-Martin-Löf randomness and -Solovay completeness

Transcription

1 Π -Martin-Löf randomness and -Solovay completeness Π Claude Sureson* Abstract Developing an analog of Solovay reducibility in the higher recursion setting, we show that results from the classical computably enumerable case can be extended to the new context. Mathematics Subject Classification: 68Q30, 03D60, 03E5. Keywords: Algorithmic Randomness, Higher Recursion, Higher Randomness. Introduction A notion of randomness adapted to the higher recursion setting was first proposed by Martin- Löf [ML70]. Hjorth and Nies then developed a whole theory, Π predicates playing the role of computable enumerability. Among different concepts of randomness, they introduced the notion of Π -Martin-Löf randomness and obtained Π versions of fundamental classical results by Chaitin, Kraft-Chaitin and Schnorr. We develop in this new context a notion of reducibility introduced by Solovay [Sol75] in the computably enumerable situation. It is formulated in approximation terms: a real x is Solovayreducible to a real y - written x s y - if there exist a constant c and a partial computable function f Q Q such that if q Q and q < y, then f (q) is defined, satisfies f (q) < x and (x f (q)) < c(y q). Solovay showed that if K is a descriptive string complexity measure associated with a universal by adjunction prefix-free machine, then x s y implies b n K(x n ) K(y n ) + b (we abuse notation in this section and identify a real with its dyadic expansion). Hence reals which are complete for Solovay reducibility must be Martin-Löf random by Schnorr s theorem. Work by Calude, Hertling, Khoussainov, Wang [CHKW98] and by Kucera, Slaman [KS0] characterize precisely the Solovay-complete reals which are left c.e. (reals whose left cut is computably enumerable): Ω-like and left c.e. Martin-Löf random reals. Proofs make use of an equivalent definition of Solovay reducibility formulated in sum terms and proposed by Downey, Hirschfeldt and Nies [DHN02] for left c.e.reals: x + s y if there exist a left c.e. real r and a natural number m such that 2 m y = x +r. * IMJ, CNRS & Université Paris 7 Denis Diderot, France. sureson@math.univ-paris-diderot.fr

2 In this article, we recast this whole development in the Π setting, replacing partially computable functions by partial Π functions, the measure K by the complexity measure K associated with a universal prefix-free Π machine and left c.e. reals by left Π reals. The two corresponding notions of Solovay reducibility (in approximation terms and in sum terms) can again be shown equivalent, and one obtains, as expected: For a left Π real x in [0,), the following are equivalent : (a) x is Π -Solovay complete (among left Π reals), (b) there is a Π prefix-free optimal machine M such that x is the measure of the open set generated by the domain of M (that is x = Ω M ), (c) there is a constant b such that for any integer n, K(x n ) > n b, (d) x is Π ML random. Inductive constructions of length ω CK, where ω CK is Church-Kleene ordinal, replace effective constructions, and reals play the role of both dyadic rationals and computable reals. Adaptations of the c.e. case to the Π case can be easy, more complex or open questions. For example we do not know whether there exists a Π equivalent to the fact that any Σ0 open subset can be generated by a prefix-free c.e (in fact computable) set of binary strings. In [HN07, 3.7,3.8], Hjorth and Nies propose a sophisticated way around. Since we restrict to left Π reals (Solovay reducibility behaves badly out of this domain), it is interesting to note that in arguments involving such reals, one generally gets Π prefix-free generating sets, for example, by extending the proof of (a) (d), one obtains: A Π ML test U n n < ω is called prefix-free if there exist uniformly Π X n, n < ω, such that U n = [X n ]. For any x 2 ω which is left Π, the following are equivalent: x is Π ML random, For any prefix-free Π ML test U n n < ω, x n<ω U n. prefix-free sets In the previous article [Sur6], we used Π -Solovay reducibility and Π -Solovay completeness as simple tools, our purpose here is to propose an accessible and global exposition of the subject. 2 Preliminaries Our references in algorithmic randomness are [DH0] and [Nie09]. Solovay s notes [Sol75] being unpublished, we appeal to the exposition given in the two previous books of the field of Solovay reducibility. For the specific domain of higher computability and randomness, we rely on [Nie09, Chapter 9], [HN07], and for higher recursion theory, on [Sac90] and [CY5]. We refer to [Mos09] and [Jec03] for descriptive set theory. 2

3 2. Some definitions and notation Cantor space 2 ω is the set of infinite sequences of 0 s and s, Baire space ω ω is the set of infinite sequences of integers; they are both endowed with the usual product topology. Concerning finite sequences (also called strings), we use the following notation: 2 <ω and ω <ω are respectively the sets of finite 0- sequences and the set of finite sequences of integers. If s is a finite sequence, then we denote by s the length of s. For an integer i ω and s ω <ω, s i is the restriction of s to the set {0,,...,i } (hence if i s, then s i = s). Definition 2. (a) For two strings s,t, we write s t if s is a prefix of t (i.e. according to previous notation, s t and s = t s ). (b) For s,t ω <ω, s lex t iff s is less or equal to t for the lexicographic ordering. iff (s t) ( i < ω s i = t i s(i) < t(i)). (c) For s,t ω <ω, s llex t iff ( s < t ) ( s = t s lex t). (llex stands for length-lexicographic). If z is an infinite sequence in Cantor or Baire space, then for i ω, z i denotes the restriction of z to the set {0,,...,i }. We also write s z if s is a finite sequence and s = z s. Given a finite set X, X is the number of elements of X The function, ω ω ω is the classical recursive bijection from ω ω onto ω defined as m,n = m+(m+n)(m+n+)/2, for m,n ω. 2.2 Σ, Π and sets The basic open sets in Cantor space are of the form [s] = {x 2 ω s x}, for s 2 <ω. If X 2 <ω, then [X] = {x 2 ω s X s x} denotes the open subset of 2 ω generated by X. Omitting the effective coding of ω <ω in ω, we shall speak directly of recursive predicates on ω <ω, 2 <ω,ω <ω ω... We assume familiarity with the arithmetical hierarchy on spaces ω (and hence ω <ω, 2 <ω ), 2 ω, ω ω or any finite product of these spaces. We recall that, for l + m, a set O ω k (2 ω ) l (ω ω ) m is Σ 0 iff there exists a recursive predicate R ω k+ (2 <ω ) l (ω <ω ) m such that ( n,ᾱ, β) O iff r ω R( n,ᾱ r, β r ). One has the following definition (and facts): 3

4 Definition 2.2 (a) For k,l ω, A ω k (2 ω ) l is Σ iff A is the projection along ω ω of an arithmetical subset B of ω ω ω k (2 ω ) l, iff A is the projection along ω ω of a Π 0 subset C of ωω ω k (2 ω ) l. (b) A set is Π if its complement is Σ. It is if it is both Σ and Π. 2.3 The correspondence between the interval [0,) and Cantor space To avoid too many definitions in the introduction, we used rational numbers in our formulation of Solovay reducibility; but we shall actually work with dyadic rationals. We thus suppose an effective coding of Q 2, the set of dyadic rationals, to be fixed (we shall often identify an element of Q 2 - i.e. of the form ±m2 n, for m,n ω - with its code). Definition 2.3 A real r in R is left Π (resp. left ) if the left cut L(r) = {q Q 2 q < r} is Π (resp. ). An element x of 2 ω is called left Π (resp. left ) if the real number 0.x = i<ω x(i)2 i is left Π (resp. left ). We shall take as canonical dyadic expansion of a real number r in [0,) the sequence x 2 ω with infinitely many 0 s such that r = 0.x. For dyadic rationals in [0,), we often use the elements of 2 <ω as codes: Definition 2.4 (a) Let u 2 <ω. It codes the dyadic rational q = i< u u(i)2 i which we denote by 0.u (0. = 0). (b) The canonical representative in 2 <ω of q [0,) Q 2 is the shortest string u so that q = 0.u. (c) We shall also denote by llex the well ordering of Q 2 [0,) transposed from 2 <ω : q = 0.u with u shortest, q llex q iff q = 0.u with u shortest, and u llex u. The correspondence between [0,) and 2 ω is based on the following (see also [Nie09,.8.]): For any x 2 ω with infinitely many 0 s and any u 2 <ω, x [u] iff 0.x [0.u,0.u+2 u ). Let µ and µ R denote respectively the uniform product measure on 2 ω and Lebesgue measure on the real line. Then one notes that µ([u]) = 2 u = µ R ([0.u,0.u+2 u )). One then deduces the following useful fact: 4

5 Lemma 2.5 If U = [X], with X 2 <ω, then for O = s X [0.s,0.s+2 s ), one has µ R (O) = µ(u), for any z 2 ω with infinitely many 0 s, z U iff 0.z O. 2.4 Well-orders on ω and characterization of Π sets. We recall a few definitions: Definition 2.6 (a) For x 2 ω, let R x ω ω be the relation defined as follows: for m,n ω (m,n) R x iff x( m,m ) = x( n,n ) = x( m,n ) =. We call the set {n < ω x( n,n ) = } the domain of R x. (b) If, for x 2 ω, R x is a linear ordering (on its domain), then for m,n ω, we write m x n for R x (m,n) and m < x n for (R x (m,n) m n). Definition 2.7 (a) Let LO = {x 2 ω R x is a linear ordering} WO = {x LO x is a well ordering.} Let O = {e ω ϕ e WO} where ϕ e e ω is an effective enumeration of the partial recursive functions. Hence if e O, ϕ e is supposed to be total. (b) If x WO, then let o.t( x ) denote the order-type of x. For ξ < ω and e O, let WO(ξ ) = {x WO o.t( x ) ξ } and WO[e] = {x WO o.t( x ) o.t( ϕe )}. (c) Let ω CK denote the Church-Kleene ordinal sup{o.t( ϕe ) e O}. WO and O are canonical Π sets. Concerning Π open sets and their generating sets, one has Fact 2.8 ( [Nie09, 9..2]) A is an open Π -subset of 2ω iff A is generated by a Π subset of 2<ω. Let ϕ z e e ω be an effective enumeration of the partial recursive functions on ω with (varying) oracle z 2 ω. The fundamental result about WO is the following one: Theorem 2.9 ( See [Nie09, 9..3]) Let B ω k (2 ω ) r, for k < ω, r, be Π. Then there exists a (total) recursive function h ω k ω such that for any ( n,z) ω k (2 ω ) r, ϕ z LO (hence is total) and h( n) ( n,z) B iff ϕ z h( n) WO. This theorem gives the notion of Π -index: 5

6 Definition 2.0 (a) If B ω is Π, then there is h total recursive such that, for any n < ω, ϕ h(n) LO and n B iff ϕ h(n) WO. Any k < ω such that h = ϕ k is called a Π -index for B. (b) If B 2 ω is Π and k < ω is such that, for any x 2ω, ϕk x LO and x B iff ϕk x WO, then k is termed a Π -index for B. (c) If B 2 ω is and k,l < ω are Π -indices for respectively B and Bc, then (k,l) is a -index for B. There is a strong connection between a Π subset of 2ω and its measure: Lemma 2. ([Sac90, IV..],[Nie09, 9..0]) (a) If B 2 ω is Π, then the left cut of µ(b) is also Π. (b) Moreover there is λ total recursive such that if k < ω is a Π -index for B, then λ(k) is a Π -index for the left cut of µ(b). The connection with set theory is the following: let L ω CK be the ω CK th level in Gödel constructible hierarchy. Relatively to the langage of Set Theory { }, L ω CK satisfies -comprehension, Σ - collection, and hence this allows definition by Σ recursion of functions which are thus Σ definable over L ω CK. As a consequence of this and of Theorem 2.9, one obtains an ω CK -enumeration of a Π subset of ω: Theorem 2.2 ( See [Nie09, 9..2]) (a) For any Π subset X of ω, there exists a function F ωck ω {ω} which is Σ definable over L ω CK such that X = range(f) ω. (b) For any X ω, X is Π iff there is a Σ formula Φ such that X = {n ω L ω CK Φ(n)} (the equivalence still holds with parameters from L ω CK in Φ). 2.5 Π ML randomness. We recall now some fundamental definitions and results of ML-randomness in the Π context: Definition 2.3 ([HN07] - see [Nie09, 9.2.3]) (a) Let U be a universal by adjunction Π prefix-free machine. One defines the corresponding descriptive string complexity notion: for y 2 <ω, let K(y) = min{ σ σ 2 <ω U(σ) = y}. (b) A Π ML test is a uniform Π sequence of open subsets of Cantor space G m m < ω such that µ(g m ) 2 m. (c) A sequence x 2 ω is Π ML random if for any Π ML test G n n < ω, x n<ω G n. Hjorth and Nies obtained a Π version of Kraft-Chaitin theorem: 6

7 Theorem 2.4 ([HN07, Thm 3.2], [Nie09, 9.2.4]) Let W ω 2 <ω be a Π bounded request set. Then there is a prefix-free Π machine M such that for any (n,t) W there is s 2 <ω such that s = n and M(s) = t. We mention here a simpler proof of this theorem by Monin in [Mon, 3.7.] inspired from Miller s exposition of Kraft-Chaitin theorem (see [DH0, 3.6.]). The Π equivalent of Schnorr s theorem is the following result: Theorem 2.5 ([HN07, Thm 3.9] - see [Nie09, 9.2.8]) For any x 2 ω, x is Π ML random iff b < ω n < ω K(x n) > n b. We shall not need this theorem and we shall in fact reobtain it in a very indirect way for left Π reals. 3 A characterization of left Π -reals In analogy with the classical c.e. case, one obtains: Lemma 3. Let r [0,). Then the following are equivalent: (a) r is left Π, (b) there exists a non-decreasing Σ definable over L ω CK sequence r ξ ξ < ω CK of reals such that r = sup{r ξ ξ < ω CK } and for any limit ordinal ξ < ω CK, r ξ = sup{r η η < ξ }, (c) there is a Π prefix-free set X 2<ω such that µ([x]) = r. Moreover if r > 2 c, for some c < ω, we can assume 0 c X. Remark 3.2 (i) By calling r [0,), we mean the existence of z 2ω with r = 0.z such that the function n z n is. (ii) Since - by 2.2 and -comprehension - the set of ω-sequences of integers in L ω CK ) is exactly the set of ω-sequences ((i.e ωω ), the reals r ξ, ξ < ω CK ω ω L ω CK of statement (b), belong to L ω CK and the assertion makes sense. (iii) The continuity condition: for ξ limit, r ξ = sup{r η η < ξ } can be omitted, but it makes the argument in (b) (c) cleaner. (i.e, Proof Let r [0,). (a) (b) Let L(r) viewed as a subset of 2 <ω be the Π left cut of r. By Theorem 2.2(a), there is 7

8 F ω CK 2 <ω {ω} Σ definable over L ω CK (and hence definable) such that L(r) is the set range(f) 2 <ω. For each ξ < ω CK, we set: r ξ = 0 if range(f ξ ) 2 <ω =, r ξ = sup{0.f(η) η < ξ and F(η) 2 <ω } otherwise. More precisely, when range(f ξ ) 2 <ω, we have r ξ = 0.z ξ where, for n < ω, z ξ n = max f or lex {F(η) n 0 n F(η) n η < ξ, F(η) 2 <ω } (we need to consider F(η) n 0 n F(η) n instead of F(η) n when F(η) < n). (b) (c) Let r > 2 c and r ξ ξ < ω CK be as in (b). Let ξ 0 < ω CK be the least ordinal ξ such that 2 c < r ξ. ξ 0 is a successor ordinal. We consider the following open intervals: I ξ0 = (2 c,r ξ0 ), for ξ 0 < ξ < ω CK, I ξ = (sup η<ξ r η,r ξ ). A necessary condition for I ξ to be nonempty is that ξ is successor. - For each ξ ξ 0, we define X(ξ ) = {s 2 <ω [0.s,0.s+2 s ) I ξ and s is minimal for with this property}. For ξ < ω CK, X(ξ ) is prefix-free (possibly empty) and I ξ is obtained as a disjoint union: I ξ = s X(ξ ) [0.s,0.s+2 s ). () - For ξ 0 η < ξ < ω CK, I η I ξ =. Hence if X(η), X(ξ ) are nonempty, their elements are incompatible. Also [0.0 c,0.0 c + 2 Oc ) = [0,2 c ) has empty intersection with all I ξ, ξ ξ 0. We deduce that if X = ξ0 ξ <ω CK Hence by 2.2, it is Π. We have X(ξ ) {0 c }, then X is prefix-free and Σ [0,r) {r ξ ξ 0 ξ < ω CK } = [0,2 c ) ξ 0 ξ <ω CK = [0.s,0.s+2 s ) (by ()). s X Hence by Lemma 2.5, µ R ([0,r) {r ξ ξ 0 ξ < ω CK }) = µ([x]). But µ R ([0,r) {r ξ ξ 0 ξ < ω CK }) = r. Therefore µ([x]) = r. (c) (a) I ξ definable over L ω CK Let r = µ([x]) with X a Π set. Then the open set [X] 2ω is also Π. By 2.(a), r = µ([x]) is left Π. Lemma 3.. Concerning the case of left reals, one notes: 8

9 Lemma 3.3 Let r be a left Π real. Then the following are equivalent: (a) r is, (b) the left cut of r is, (c) any nondecreasing sequence r ξ ξ < ω CK Σ -definable over L ω CK each ξ < ω CK such that r = sup ξ <ω CK r ξ is eventually constant. with r ξ for Proof - (a) (b) is as in the computable case (see [DH0, 5..]). - We show (a) (c). Let us assume z 2 ω is with infinitely many s such that r = 0.z and z ξ ξ < ω CK is Σ -definable over L ω CK with - ξ < ω CK, z ξ 2 ω is, - 0.z ξ ξ < ω CK nondecreasing, - 0.z = sup ξ <ω CK 0.z ξ. We claim that for any n < ω, there is ξ < ω CK such that 0.z n = 0.z ξ n. Let us suppose for a contradiction that for some n < ω and all ξ < ω CK, 0.z ξ n < 0.z n. Then there is t 2 n such that 0.t = max ξ <ω CK 0.z ξ n < 0.z n. This implies 0.t ω 0.z n. But since z involves infinitely many s, necessarily 0.z n < 0.z. Hence 0.t ω < 0.z For any ξ < ω CK, 0.z ξ 0.t ω. Hence sup ξ <ω CK 0.z ξ 0.t ω < 0.z. We reached a contradiction. We can thus define the function f ω ω CK n least ξ < ω CK s.t. 0.z ξ n = 0.z n. f is Σ -definable over L ω CK. By Σ -collection, there must exist ξ 0 < ω CK such that range( f ) ξ 0. But then 0.z = 0.z ξ0. lemma Π -Solovay reducibility and Π -Solovay completeness As in the c.e. case (see [DH0, 9.., 9..8] and [Nie09, , ]), there are two notions of Π -Solovay reducibility which will be shown equivalent. Definition 4. (a) For x,y 2 ω, x is Π -Solovay reducible to y if there exist m < ω and a partial Π function f whose domain and range are included in Q 2 such that for any q Q 2 with q < 0.y, f (q) is defined < 0.x and (0.x f (q)) < 2 m (0.y q). One writes x s y. (b) Let x,y 2 ω be left Π. x + s y iff there exist r R left Π and m < ω such that 2m 0.y = 0.x +r. To show equivalence of the two notions as in [DHN02], we first obtain a useful characterization of s (see [DH0, 9..7]), replacing dyadic rationals by reals: 9

10 Lemma 4.2 Let x,y 2 ω be left Π and by Lemma 3., let x ξ ξ < ω CK, y ξ ξ < ω CK be non- and decreasing Σ definable over L ω CK y ξ. Then 0.y = sup ξ <ω CK x s y sequences of reals, such that 0.x = sup ξ <ω CK there are m < ω, g ω CK ω CK strictly increasing, Σ definable over L ω CK such that for any ξ < ω CK, 0.x x g(ξ ) 2 m (0.y y ξ ). To simplify notation, we denote the reals converging towards 0.x and 0.y by x ξ and y ξ rather than by 0.x ξ and 0.y ξ. Proof : We assume m < ω and the Π function f witness the fact that x s y. Let us distinguish two cases: Case : y is left Π but not. Let ξ < ω CK. To define g by Σ recursion, we assume g ξ is Σ defined over L ω CK η < ξ, 0.x x g(η) (0.y y η ). x ξ and satisfies We want to define g(ξ ) so that the above inequality holds for ξ. The y ξ s which are must be < 0.y. We can thus consider the least q Q 2 for llex such that y ξ q < 0.y, and define g(ξ ) as the least δ > range(g ξ ) such that x δ f (q). One derives: 0.x x g(ξ ) 0.x f (q) < 2 m (0.y q) 2 m (0.y y ξ ). Case 2: y is. Let y 2 ω be so that 0.y = 0.y and y admits infinitely many s. Since, for any n < ω, 0.x f (0.y n ) < 2 m (0.y 0.y n ), 0.x must be equal to sup n<ω f (0.y n ) and x must be. By Lemma 3.3, there is ξ 0 < ω CK such that 0.x = x ξ0. But then taking g(ξ ) = ξ 0 + ξ makes g trivially satisfy the expected inequality. : We assume m < ω and g ω CK ω CK satisfy the righthand side of the equivalence in Lemma 4.2. To define f, for each q Q 2 with q < 0.y, we consider the least ξ < ω CK such that q < y ξ and set f (q) = x g(ξ ). Lemma 4.2 Remark 4.3 By the arguments in the proof of Lemma 4.2 ( case 2), one notes that the class of reals which are minimal for s is exactly the class of reals. Our goal is now to prove in analogy with [DHN02]: Proposition 4.4 For any x,y 2 ω which are left Π, x s y x + s y. Proof Let x,y 2 ω be left Π. 0

11 We show first the easier x + s y x s y. Let m < ω and r R left Π be so that 2m 0.y = 0.x+r. If L(0.x) and L(r) denote respectively the Π left cuts of 0.x and r, then by 2.2, there exist F x,f r ω CK Q 2 {ω} Σ definable over such that range(f x ) Q 2 = L(0.x) and range(f r ) Q 2 = L(r). L ω CK For any q < 0.y, there must exist q L(0.x), q L(r) such that 2 m q < q +q. Hence for any q < 0.y, we consider the least couple (for lexicographic order) (ξ,η) (ω CK ) 2 such that 2 m q < F x (ξ )+F r (η) and set f (q) = F x (ξ ). One checks that f and m satisfy the conditions of Definition 4.(a) for x s y : f is Σ definable over L ω CK, and for any q < 0.y, 2 m (0.y q) = 0.x +r 2 m q We now prove x s y x + s y. > 0.x +r F x (ξ ) F r (η) > (0.x f (q))+(r F r (η)) > 0.x f (q). We cannot import the sum argument of [DHN02], but by the previous lemma, we assume x ξ ξ < ω CK, y ξ ξ < ω CK are nondecreasing Σ definable over L ω CK sequences of reals such that sup ξ <ω CK x ξ = 0.x and sup ξ <ω CK y ξ = 0.y, m < ω and g ω CK ω CK, Σ definable over L ω CK, strictly increasing are such that for any ξ < ω CK, (0.x x g(ξ ) ) 2 m (0.y y ξ ). () We suppose moreover that m < ω is such that 0.x < 2 m 0.y (2) Our goal is either to build inductively a function h ω CK ω CK strictly increasing, Σ definable over L ω CKsuch that - x g(h(0)) < 2 m y h(0), - for any η < ω CK, Φ(η) holds where Φ(η) η < η x g(h(η)) x g(h(η )) 2 m (y h(η) y h(η )), or to obtain a real r such that 2m 0.y = 0.x +r. Step 0 There must exist δ < ω CK such that x g(δ) < 2 m y δ. Otherwise we would get sup δ<ω CK x g(δ) 2 m sup δ<ω CK y δ, that is 0.x 2 m 0.y which would contradict (2). We thus take the least such δ for h(0). Step δ > 0 We assume h δ is Σ definable over L ω CK, strictly increasing, and such that for any η < δ, Φ(η) holds. Let η < δ be fixed. For any η < η < δ, Φ(η) gives us: x g(h(η)) x g(h(η )) 2 m (y h(η) y h(η )). (4) (3)

12 Let us consider the reals: x δ = sup η<δ x g(h(η)) and y δ = sup η<δ y h(δ). By taking the supremum on η < δ in (4), we derive: x δ x g(h(η )) 2 m (y δ y h(η )). (5) Now Φ(η), for η < δ gives x g(h(η)) x g(h(0)) 2 m (y h(η) y h(0) ) and hence, 2 m y h(η) x g(h(η)) 2 m y h(0) x g(h(0)) > 0. (The last inequality holds by step 0). Hence by taking the suprema on η < δ, we obtain Now by (), 0.x x g(h(η)) (0.y y h(η) ) and hence 2 m y δ x δ 0. (6) 0.x x δ 2m (0.y y δ ) (7) Case : For any θ > range(h δ ), x g(θ) x δ > 2m (y θ y δ ). By taking suprema, we obtain 0.x x δ 2m (0.y y δ ). (8) Combined with (7), this gives 0.x x δ = 2m (0.y y δ ) and 2m 0.y = 0.x +(2 m y δ x δ ). By (6), r δ = 2 m y δ x δ is a positive real. Therefore x + s y. Case 2: There exists θ > range(h δ ) such that x g(θ) x δ 2m (y θ y θ ). We define h(δ) as the least such θ. Then one has x g(h(δ)) x δ 2m (y h(δ) y δ ). Combining this with the inequality (5), one deduces x g(h(δ)) x g(h(η )) 2 m (y h(δ) y h(η )). This holds for all η < δ, hence Φ(δ) is satisfied. Hence either for some δ < ω CK, the construction failed and we obtained 2 m 0.y = 0.x +r, for a real r or we have defined h ω CK ω CK with the right properties. In this last situation, since Φ(η) holds for any η < ω CK, we obtain for any 0 < η < η < ω CK, 2 m y h(η) x g(h(η)) 2 m y h(η ) x g(h(η )) 2 m y h(0) x g(h(0)) > 0. For η < ω CK, let r η = 2 m y h(η) x g(h(η)). r η η < ω CK sequence of positive reals. Hence r = sup η<ω CK L ω CK η < ω CK, we deduce Therefore x + s y. r = lim η<ω CK r η = 2 m 0.y 0.x. is a nondecreasing Σ definable over r η is left Π. Taking the limit for Proposition 4.4 For further use, let us state a direct consequence of (3): 2

13 Lemma 4.5 Let x, y be left Π sequences such that x s y. Then there exist nondecreasing Σ definable over L ω CK, sequences of reals x ξ ξ < ω CK, y ξ ξ < ω CK and m < ω such that 0.x = sup ξ <ω CK x ξ, 0.y = sup ξ <ω CK y ξ and for any η < ξ < ω CK, x ξ x η 2 m (y ξ y η ). We now consider the notion of completeness associated with Π -Solovay reducibility: Definition 4.6 Let x 2 ω be left Π. Then x is Π -solovay complete if for any y 2ω which is left Π, y s x (or equivalently y + s x). 5 Characterization of Π -Solovay completeness Let K denote the optimal descriptive complexity defined in 2.3. In correspondence with results of Calude, Hertling, Khoussainov, Wang [CHKW98] and Kucera, Slaman [KS0] in the c.e. case, one obtains: Theorem 5. Let x 2 ω be left Π. The following are equivalent: (a) x is Π -Solovay complete, (b) there is a prefix-free Π machine V universal by adjunction such that 0.x = Ω V = µ([dom(v )]), (c) there is a prefix-free optimal Π machine V such that 0.x = Ω V, (d) there is b < ω such that for any n < ω, K(x n ) > n b, (e) x is Π ML random. Proof We shall prove the following implications: (a) (b) (c) (d) (a) and (a) (e); the remaining implication (e) (d) is the easy direction of Theorem 2.5 (based on the fact that if R b = [{s 2 <ω K(s) s b}], then (R b ) b ω is a Π ML set). We shall not need the sophisticated (d) (e) of Theorem 2.5 because as mentioned above, when dealing with left Π -reals, one can get Π prefix-free sets. By Proposition 4.4, we shall use indifferently s or + s. (a) (b) Let x 2 ω be Π -Solovay complete. Let M be a prefix-free Π machine which is universal by adjunction and let Ω M = µ([dom(m)]). Then Ω M is left Π. Since x is Π -Solovay complete, there exist m < ω and a left Π real r such that 2 m 0.x = Ω M +r. () We can assume moreover that 0.x +2 m <. Let us note that 2 m 2 m +r2 m < 2 m +0.x <. 3

14 Hence by Lemma 3.(c) applied to the left Π real 2 m + r2 m, there exists a prefix-free Π set X 2 <ω such that 0 m X and µ([x]) = 2 m r +2 m. We now define the machine V in a way very similar to [CHKW98]: dom(v ) = 0 m dom(m) (X {0 m }) and ( for t dom(m), V (0m t) = M(t), for s X {0 m }, V (s) = s. Then V is a prefix-free universal by adjunction Π machine such that (b) (c) is immediate. Ω V = 2 m Ω M +(µ([x]) 2 m ) = 2 m Ω M +2 m r = 0.x (by ()). (c) (d) This is the Π version of Chaitin s theorem ( [Nie09, 3.2.], [DH0, 6..3]); the result is mentioned without proof in [HN07] and [Nie09, p.376]. Since this article aims at an accessible exposition of Π -Solovay completeness, we supply some arguments. Let V be the Π prefix-free optimal machine such that 0.x = Ω V (x with infinitely many 0 s). Let V ξ ξ < ω CK be the contruction of V obtained (by 2.0(a)) from an index k < ω for V : V = {(s,t) (2 <ω ) 2 ϕ ϕk (s,t) WO}, for ξ < ω CK, V ξ = {(s,t) (2 <ω ) 2 o.t( ) < ξ }. (2) ϕϕk (s,t) Then one defines the (plain) Π machine M as follows: for s 2 <ω, let ξ s < ω CK be the least ordinal ξ (if it exists) such that 0.s µ([dom(v ξ ])) < 0.s+2 s. M(s) is the least t for llex such that t range(v ξ 2 s ) where 2 m = {u 2 <ω u m}, for m < ω. One cannot simply take as in the classical case t least such that t range(v ξ ) since range(v ξ ) might be 2 <ω. One obtains a constant c < ω such that for any n < ω, n < K V (M(x n )) K M V (M(x n ))+c K V (x n )+c. Using again the optimality of V, we have K = K V + O() and we deduce the existence of b < ω such that for any n < ω, K(x n ) > n b. To assert that M is a Π -machine, we relied on the fact that the function F ξ µ([dom(v ξ ]) is Σ definable over L ω CK or equivalently on the fact that the relation (in (q,ξ ) Q 2 ω CK ) q < µ([dom(v ξ ]) is definable over L ω CK. We provide some explanation the reader may choose to skip: the uniform definability in ξ relies on Sacks result 2.(b) and on properties of the Constructible hierarchy defined in (Kripke-Platek) set theory. Corresponding to the construction V ξ ξ < ω CK of V with respect to the index k (see (2)), one considers, for e O, V e = {(s,t) (2 <ω ) 2 o.t( ϕϕk (s,t) ) < o.t( ϕ e )}. 4

15 By [Mos09, 4.A.2] or [Nie09, 9..4], the relation (in (e,x) O 2 ω ) x [dom(v e )] is. By the fundamental result 2.9, there exist two recursive functions g,h such that for any e in O, (g(e),h(e)) is a -index for [dom(v e )]. One can then deduce from Lemma 2. that the relation q < µ([dom(v e )]) is (in q Q 2, e O). It remains to associate with each ξ < ω CK some e O such that o.t( ϕe ) = ξ in a Σ way over L ω CK. If ξ < ω CK, then there exists e O such that ξ = o.t( ϕe ). Given such a fixed couple (ξ,e), the isomorphism G (ξ, ) (dom( ϕe ), ϕe ) can be defined by Σ recursion over L ω CK. Hence G is Σ definable over L ω CK. By comprehension, G belongs to L ω CK. Now let us consider the 0 formula Φ(ξ,e,y): y is a transitive set ξ y (ω ω) y G y G isomorphism (ξ, ) (dom(r e ),R e ) where R e = {(m,n) ω ω ϕ e ( m,n ) = ϕ e ( m,m ) = ϕ e ( n,n ) = } and a set u is transitive if x u x u. We can now define F ω CK O as follows: ξ Ord e < ω F(ξ ) = e iff [ θ Ord y (y = L θ Φ(ξ,e,y) e < e Φ(ξ,e,y)] [ θ < θ y (y = L θ e < ω Φ(ξ,e,y ))]. By [Dev84, p.70], the relation η Ord x = L η is in any (transitive) model of Kripke-Platek set theory. L ω CK is such a model. Therefore F is definable over L ω CK So finally, the relation (in (q,ξ )) is over L ω CK. (d) (a) and satisfies: ξ < ω CK, F(ξ ) O o.t( ϕf(ξ ) ) = ξ. q < µ([dom(v ξ ]) iff e < ω (e = F(ξ ) q < µ([dom(v e ])) Let x 2 ω be left Π and such that there is b < ω with n < ω K(x n) > n b. We keep ideas from Kucera and Slaman [KS0] who assumed x to be ML-random. Since the open sets involved in the Π ML-tests can be naturally generated by Π prefix-free sets, we show (d) (a) (rather than appealing to Theorem 2.5 for (d) (e) and checking (e) (a)). Let y 2 ω be left Π and x as above, we aim at proving y s x. By Lemma 3., let x ξ ξ < ω CK and y ξ ξ < ω CK be Σ definable over L ω CK y ξ. sequences of reals such that 0.x = sup ξ <ω CK x ξ and 0.y = sup ξ <ω CK We define simultaneously (and uniformly in n < ω) by Σ -recursion over L ω CK sequences of reals bn ξ ξ < ωck, r n ξ ξ < ωck such that ξ < ω CK the following way: for each n < ω, ξ = 0: let b n 0 = y 0 and r0 n = x n y 0. nondecreasing two nondecreasing, b n ξ y ξ and x ξ r n ξ in ξ : We assume the nondecreasing b n η η < ξ and rη n η < ξ have been defined and we distinguish two cases: case : x ξ η<ξ [x η,rη), n let then b n ξ = sup η<ξ bn η and r n ξ = x ξ. case 2: x ξ η<ξ [x η,r n η), let then b n ξ = y ξ and r n ξ = x ξ +2 2n (y ξ sup η<ξ b n η). 5

16 One can check by induction on ξ < ω CK the following properties: b n ξ ξ < ωck and r n ξ ξ < ωck are nondecreasing, if ξ is in case, b n ξ = sup η<ξ bn η is < y ξ, if ξ is in case 2, b n ξ = y ξ is > sup η<ξ b n η. Hence the interval [x ξ,r n ξ ) is nonempty iff ξ is in case 2. Let us set For ξ < ω CK, O n (ξ ) = η<ξ [x η,rη) n and O n = ξ <ω CK O n (ξ ). In both case and 2, one has r n ξ = x ξ + 2 2n (b n ξ sup η<ξ bn η). Hence the intervals [x η,rη) n being disjoint, for η < ξ < ω CK, one checks that for all ξ < ω CK, µ R (O n (ξ )) 2 2n b n ξ. Therefore µ R (O n ) 2 2n. (3) We introduce the open subsets of 2 ω corresponding to the O n s modulo the x ξ s: for ξ < ω CK, Y n (ξ ) = {s 2 <ω s minimal for such that [0.s,0.s+2 s ) (x ξ,r n ξ )} and Y n = ξ <ω CK Y n (ξ ). Y n is Π. Since for η < ξ < ωck, the intervals [x η,rη) n and [x ξ,r n ξ ) are disjoint, two elements of respectively Y n (η) and Y n (ξ ) are incompatible. Therefore Y n is prefix free. Now for ξ < ω CK, (x ξ,r n ξ ) = s Y n(ξ )[0.s,0.s+2 s ). Hence s Yn [0.s,0.s+2 s ) = ξ <ω CK(x ξ,r n ξ ) = O n {x ξ ξ < ω CK }. By Lemma 2.5, we deduce: µ([y n ]) = µ R (O n {x ξ ξ < ω CK }) = µ R (O n ) 2 2n (by (3)). (4) For any z 2 ω with infinitely many 0 s, z [Y n ] iff 0.z O n {x ξ ξ < ω CK }. (5) In order to argue with K in place of the Π ML test [Y n] n < ω, we consider the bounded request set L = {( s m +,s) s Y m, m < ω}. By (4) and the prefix-free character of Y n, the weight of L is (this is why we used 2 2n rather than 2 n ) and L is Π. By the Π KC theorem 2.4, there exists a prefix free Π machine M associated with L. Hence there is c < ω such that for any s Y m, K(s) K M (s)+c s m++c. But by hypothesis on x, there is b < ω such that for any n < ω, K(x n ) > n b. Let m 0 c+b+ be fixed. We deduce that for any n < ω, x n Y m0, that is x [Y m0 ]. x cannot be equal to one of the x ξ s (because it would be which contradicts the fact that for any n < ω, K(x n ) > n b). Hence by (5), 0.x O m0. Claim 5.2 Since 0.x O m0, case 2 must occur unboundedly often in constructing O m0. Proof Let us assume for a contradiction that case occurs for every ξ ξ 0. That is 6

17 ξ ξ 0, x ξ η<ξ0 [x η,r m 0 η ). Therefore 0.x sup η<ξ0 r m 0 η. But since 0.x O m0 = η<ξ0 [x η,r m 0 η ), we deduce 0.x sup η<ξ0 r m 0 η. Therefore 0.x = sup η<ξ0 r m 0 η would be, a contradiction. Claim 5.2 We can thus define, relatively to m 0, by Σ recursion over L ω CK follows: h(0) = 0, h(η +) is the least δ > h(η) in case 2, for η limit, h(η) = δ<η h(δ). the function h ω CK ω CK as By Σ -collection, the construction is possible at limit stages: δ<η h(δ) < ω CK. The function h is strictly increasing and enumerates all δ < ω CK which are in case 2. One can check by induction on δ < ω CK that b m 0 = sup δ h(η) δ y h(η). Hence for any δ < ω CK, one has x h(δ +) r m 0 h(δ ) = x h(δ ) +2 2m 0 (y h(δ ) sup θ<δ y h(θ) ) (6) From (6), δ being fixed, one derives by induction on η δ, x h(η) x h(δ) +2 2m 0 ( sup y h(η ) sup y h(δ )). (7) η <η δ <δ Taking successor ordinals η = η +, δ = δ + such that η > δ, gives x h(η +) x h(δ +) +2 2m 0 (y h(η ) y h(δ )). By considering the limits on η < ω CK, one gets for any δ < ω CK, 0.x x h(δ +) 2 2m 0 (0.y y h(δ )). One then easily derives y s x. Since y was arbitrary, this gives the Π -Solovay completeness of x. (a) (e) We needed this implication in [Sur6] and proposed there a proof inspired from [Nie09, ] based on the completeness for + s. Our argument here is simpler and involves s. We argue by contradiction, assuming x not Π ML random and we show that if y s x then y cannot be Π ML random either. By Remark 4.3, we can assume x and y are not and by Lemma 4.5, let m < ω, x ξ ξ < ω CK and y ξ ξ < ω CK nondecreasing Σ definable over L ω CK and 0.x = sup ξ <ω CK x ξ, 0.y = sup ξ <ω CK for any η < ξ < ω CK, 0 x ξ x η 2 m (y ξ y η ). y ξ sequences of reals be such that 7

18 Let U n n < ω be a Π ML test such that x n<ω U n and let X n 2 <ω, for n < ω, be uniformly Π such that U n = [X n ]. We consider a (uniform) Π construction of X n: Let F ω ω CK 2 <ω {ω} be Σ definable over L ω CK for n < ω, X n = range(f {n} ω CK so that: ) 2 <ω. For n < ω, ξ < ω CK, we set X n,ξ = range(f {n} ξ ) 2 <ω. Let n < ω be fixed. For each ξ < ω CK, if there is s X n,ξ such that x ξ [0.s,0.s + 2 s ), then s n ξ is the least such string for llex in X n,ξ. One considers the open interval (y O n (ξ ) = ξ,y ξ +2 m sn ξ ) if s n ξ defined, otherwise, and let O n = ξ <ω CK O n (ξ ). We note that the set {ξ < ω CK s n ξ is defined} is definable over L ω CK. We shall check that for all n < ω, 0.y O n and µ R (O n ) 2 m+ µ(u n ). Claim 5.3 For each n < ω, 0.y O n. Proof Since x U n, there exists s X n such that s x, and hence 0.x (0.s,0.s + 2 s ) (we consider the open interval because x does not belong to Q 2 ). Let ξ 0 < ω CK be such that s X n,ξ0. There is ξ ξ 0 such that for any ξ ξ, x ξ (0.s,0.s+2 s ). Hence for ξ ξ, s n ξ is defined and sn ξ llex s. Since x ξ belongs to both intervals [0.s,0.s + 2 s ) and [0.s n ξ,0.sn ξ + 2 sn ξ ), the strings s and s n ξ must be compatible and hence sn ξ s. This implies for ξ ξ, [0.s,0.s+2 s ) [0.s n ξ,0.sn ξ +2 sn ξ ). For ξ ξ, both 0.x and x ξ belong to the interval [0.s n ξ,0.sn ξ + 2 sn ξ ). Therefore 0 0.x x ξ < 2 sn ξ. This gives for ξ ξ, 0.y y ξ 2 m (0.x x ξ ) < 2 m sn ξ and Hence since y is not, 0.y O n(ξ ) for ξ ξ. y ξ 0.y < y ξ +2 m sn ξ. Claim 5.3 Claim 5.4 For n < ω, µ R (O n ) 2 m+ µ(u n ). Proof It is possible to argue with maximal open or half-open subintervals of the subset of the real line A n = t Xn [0.t,0.t +2 t ), but we propose an argument in the manner of [Sur6, 5.2]. For n < ω, let X n be the prefix-free subset of X n consisting of the strings of X n which are minimal 8

19 for (it is not necessarily Π ). Let us note that O n = ξ <ω CK O n (ξ ) = O n (ξ ) = ( s n ξ defined t X n t s n ξ s n ξ defined O n (ξ )). (8) We must bound the measure µ R ( t s n ξ O n(ξ )), for t X n. Let t X n be fixed. We set Ord(t) = {ξ < ω CK t s n ξ }. Let ξ η be both in Ord(t). From we derive x ξ [0.s n ξ,0.sn ξ +2 sn ξ ) [0.t,0.t +2 t ), x η [0.s n η,0.s n η +2 sn η ) [0.t,0.t +2 t ), 0 y η y ξ 2 m (x η x ξ ) 2 m 2 t 2 m t. (9) Let ξ t be the least element of Ord(t) (if nonempty). If ξ Ord(t), then t s n ξ. We thus deduce: O n (ξ ) = (y ξ,y ξ +2 m sn ξ ) (y ξt, sup ξ Ord(t) y ξ +2 m t ). Hence t s n O n(ξ ) = ξ ξ Ord(t) O n (ξ ) (y ξt,sup ξ Ord(t) y ξ +2 m t ) and µ R ( O n (ξ )) 2 m t + sup (y ξ y ξt ), t s n ξ ξ Ord(t) 2 m t +2 m t (by (9)) 2 m+ t. (0) From (8) and (0), we obtain: µ R (O n ) t X n µ R ( t s n ξ O n(ξ )) t X n 2 m+ t 2 m+ µ([x n]) 2 m+ µ(u n ). To conclude, let us consider the open subsets of 2 ω corresponding to the O n s: Claim 5.4 Y n ξ = {t 2<ω [0.t,0.t +2 t ) O n (ξ )} for n < ω, ξ < ω CK, and Y n = ξ <ω CK Y n ξ. Then O n = t Yn [0.t,0.t + 2 t ). Hence by Lemma 2.5 and Claims 5.3, 5.4, we deduce y [Y n ] and µ([y n ]) = µ R (O n ) 2 m+ µ(u n ). Therefore [Y m++n ] n < ω is a Π ML test such that y n<ω[y m++n ]. This concludes the proof of (a) (e). 9

20 (e) (d) This implication is a direct consequence of the fact that if R b = [{s K(s) s b}], for b < ω, then R b b < ω is a Π ML test. This ends the proof of the theorem. Theorem 5. Let us note that since the open sets constructed in the proof of (a) (e) can be expressed in a Σ definable way over L ω CK, as unions of disjoint intervals (because of the nondecreasing character of the sequence of left end-points y ξ, for ξ < ω CK ), one can deduce: Corollary 5.5 A Π ML test U n n < ω is called prefix-free if there exist uniformly Π X n, n < ω, such that U n = [X n ]. For any x 2 ω which is left Π, the following are equivalent: x is Π ML random, For any prefix-free Π ML test U n n < ω, x n<ω U n. prefix-free sets Hence to prove Π ML randomness for left Π reals, one can work with open sets generated by prefix-free sets. Π Proof We assume x n<ω U n for a Π ML test U n n < ω, and exhibit a prefix-free Π ML test U n n < ω such that x n<ω U n. We suppose x is not and apply the argument of (a) (e) to x itself (taking m = 0). For ξ < ω CK, we define - when possible - s n ξ as in the proof of (a) (e) and set (x O n (ξ ) = ξ,x ξ +2 sn ξ ) if s n ξ defined, otherwise, O n = ξ <ω CK O n (ξ ). Then as above, for any n < ω, we obtain 0.x O n and µ R (O n ) 2µ(U n ). For ξ < ω CK and s n ξ defined, let x ξ = max(x ξ,sup{x η + 2 sn η η < ξ s n η defined}), and let I n (ξ ) = [x ξ,x ξ +2 sn ξ ) (by convention, [a,b) = if a b). If η < ξ < ω CK and I n (η), I n (ξ ) are nonempty, then they are disjoint and I n (ξ ) is above I n (η). Also one checks by induction on ξ < ω CK : {[x η,x η +2 sn η ) η < ξ s n η defined} = {I n (η) η < ξ s n η defined}. Hence {[x ξ,x ξ +2 sn ξ ) s n ξ defined} = {I n(ξ ) s n ξ defined}. Therefore we deduce the equality modulo reals: O n = {(x ξ,x ξ +2 sn ξ ) s n ξ defined} = {(x ξ,x ξ +2 sn ξ ) s n ξ defined}. Hence 0.x {(x ξ,x ξ +2 sn ξ ) s n ξ defined} and µ R ( {(x ξ,x ξ +2 sn ξ ) s n ξ defined}) = µ(o n) 2µ(U n ). 20

21 To return to Cantor space, it suffices to consider Y n = {t 2 <ω ξ (s n ξ defined t minimal for such that [0.t,0.t +2 t ) (x ξ,x ξ +2 sn ξ ))}. Then Y n is uniformly Π, prefix-free and by Lemma 2.5 such that x [Y n ] and µ([y n ]) = µ R ( {(x ξ,x ξ +2 sn ξ ) s n ξ defined}) 2µ(U n). Hence if U n = [Y n ], for n < ω, then U n+ n < ω is the expected prefix-free Π ML test. Corollary 5.5 References [CHKW98] C Calude, P. Hertling, B. Khoussainov, and Y. Wang. Recursively enumerable reals and Chaitin Ω numbers. STACS 98. 5th annual symposium on theoretical aspects of computer science. Paris, France, February 25 27, 998. Proceedings, pages , 998. [CY5] C.T. Chong and L. Yu. Recursion Theory, Computational aspects of definability. De Gruyter, 205. [Dev84] K. Devlin. Constructibility. Perspectives in Mathematical Logic. Berlin etc.; Springer, 984. [DH0] [DHN02] [HN07] R. Downey and D. Hirschfeldt. Algorithmic Randomness and Complexity. Springer- Verlag, 200. R. Downey, D. Hirschfeldt, and A. Nies. Randomness, computability, and density. SIAM J. Comput., 3(4):69 83, G. Hjorth and A. Nies. Randomness via effective descriptive set theory. J. Lond. Math. Soc., II. Ser., 75(2): , [Jec03] T. Jech. Set theory. The third millennium edition. Berlin: Springer, [KS0] [ML70] A. Kucera and T. Slaman. Randomness and recursive enumerability. SIAM J. Comput., 3():99 2, 200. Per Martin-Löf. On the notion of randomness. In Intuitionism and Proof Theory (Proc. Conf., Buffalo, N.Y., 968), pages North-Holland, Amsterdam, 970. [Mon] B. Monin. Higher computability and randomness, Ph.D. Université Paris 7, [Mos09] Y. Moschovakis. Descriptive Set Theory. Amer. Math. Soc., 2 edition, [Nie09] A. Nies. Computability and Randomness. Oxford University Press,

22 [Sac90] G. Sacks. Higher recursion theory. Berlin, Springer-Verlag, 990. [Sol75] R. Solovay. Draft of paper on Chaitin s work. Unpublished notes [Sur6] C. Sureson. Π -Martin-Löf random reals as measures of natural open sets. Theor. Comput. Sci., 653:26 4,