fact that the array re-cursive r.e. wtt-degrees form an ideal now follows from Observation 2.5.One would perhaps hope that the previous result could be extended to classeswider than the class of r.e. sets and their weak truth-table degrees. Of course, if A T ( tu ) K , then A wtt K and so A is ω -r.e.; however, we now show that thereare ω -r.e. sets A which are anti-complex and yet A T ( tu ) K . This shows that thecondition B T A ⊕ K for the bound for A with tiny use cannot in general beimproved to B wtt A ⊕ K .We first need a lemma which again is not new, but which is not found in standardreferences (an approximation, insufficient for our purposes, is Theorem 9.14.6 in [3]).Let Ω be the halting probability — any left-r.e. Martin-L¨of random real would do. Lemma 4.5.
For any r.e. set A , there is a reduction of A to Ω with use boundedbelow n . Indeed, we can even get a bound of h ( n ) where h is such that P n − h ( n ) is finite,such as log n + 2 log log n . Proof.
Let h Ω s i be an effective, increasing approximation of Ω and, similarly, let h A s i be an effective enumeration of A . Let h be a function such that P n − h ( n ) isfinite.If n is the smallest number which enters A at stage s , we enumerate the interval[Ω s , Ω s +2 − h ( n ) ] into a Solovay test G which we enumerate. Since n enters A at mostonce, the total measure of G is at most P n − h ( n ) , which is finite by assumption.Ω is random, so it belongs to only finitely many of the intervals in G . To compute A ( n ) from Ω ↾ h ( n ), find a stage t at which Ω t ↾ h ( n ) = Ω ↾ h ( n ); we claim that A ( n ) = A t ( n ). If n enters A at a later stage s , then [Ω s , Ω s + 2 − h ( n ) ] is in G ,but Ω − Ω t − h ( n ) and Ω t Ω s Ω, so we conclude that Ω is in the interval[Ω s , Ω s + 2 − h ( n ) ]. Thus we can get a wrong answer for only finitely many numbers n , and we can find a reduction as required. (cid:3) Proposition 4.6.
There is an anti-complex ω -r.e. set which is not reducible to K with tiny use. Indeed, as the proof shows, there is such a set which is also the difference of twoleft-r.e. reals. (We cannot get a left-r.e. real, because every left-r.e. real is weaktruth-table equivalent to an r.e. set.)
Proof.
By [11, Theorem 4.1], there is a coinfinite r.e. set A such that every supersetof A is Schnorr trivial (indeed, any dense simple set would do). Let B = A ∪ Ω. B is Schnorr trivial and thus anti-complex. B is also ω -r.e., since it is a Booleancombination of two sets which are wtt-reducible to K .Now assume for a contradiction that B T ( tu ) K . Then there is some reductionΓ Ω = B such that for all n , | Γ(Ω ↾ n ) | > n because Ω has the same wtt-degree as K . By Lemma 4.5, there is a reduction ∆ Ω = A with the same property, as thereis a reduction from A to Ω with use below 2 log n .We use the functionals Γ and ∆ to define a recursive martingale which willsucceed on Ω, contradicting the fact that Ω is random. The martingale d is defined EDUCIBILITIES WITH TINY USE 15 by induction on the length of the binary strings which form its domain. We startwith the value 1. If d ( σ ) is defined, we first calculate ∆( σ )( n ) and Γ( σ )( n ), where n = | σ | (if either | Γ( σ ) | n or ∆( σ ) | n , then we know that σ cannot be an initialsegment of Ω, so we can stop all betting). If ∆( σ )( n ) = 1, then we hedge our bets,that is, we let d ( σ
0) = d ( σ
1) = d ( σ ). Otherwise, we put all of the capital we haveon the outcome Γ( σ )( n ), because in this case, if σ = Ω ↾ n , then A ( n ) = 0 and so B ( n ) = Ω( n ). Thus we let d ( σi ) = 2 d ( σ ) and d ( σ (1 − i )) = 0, where i = Γ( σ )( n ).Since A is coinfinite, there are infinitely many n at which we double our moneybetting along Ω, so lim n d (Ω ↾ n ) = ∞ as required for the contradiction. (cid:3) Sets bounding nonrecursive sets with tiny use
We now turn to investigate the ranges of the relations T ( tu ) and uT ( tu ) (where thedomain is restricted to the class of nonrecursive sets to avoid triviality). Unlike theirdomains, these ranges are not equal, because as we observed earlier, if A uT ( tu ) B and A is nonrecursive, then B is high, whereas we will shortly see that there arenonhigh sets which bound nonrecursive sets with tiny use. First, we prove someresults on the range of uT ( tu ) .5.1. High degrees.
Unlike for Turing reducibility, with weak truth-table reducibil-ity we have to be careful when we deal with functions (elements of the Baire space ω ω ) and sets (elements of the Cantor space { , } ω ). For example, a function is al-ways Turing equivalent to its graph, but if it is not bounded by a recursive function,it may not be wtt-equivalent to its graph. Our primary interest is to investigate T ( tu ) on sets , and so far we have not treated functions as oracles in computationswith recursive or tiny use. However, as a technical tool, we can extend the defi-nitions of T ( tu ) and uT ( tu ) to include functions as oracles in the standard way;weak truth table invariance still holds. In this context we have the following result. Observation 5.1.
Let G ( f ) be the graph of f . If f is a dominant function, then G ( f ) uT ( tu ) f . This allows us to characterise the range of uT ( tu ) . Lemma 5.2.
Let B ∈ { , } ω . There is some nonrecursive set A such that A uT ( tu ) B if and only if there is some dominant function f wtt B .Proof. In the proof of Proposition 2.10 we noticed that if there is some nonrecur-sive set A such that A uT ( tu ) B witnessed by some reduction Φ, then the map n
7→ | Φ( B ↾ n ) | is dominant and is weak truth-table reducible to B . In the otherdirection, suppose that f is dominant and that f wtt B . Let A be the graph of f .Then A uT ( tu ) f ; together with f wtt B we get A uT ( tu ) B from Observation2.4. (cid:3) We know that every high Turing degree contains a dominating function, but theweak truth-table degree of that function may not contain any set.
Lemma 5.3.
Let f be a function such that n C ( f ( n )) is bounded by somerecursive function. Then f is wtt-equivalent to some set.Proof. Let g be a recursive function which bounds C ( f ( n )). Let A be the setof pairs ( n, u ) where u is the first number below g ( n ) which is discovered in someeffective enumeration of the universal machine U to be mapped by U to f ( n ). Then A ≡ wtt f . (cid:3) Proposition 5.4.
Every high Turing degree contains a dominant function ˆ f suchthat C ( ˆ f ( n )) + n .Proof. Let g be a dominant function; we first find an f T g with the desiredproperties.Once again, let h Ω s i be an effective increasing approximation of Ω. Define f byletting f ( n ) be the least s g ( n ) such that Ω s ↾ n = Ω g ( n ) ↾ n . It is certainly truethat f T g .First we show that C ( f ( n )) + n . Let M be a machine that on an input σ oflength n outputs the least stage s such that σ = Ω s ↾ n if such a stage exists. Thenfor all n , M (Ω g ( n ) ↾ n ) = f ( n ), so C M ( f ( n )) n as required.Next, let h be an order function. We first note that H (Ω h ( n ) ↾ n ) H ( n ) (asbefore, H denotes prefix-free Kolmogorov complexity), and since Ω is random, foralmost all n , Ω h ( n ) ↾ n = Ω ↾ n , as H (Ω ↾ n ) > + n . Thus we can let ˆ h ( n ) be theleast s > h ( n ) such that Ω s ↾ n = Ω h ( n ) ↾ n ; this too is a recursive function, definedon almost every input.Since g is dominant, for almost all n , g ( n ) > ˆ h ( n ), which implies that Ω g ( n ) ↾ n = Ω h ( n ) ↾ n since the approximation Ω s ↾ n does not return to old values, and so f ( n ) > ˆ h ( n ) > h ( n ) for almost all n . Thus f is dominant.Next, we code a set A in the Turing degree of g into f to get a function whichis Turing equivalent to g . We let ˆ f ( n ) = 2 f ( n ) + A ( n ). Then A T ˆ f andˆ f T f ⊕ A T g , so ˆ f ≡ T g . Since ˆ f bounds f , ˆ f is dominant, and C ( ˆ f ( n )) + C ( f ( n )) + n . (cid:3) Corollary 5.5.
Every high Turing degree contains sets A and B such that A uT ( tu ) B .Proof. Let a be a high Turing degree. By Proposition 5.4 and Lemma 5.3, there issome dominant f ∈ a which is wtt-equivalent to some set B , so of course B ∈ a .By Lemma 5.2, there is some set A such that A uT ( tu ) B . Indeed, we can take A to be the graph of f . A is thus Turing equivalent to f , so A ∈ a . (cid:3) We can improve on the corollary in case, for example, the high degree is alsogeneralised low.
Theorem 5.6. If a is a Turing degree such that a ∨ ′ > T ′′ , then for every B ∈ a there is some A ∈ a such that A uT ( tu ) B . The point is that under the assumption that every B ∈ a is wtt-equivalent to somedominant function f , we can let A be the graph of f . This means that we can provethe following equivalent fact instead. Proposition 5.7. If a is a Turing degree such that a ∨ ′ > T ′′ , then every B ∈ a is wtt-equivalent to some dominating function.Proof. Let B ∈ a , and let h ϕ e i be an enumeration of all partial recursive functions.Then Tot, the collection of all indices e such that ϕ e is total, has Turing degree ′′ ,so there is some Turing reduction Φ B ⊕ K = Tot.We show that there is a set E ⊆ Tot which is recursively enumerable in B andsuch that for every (total) recursive function g there is some e ∈ E such that g = ϕ e . The set E is enumerated as follows: at stage s , if Φ B ⊕ K s ( e ) ↓ = 1, then weenumerate g ( e, σ ) into E with use B ↾ u , where u is the use of the computation, EDUCIBILITIES WITH TINY USE 17 σ = K s ↾ u and the instructions for calculating ϕ g ( e,σ ) are as follows. We emulate ϕ e as long as σ is an initial segment of K t for stages t > s , waiting for computationsto converge, but if at some stage we observe that σ is no longer an initial segment of K , we make ϕ g ( e,σ ) total by immediately converging on all inputs for which we havenot yet given an output and giving the answer 0. The function g is thus recursive.Now E is used to construct a dominant function f wtt B : we let f ( n ) be themaximum of the values ϕ e ( n ) for the e that are enumerated into E by stage n with B -use at most n .Again, we can modify f to be a dominant function ˆ f ≡ wtt B by coding B intoˆ f , say again by letting ˆ f ( n ) = 2 f ( n ) + B ( n ). (cid:3) We do not know much in general about the range of T ( tu ) . We give some partialresults in the following subsections.5.2. Recursively enumerable degrees.
The techniques of the previous subsec-tion can be improved to yield the following result.
Theorem 5.8.
For every nonrecursive r.e. set B there is a nonrecursive r.e. set A such that A T ( tu ) B . In order to prove this theorem, we take dominant functions which have decentapproximations. Let h be a high r.e. Turing degree. By standard manipulations,we can get a dominant function f T h with an approximation with the following“nice” properties. • The approximation is increasing: for all n and s , f s ( n ) > f s − ( n ). • If f s ( n ) = f s − ( n ), then f s ( n ) = s . • For all s there is at most one n such that f s ( n ) = s (this is done by delayingchanges in the approximation). Proof of Theorem 5.8.
Let B be a nonrecursive r.e. set. By a standard cone-avoiding addition to the Sacks jump inversion theorem, there is a high r.e. degree h which does not compute B . Let f T h be an ω -r.e. dominant function withan approximation h f s i as described above. Let g be a recursive bound on themind-change number of the approximation h f s i .Enumerate a set A as follows: for all n ∈ B , if n enters B at stage s , enumerate f s ( n ) into A .Suppose that we want to compute A from B . To find out if t ∈ A , we first goto stage t and see if f t ( n ) = t for some n t — if not, then t is certainly not in A . If so, then t is in A if and only if for the unique n = n ( t ) such that f t ( n ) = t , n enters B at a later stage s before f s ( n ) changes. This gives a reduction of A to B with identity use.In fact, A T ( tu ) B . The idea is the following. Suppose again that we want tofind out whether t is in A and that we find that f t ( n ) = t . If we knew that f ( n ) > t ,in other words, that there is a later stage s at which we have f s ( n ) = f t ( n ), thenwe could wait for that stage and see if n entered B before that stage or not. Ofcourse, we cannot always do this, because it may happen that t = f t ( n ) = f ( n ) (orelse A would be recursive, whereas later we show it is not). But now suppose that h is an order function and that we want to reduce A to B with use bounded by h .Then if n = n ( t ) < h ( t ), then we can consult B ( n ) as before to compute A ( t ). If h ( t ) n , then h − ( n ) > t ; since f is dominant, f ( n ) > t except for finitely many n , so we can employ the second tactic of waiting for f s ( n ) to change in order to compute A ( t ). In the second case we do not consult B at all, so overall we get areduction with use bounded by h .Finally, to show that A is not recursive, we see that B T A ⊕ f and recall that B T f . To find B ( n ), we calculate the least stage t at which f t ( n ) = f ( n ); if n / ∈ B t , then n ∈ B if and only if t ∈ A . (cid:3) Just as we did for tt , we can apply the “tiny use” operator to many-one reducibilityand say that A m ( tu ) B if for every order function h there is a recursive function f dominated by h such that A = f − B . The previous proof can be slightly modifiedto show that for every nonrecursive r.e. set B , there are an r.e. set ˆ B which iswtt-equivalent to B and a nonrecursive r.e. set A such that A m ( tu ) ˆ B . We simplyenumerate ( n, m ) into ˆ B if n is enumerated into B at stage s and f s ( n ) is the m th value we see for f t ( n ) by stage s , that is, m = |{ f t ( n ) : t s }| . Then, given t , wecan find n and m ; then t ∈ A if and only if ( n, m ) ∈ ˆ B and, as described above, thiscan be done with tiny use. To get B ≡ wtt ˆ B rather than just Turing equivalence,we use a function f which is ω -r.e., the existence of which is guaranteed by theproof of Proposition 5.4.Theorem 5.8 cannot be extended to all ∆ sets, as Downey, Ng and Solomon [9]constructed a ∆ set which has minimal wtt-degree.Finally, there is an r.e. set B which has minimal tt-degree [19]. For such B ,there can be no nonrecursive A tt ( tu ) B . Thus we cannot improve T ( tu ) in thetheorem to tt ( tu ) , or, in the comments after the proof, get ˆ B ≡ tt B rather thanˆ B ≡ wtt B .5.3. Hyperimmune-freeness.
Most hyperimmune-free Turing degrees do not con-tain any set B for which there is a nonrecursive A such that A T ( tu ) B . For exam-ple, a Turing degree which is both minimal and hyperimmune free does not containsuch sets because if deg T ( X ) is hyperimmune free and minimal, then deg wtt ( X ) isalso minimal, as every nonrecursive Y wtt X is Turing equivalent to X and thusalso wtt-equivalent to X . Similarly, if X is Martin-L¨of random and deg T ( X ) ishyperimmune free, then every nonrecursive Y wtt X is truth-table equivalent to aMartin-L¨of random set and so cannot be anti-complex, so again we get that every A T ( tu ) X is recursive.Thus in the realm of the hyperimmune-free degrees, generic sets (in the senseof either recursive Sacks forcing or forcing with sets of positive measure) do notcompute nonrecursive sets with tiny use.On the other hand, there is a hyperimmune-free B with a nonrecursive A T ( tu ) B . This follows from the hyperimmune-free basis theorem and the following theo-rem. Theorem 5.9.
There is a Π -class with no recursive elements consisting of sets B for which there are nonrecursive sets A such that A m ( tu ) B . (We remark that it is already known that there is a Π -class with no recursive ele-ments which consists of anti-complex sets; for example, there is one which consistsof Schnorr trivial sets [11] and one which consists of sets A such that deg T ( A ) isr.e. traceable.) Proof.
We imitate part of the the proof of Theorem 5.8. Again, let h f s i be an ω -r.e.approximation for a dominant function f with the properties discussed above; say g EDUCIBILITIES WITH TINY USE 19 is a recursive function which bounds the number of possible values m ( n ) = |{ f s ( n ) : s < ω }| .For n < ω and k m ( n ), let π ( n, k ) be the k th value of f s ( n ). Thus π ( n, m ( n )) = f ( n ). Now let D = { ( n, k ) : k < m ( n ) } and E = π [ D ]; both are r.e. sets.Furthermore, E is nonrecursive as f T D T E : for each n , we find π ( n, k )recursively in k and so, consulting E , determine if k < m ( n ) or not.We can thus split E into a pair E and E of recursively inseparable r.e. sets.Let P be a Π -class of sets which separate D = π − E from D = π − E (setsthat contain D and are disjoint from D ). Let B be any element of P and let A = π [ B ]. Then A separates E and E , so A is not recursive. We claim that A T ( tu ) B , indeed that A m ( tu ) B .The argument is similar to that of the proof of Theorem 5.8. For any t , t ∈ A ifand only if t is the k th value of f s ( n ) for some (unique) n which can be effectivelyobtained from t ; the question is whether ( n, k ) ∈ B . As before, if h is any orderfunction, then for all but finitely many t , either ( n, k ) < h ( t ) or k < m ( n ). In thelatter case, ( n, k ) ∈ D , so we do not need to consult B about ( n, k ), as the value B ( n, k ) is decided by which one of D or D the pair ( n, k ) is enumerated into, afact which is revealed to us with sufficient patience. (cid:3) References [1] Adam R. Day. Process and truth-table characterizations of randomness. Manuscript, 2010.[2] Rod Downey, Denis R. Hirschfeldt, Andr´e Nies and Sebastiaan A. Terwijn. Calibrating ran-domness.
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J.N.Y. Franklin, Department of Mathematics, 6188 Kemeny Hall, Dartmouth Col-lege, Hanover, NH 03755-3551, USA
E-mail address : [email protected] N. Greenberg, Victoria University, PO Box 600, Wellington 6140, New Zealand
E-mail address : [email protected] F. Stephan, Department of Computer Science and Department of Mathematics, Na-tional University of Singapore, Singapore 117543, Republic of Singapore
E-mail address : [email protected] G. Wu, School of Physical and Mathematical Sciences, Nanyang Technological Uni-versity, Singapore 639798, Republic of Singapore.
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