On the incomputability of computable dimension
aa r X i v : . [ c s . L O ] F e b ON THE INCOMPUTABILITY OF COMPUTABLE DIMENSION
LUDWIG STAIGERInstitut für Informatik, Martin-Luther-Universität Halle-Wittenberg, von-Seckendorff-Platz 1, D–06099 Halle (Saale), Germany e-mail address : [email protected]
Abstract.
Using an iterative tree construction we show that for simple computable sub-sets of the Cantor space Hausdorff, constructive and computable dimensions might beincomputable.
Computable dimension along with constructive dimension was introduced by Lutz [Lut03a,Lut03b] as a means for measuring the complexity of sets of infinite strings ( ω -words). Sincethen and prior to this constructive and computable dimension were investigated in connec-tion with Hausdorff dimension (for a detailed account see [DH10, Section 13]). The resultsof [Hit05, Sta93, Sta07] show that the Hausdorff, constructive and computable dimensionsof automaton definable sets of infinite strings (regular ω -languages) are computable. In con-trast to this Ko [Ko98] derived examples of computable ω -languages with an incomputableHausdorff dimension.In this paper we derive examples of computable ω -languages of a simple structure whichhave not only incomputable Hausdorff dimension but also incomputable computable dimen-sion. To this end we use an iteration of finite trees which resembles the tree construction ofFurstenberg [Fur70] (see also [MSS18])Lutz [Lut03a, Lut03b] defines computable and constructive dimension via σ -(super)gales.Terwijn [Ter04, CST06] observed that this can also be done using Schnorr’s concept of com-bining martingales with (exponential) order functions [Sch71, Section 17]. For the com-putable ω -languages constructed in this paper we can show that Schnorr’s concept is insome details more precise than Lutz’s approach.1. Notation
In this section we introduce the notation used throughout the paper. By N = { , , , . . . } we denote the set of natural numbers, by Q the set of rational numbers, and R are the realnumbers.Let X be an alphabet of cardinality | X | ≥ . By X ∗ we denote the set of finite wordson X , including the empty word e , and X ω is the set of infinite strings ( ω -words) over X .Subsets of X ∗ will be referred to as languages and subsets of X ω as ω -languages .For w ∈ X ∗ and η ∈ X ∗ ∪ X ω let w · η be their concatenation . This concatenationproduct extends in an obvious way to subsets W ⊆ X ∗ and B ⊆ X ∗ ∪ X ω . We denote by Preprint submitted toLogical Methods in Computer Science c (cid:13)
L. Staiger CC (cid:13) Creative Commons
L. STAIGER | w | the length of the word w ∈ X ∗ and pref ( B ) is the set of all finite prefixes of strings in B ⊆ X ∗ ∪ X ω .It is sometimes convenient to regard X ω as Cantor space, that is, as the product spaceof the (discrete space) X . Here open sets in X ω are those of the form W · X ω with W ⊆ X ∗ . Closed are sets F ⊆ X ω which satisfy the condition F = { ξ : pref ( ξ ) ⊆ pref ( F ) } .For a computable domain D , such as N , Q or X ∗ , we refer to a function f : D → R as left-computable (or approximable from below ) provided the set { ( d, q ) : d ∈ D ∧ q ∈ Q ∧ q < f ( d ) } is computably enumerable. Accordingly, a function f : D → R is called right-computable (or approximable from above ) if the set { ( d, q ) : d ∈ D ∧ q ∈ Q ∧ q > f ( d ) } is computablyenumerable, and f is computable if f is right- and left-computable. If we refer to a function f : D → Q as computable we usually mean that it maps the domain D to the domain Q ,that is, it returns the exact value f ( d ) ∈ Q . If D = N we write f as a sequence ( q i ) i ∈ N .A real number α ∈ R is left-computable, right computable or computable provided theconstant function c α ( t ) = α is left-computable, right-computable or computable, respectively. α ∈ R is referred to as computably approximable if α = lim i →∞ q i for a computable sequence ( q i ) i ∈ N of rationals. It is well-known (see e.g. [ZW01]) that there are left-computable whichare not right-computable and vice versa, and that there are computably approximable realswhich are neither left-computable nor right-computable.The following approximation property is easily verified. Property 1.1.
Let ( q i ) i ∈ N be a computable family of rationals converging to α and let ( q ′ i ) i ∈ N , q ′ i > , be a computable family of rationals converging to . If α is not right-computable then there are infinitely many i ∈ N such that α − q i > q ′ i .For, otherwise, α as the limit of ( q i + q ′ i ) i ∈ N would be right-computable.2. Gales and Martingales
Hausdorff [Hau18] introduced a notion of dimension of a subset Y of a metric space whichis now known as its Hausdorff dimension , dim Y ; Falconer [Fal03] provides an overviewand introduction to this subject. In the case of the Cantor space X ω , Lutz [Lut03b] (seealso [DH10, Section 13.2]) has found an equivalent definition of Hausdorff dimension viageneralisations of martingales.Following Lutz a mapping d : X ∗ → [0 , ∞ ) will be called an σ -supergale provided ∀ w ( w ∈ X ∗ → | X | σ · d ( w ) ≥ X x ∈ X d ( wx )) . (2.1)A σ -supergale d is called an σ -gale if, for all w ∈ X ∗ , Eq. (2.1) is satisfied with equality.( Super ) Martingales are -(super)gales.From Eq. (2.1) one easily infers that if d, V : X ∗ → [0 , ∞ ) satisfy ∀ w ( w ∈ X ∗ → V ( w ) | X | (1 − σ ) ·| w | = d ( w )) (2.2)then d is a σ -(super)gale if and only if V is a (super)martingale. Thus (super)gales can beviewed as a combination of (super)martingales with exponential order functions in the senseof Schnorr [Sch71, Section 17] (see also [Ter04, CST06] or [DH10, Section 13.3]).Following Lutz [Lut03b] we define as follows. Definition 2.1.
Let F ⊆ X ω . Then α is the Hausdorff dimension dim F of F provided NCOMPUTABILITY OF DIMENSION 3 (1) for all σ > α there is a σ -supergale d such that ∀ ξ ( ξ ∈ F → lim sup w → ξ d ( w ) = ∞ ) , and (2) for all σ < α and all σ -supergales d it holds ∃ ξ ( ξ ∈ F ∧ lim sup w → ξ d ( w ) < ∞ ) .If the ω -language F ⊆ X ω is closed in Cantor space and satisfies a certain balancecondition Theorem 4 of [Sta89] shows that the calculation of its Hausdorff dimension canbe simplified. For the purposes of our investigations the following special case will suffice. Proposition 2.2.
Let F ⊆ X ω be non-empty and satisfy the conditions (1) F = { ξ : pref ( ξ ) ⊆ pref ( F ) } and (2) | pref ( F ) ∩ w · X k | = | pref ( F ) ∩ v · X k | for all k ∈ N and w, v ∈ pref ( F ) with | w | = | v | .Then dim F = lim inf n →∞ log | X | | pref ( F ) ∩ X n | n . Iterative Tree Construction
The aim of this section is, given a sequence of rationals ( q i ) i ∈ N , < q i < , to construct an ω -language F ⊆ X ω with Hausdorff dimension dim F = lim inf i →∞ q i satisfying the conditions(1) and (2) of Proposition 2.2.3.1. Preliminaries.
As a preparation we show how to find sequences of natural numbers ( k i ) i ∈ N and ( ℓ i ) i ∈ N with appropriate properties such that q i = k i /ℓ i . Lemma 3.1.
Let ( q i ) i ∈ N , < q i < , q i = q i +1 , be a family of positive rationals. Thenthere are families of natural numbers ( k i ) i ∈ N , ( ℓ i ) i ∈ N , ( κ i ) i ∈ N , ( p i ) i ∈ N and ( r i ) i ∈ N , such that q i = k i /ℓ i , q i +1 = r i · k i + κ i · ℓ i r i · ℓ i + p i · ℓ i where κ i = (cid:26) , if q i > q i +1 and p i , if q i < q i +1 . Moreover, for ≤ t ≤ p i · ℓ i we have q i ≥ r i · k i r i · ℓ i + t ≥ q i +1 , if q i > q i +1 and (3.1) q i ≤ r i · k i + tr i · ℓ i + t ≤ q i +1 , if q i < q i +1 . (3.2) Proof.
Let q i = k i /ℓ i and q i +1 = a/b · q i = a · k i b · ℓ i , with a, b ∈ N \ { } , a = b . Since > q i +1 we have b · ℓ i − a · k i = a · q i q i +1 · (1 − q i +1 ) · ℓ i > .Assume q i > q i +1 . Then b > a and the equation r i · k i + κ i · ℓ i r i · ℓ i + p i · ℓ i = a · k i b · ℓ i (3.3)has the solutions r i = a , and p i = ( b − a ) = a · ( q i q i +1 − and κ i = 0 .If q i < q i +1 then a > b and r i := b · ℓ i − a · k i = a · ( q i q i +1 · ℓ i − k i ) = a · q i · ( q i +1 − · ℓ i and p i = κ i := ( a − b ) · k i = a · q i · (1 − q i q i +1 ) · ℓ i are solutions of Eq. (3.3).In view of κ i = 0 Eq. (3.1) is obvious. Eq. (3.2) follows inductively from k +1 ℓ +1 ≥ kℓ whenever ≤ k < ℓ . Here lim sup w → ξ d ( w ) is an abbreviation for lim n →∞ sup { d ( w ) : w ∈ pref ( ξ ) ∧ | w | ≥ n } . L. STAIGER
If the family ( q i ) i ∈ N is a computable one then the families in Lemma 3.1 can be chosento be computable. In addition, the values ℓ i and ℓ i +1 /ℓ i can be made arbitrarily large.3.2. Tree construction.
The ω -language F will be the limit of the following sequence offinite trees T i . These trees have a property similar to the one in Proposition 2.2 (2) whichis referred to as spherical symmetry in [Fur70].We define the following auxiliary languages T i ⊆ X ℓ i and U i ⊆ X p i · ℓ i .Let T := X k · ℓ − k or T := 0 ℓ − k · X k and set T i +1 := T r i i · U i with U i := (cid:26) X p i · ℓ i , if q i +1 ≥ q i and { u i } , otherwise (3.4)where u i ∈ X p i is a fixed word. Then ℓ i +1 = ( r i + p i ) · ℓ i . Thus T i +1 consists of a concatenationof r i copies of T i plus an appendix U i of length p i · ℓ i . The values r i and p i are referred toas repetition or prolongation factors, respectively.By induction one proves | T i | = | X | q i · ℓ i . (3.5) Property 3.2.
The trees T i have the following properties. Let ℓ ≤ ℓ i .(1) Prefix property: pref ( T i +1 ) = S r i − j =0 T ji · pref ( T i ) ∪ T r i i · pref ( U i ) ,(2) Extension property: pref ( T i ) ∩ X ℓ = pref ( T i +1 ) ∩ X ℓ , and(3) Spherical symmetry: pref ( T i ) ∩ X ℓ = ( pref ( T i ) ∩ X ℓ − ) · X or | pref ( T i ) ∩ X ℓ | = | pref ( T i ) ∩ X ℓ − | . The infinite tree.
We define our ω -language F having the properties mentioned inProposition 2.2 as F := T i ∈ N T i · X ω where the family ( T i ) i ∈ N satisfies Eq. (3.4).Before we proceed to further properties of ( T i ) i ∈ N and F we mention a general property. Lemma 3.3.
Let T i ⊆ X ∗ , T i +1 ⊆ T i · X · X ∗ , T i ⊆ pref ( T i +1 ) and F := T i ∈ N T i · X ω .Then pref ( F ) = S i ∈ N pref ( T i ) .If, moreover, all T i are finite then F := { ξ : ξ ∈ X ω ∧ pref ( ξ ) ⊆ S i ∈ N pref ( T i ) } .Proof. In view of T i +1 ⊆ T i · X · X ∗ we have T i +1 · X ω ⊆ T i · X ω and also | w | ≥ i for w ∈ T i .If w ∈ pref ( F ) then w ∈ pref ( ξ ) where ξ ∈ F ⊆ T i · X ω for i > | w | . Consequently, w ∈ pref ( T i ) .Using the condition T i ⊆ pref ( T i +1 ) , by induction we obtain that for every w ∈ pref ( T i ) there is an infinite chain ( w j ) j ≥ i such that w j ∈ T j and w ⊑ w i ⊏ w i +1 ⊏ · · · . Thus thereis a ξ ∈ F with w ⊏ ξ .If the languages T i are finite F = T i ∈ N T i · X ω is closed in the product topology of thespace X ω which implies F := { ξ : ξ ∈ X ω ∧ pref ( ξ ) ⊆ pref ( F ) } .Lemma 3.3 shows that F := { ξ : ξ ∈ X ω ∧ pref ( ξ ) ⊆ S i ∈ N pref ( T i ) } for the family ( T i ) i ∈ N defined in Section 3.2.From the spherical symmetry of T i (see Property 3.2 (3)) the ω -language F = T i ∈ N T i · X ω inherits the following balance property of Proposition 2.2 (2). NCOMPUTABILITY OF DIMENSION 5
Lemma 3.4.
Let F = T i ∈ N T i · X ω where the T i are defined by Eq. (3.4). Then for all k ∈ N and w, v ∈ pref ( F ) with | w | = | v | we have | w · X k ∩ pref ( F ) | = | v · X k ∩ pref ( F ) | . Proof.
We proceed by induction on k . Let k = 1 . Then for all w, v ∈ pref ( F ) with | w | = | v | either pref ( F ) ∩ X | u | +1 = ( pref ( F ) ∩ X | u | ) · X or | pref ( F ) ∩ X | u | +1 | = | pref ( F ) ∩ X | u | | ( u ∈ { w, v } ).In the first case we have | w · X ∩ pref ( F ) | = | X | = | v · X ∩ pref ( F ) | and in the second | w · X ∩ pref ( F ) | = 1 = | v · X ∩ pref ( F ) | .Let the assertion be proved for k and all pairs u, u ′ ∈ pref ( F ) of the same length. Let w, v ∈ pref ( F ) with | w | = | v | and consider words w ′ , v ′ ∈ X k such that w · w ′ , v · v ′ ∈ pref ( F ) .Then from the spherical symmetry we obtain either pref ( F ) ∩ X | u | +1 = ( pref ( F ) ∩ X | u | ) · X or | pref ( F ) ∩ X | u | +1 | = | pref ( F ) ∩ X | u | | for u ∈ { w · w ′ , v · v ′ } and we proceed as above.Since, by our assumption |{ w ′ : | w ′ | = k ∧ w · w ′ ∈ pref ( F ) }| = |{ v ′ : | v ′ | = k ∧ v · v ′ ∈ pref ( F ) }| , the assertion follows.As a consequence of Lemmas 3.3, 3.4 and Proposition 2.2 we obtain the following. Corollary 3.5.
Let F = T i ∈ N T i · X ω where the T i are defined by Eq. (3.4). Then dim F =lim inf n →∞ log | X | | pref ( F ) ∩ X n | n . Next we investigate in more detail the structure function s F : N → N where s F ( ℓ ) := | pref ( F ) ∩ X ℓ | . First, Lemma 3.3 implies pref ( F ) ∩ X ℓ = pref ( T i ) ∩ X ℓ whenever ℓ ≤ ℓ i . (3.6)From Eqs. (3.4) and (3.5) and the properties of the tree family ( T i ) i ∈ N we obtain for theintervals ℓ i ≤ ℓ ≤ ℓ i +1 : Lemma 3.6.
Let F = T i ∈ N T i · X ω where the T i are defined by Eq. (3.4). Then the structurefunction s F : N → N satisfies the following relations. (1) In the interval [ j · ℓ i , ( j + 1) · ℓ i ] where j < r i : s F ( j · ℓ i + t ) = s F ( ℓ i ) j · s F ( t ) for ≤ t ≤ ℓ i (2) In the subinterval [ j · ℓ i + j ′ · ℓ i − , j · ℓ i + ( j ′ + 1) · ℓ i − ] where j ′ < r i − : s F ( j · ℓ i + j ′ · ℓ i − + t ) = s F ( ℓ i ) j · s F ( ℓ i − ) j ′ · s F ( t ) for ≤ t < ℓ i − . (3) In the interval [ r i · ℓ i , ℓ i +1 ] : s F ( r i · ℓ i + t ) = ( s F ( ℓ i ) r i , if | U i | = 1 and s F ( ℓ i ) r i · | X | t , if U i = X p i · ℓ i for ≤ t ≤ p i · ℓ i . L. STAIGER
This yields the following connection to the values q i . In order to connect our consider-ations to the application of Proposition 2.2 we consider the values of log | X | s F ( n ) n instead of s F ( n ) .From Eqs. (3.6) and (3.5) we obtain log | X | s F ( j · ℓ i ) j · ℓ i = q i . (3.7)Now we use the identities of Lemma 3.6 and Eqs. (3.1) and (3.2) to bound log | X | s F ( ℓ ) ℓ in therange ℓ i ≤ ℓ ≤ ℓ i +1 = r i · ℓ i + n i · ℓ i .For ℓ i ≤ ℓ < r i · ℓ i we have ℓ = j · ℓ i + j ′ · ℓ i − + t where ≤ t < ℓ i − , and Lemma 3.6 (1)and (2) yield log | X | s F ( ℓ ) ℓ ≥ j · ℓ i ℓ · q i + j ′ · ℓ i − ℓ · q i − ≥ j · ℓ i + j ′ · ℓ i − ℓ · min { q i − , q i } (3.8) ≥ (1 − ℓ i − ℓ i ) · min { q i − , q i } If r i · ℓ i ≤ ℓ ≤ ℓ i +1 , that is, for ℓ = r i · ℓ i + t where t ≤ ℓ i +1 − r i · ℓ i , following Eqs. (3.1) and(3.2), respectively, we have according to Lemma 3.6 (3) q i ≥ log | X | s F ( ℓ ) ℓ = log | X | s F ( r i · ℓ i ) r i · ℓ i + t ≥ q i +1 if q i > q i +1 (3.9) q i ≤ log | X | s F ( ℓ ) ℓ = log | X | s F ( r i · ℓ i ) + tr i · ℓ i + t ≤ q i +1 if q i < q i +1 (3.10)The considerations in Eqs. (3.7), (3.8), (3.9) and (3.10) show the following. Lemma 3.7.
If the sequence ( ℓ i ) i ∈ N is chosen in such a way that lim inf i →∞ ℓ i − ℓ i = 0 then lim inf ℓ →∞ log | X | s F ( ℓ ) ℓ = lim inf i →∞ q i . Proof.
In view of Eq. (3.7) the limit cannot exceed lim inf i →∞ q i .On the other hand, by Eqs. (3.8), (3.9) and (3.10), for ℓ i ≤ ℓ ≤ ℓ i +1 , the intermediatevalues satisfy log | X | s F ( ℓ ) ℓ ≥ (1 − ℓ i − ℓ i ) · min { q i − , q i , q i +1 } .3.4. Monotone families ( q i ) i ∈ N . If the sequence ( q i ) i ∈ N is monotone we can simplify theabove considerations of Eq. (3.8). Proposition 3.8.
Let the sequence ( q i ) i ∈ N be monotone and lim i →∞ q i = α . (1) If ( q i ) i ∈ N is decreasing and T = X k · ℓ − k then s F ( ℓ ) ≥ | X | α · ℓ , for all ℓ ∈ N . (2) If ( q i ) i ∈ N is increasing and T = 0 ℓ − k · X k then s F ( ℓ ) ≤ | X | α · ℓ , for all ℓ ∈ N .Proof. If ( q i ) i ∈ N is decreasing we start with T = X k · ℓ − k and have s F ( ℓ ) ≥ | X | q · ℓ ≥| X | α · ℓ for ℓ ≤ ℓ . Then we use Eqs. (3.6) and (3.4) and proceed by induction. s F ( j · ℓ i + t ) = s F ( j · ℓ i ) · s F ( t ) ≥ | X | q i · ℓ i · | X | α · t ≥ | X | α · ℓ for j < r i . In the range r i · ℓ i ≤ ℓ ≤ ℓ i +1 we have according to Eq. (3.9) s F ( ℓ ) ≥ | X | q i +1 · ℓ ≥ | X | α · ℓ . NCOMPUTABILITY OF DIMENSION 7 If ( q i ) i ∈ N is increasing we start with T = 0 ℓ − k · X k and have s F ( ℓ ) ≥ | X | q · ℓ ≤ | X | α · ℓ for ℓ ≤ ℓ . Again we use Eqs. (3.6) and (3.4) and proceed by induction. s F ( j · ℓ i + t ) = s F ( j · ℓ i ) · s F ( t ) ≤ | X | q i · ℓ i · | X | α · t ≤ | X | α · ℓ for j < r i . In the range r i · ℓ i ≤ ℓ ≤ ℓ i +1 we have according to Eq. (3.10) s F ( ℓ ) ≤ | X | q i +1 · ℓ ≤ | X | α · ℓ .4. Incomputable dimensions
Hausdorff dimension.
In this section we provide the announced examples. First wehave the following.
Lemma 4.1.
If the sequence ( q i ) i ∈ N of rationals < q i < , q i = q i +1 , is computablethen one can construct an ω -language F ⊆ X ω according to the tree construction such that pref ( F ) is a computable language.Proof. Construct from ( q i ) i ∈ N the numerator and denominator sequences ( k i ) i ∈ N and ( ℓ i ) i ∈ N and the corresponding sequences for the repetition and prolongation factors ( r i ) i ∈ N and ( p i ) i ∈ N . Then in view of Eq. (3.4) the assertion is obvious.Our lemma shows that the ω -language F ⊆ X ω has a very simple computable structure(compare with [Sta07, Section 4.2]).Next we show that the Hausdorff dimension of a computable ω -language F ⊆ X ω as inLemma 4.1 may be incomputable. Theorem 4.2.
If the sequence ( q i ) i ∈ N of rationals < q i < , q i = q i +1 , is computable and α = lim inf i →∞ q i then there is an ω -language F ⊆ X ω such that pref ( F ) is a computablelanguage and dim F = α .Proof. Construct from ( q i ) i ∈ N the numerator and denominator sequences ( k i ) i ∈ N and ( ℓ i ) i ∈ N such that lim inf i →∞ ℓ i ℓ i +1 = 0 . Then the assertion follows from Lemmas 3.7, 4.1 and Corol-lary 3.5.Theorem 3.4 of [Ko98] proves a similar result where the achieved Hausdorff dimension α is a computably approximable number. In [ZW01] it is shown that there are reals whichare not computably approximable of the form lim inf i →∞ q i where ( q i ) i ∈ N is a computablesequence.4.2. Computable dimension.
If we require the supergales in Definition 2.1 to be com-putable mappings we obtain the definition of computable dimension dim comp F of [Hit05,Lut03b]. In view of Eq. (2.2) we may, as in Section 13.15 of [DH10], define the computabledimension of an ω -language E ⊆ X ω via martingales. Definition 4.3.
Let F ⊆ X ω . Then α is the computable dimension of F provided(1) for all σ > α there is a computable martingale V such that ∀ ξ ( ξ ∈ F → lim sup w → ξ V ( w ) | X | (1 − σ ) ·| w | = ∞ ) , and(2) for all σ < α and all computable martingales V it holds ∃ ξ ( ξ ∈ F ∧ lim sup w → ξ V ( w ) | X | (1 − σ ) ·| w | < ∞ ) . L. STAIGER
The inequality dim F ≤ dim comp F is immediate.We associate with every non-empty ω -language E ⊆ X ω a martingale V E in the followingway. Definition 4.4. V E ( e ) := 1 V E ( wx ) := ( | X || pref ( E ) ∩ w · X | · V E ( w ) , if x ∈ X and wx ∈ pref ( E ) , and , otherwise.In view of the spherical symmetry, for F defined as in Section 3.3, we obtain V F ( w ) = | X | | w | /s F ( | w | ) , if w ∈ pref ( F ) . (4.1)Moreover, if pref ( F ) is computable then s F and V F are computable mappings. Theorem 4.5.
If the sequence ( q i ) i ∈ N of rationals < q i < , q i = q i +1 , is computable and α = lim inf i →∞ q i then there is an ω -language F ⊆ X ω such that pref ( F ) is a computablelanguage and dim F = dim comp F = α .Proof. We use the ω -language F defined in the proof of Theorem 4.2 and the associatedcomputable martingale V F .Let σ > α = lim inf i →∞ q i . Then ( σ − q i ) > ( σ − α ) / > for infinitely many i ∈ N .Since s F ( ℓ i ) = | X | q i · ℓ i (see Eq. (3.7)), we have V F ( w ) / | X | (1 − σ ) ·| w | = | X | ( σ − q i ) ≥ | X | ( σ − α ) / for w ∈ pref ( F ) ∩ X ℓ i . This shows lim sup w → ξ V F ( w ) / | X | (1 − σ ) ·| w | = ∞ for all ξ ∈ F , thatis, dim comp F ≤ α .The other inequality follows from dim F ≤ dim comp F and Theorem 4.2.In certain cases we can achieve even the borderline value lim sup w → ξ V F ( w ) | X | (1 − dim F ) ·| w | = lim sup n →∞ | X | dim F · n s F ( n ) = ∞ for all ξ ∈ F . (4.2)
Theorem 4.6.
Let ( q i ) i ∈ N , < q i < , q i = q i +1 , be a computable sequence of rationals with lim inf i →∞ q i = α . If α is not right-computable then there is an ω -language F ⊆ X ω suchthat pref ( F ) is a computable language, dim F = dim comp F = α and Eq. (4.2) is satisfied.Proof. We construct F as in the proof of Theorem 4.2 requiring additionally that ℓ i ≥ i .Then pref ( F ) is computable and dim F = dim comp F = α . In view of Property 1.1 thereare infinitely many i ∈ N with α − i > q i and, consequently, s F ( ℓ i ) = | X | q i · ℓ i ≤ | X | α · ℓ i − ℓ i /i .This shows lim sup n →∞ | X | α · n s F ( n ) ≥ lim sup i →∞ | X | ℓ i /i = ∞ .4.3. Comparison of gales and martingales.
In this final part we compare the precisionwith which (super)gales and martingales achieve the value of computable dimension of asubset E ⊆ X ω . In Theorem 4.6 we have seen that there are ω -languages F ⊆ X ω suchthat dim comp F = α and lim sup w → ξ V F ( w ) / | X | (1 − α ) ·| w | = ∞ for all ξ ∈ F , that is, thecomputable martingale V F “matches” exactly the value of the computable dimension of F .The following theorem shows that this is, in some cases, not possible for supergales.First, observe that, for σ ′ ≥ σ any σ -supergale d : X ∗ → [0 , ∞ ) is also a σ ′ -supergale.Thus computable σ -supergales exist for all σ ∈ [0 , . NCOMPUTABILITY OF DIMENSION 9
We define the cut point χ d of a supergale d as the smallest value σ for which d can bean σ -supergale. χ d := inf n σ : ∀ w (cid:16) | X | σ · d ( w ) ≥ X x ∈ X d ( wx ) (cid:17)o . (4.3)If d is a computable mapping then χ d as sup { q : q ∈ Q ∧ ∃ w ( | X | q · d ( w ) < P x ∈ X d ( wx )) } is a left-computable real number. For computable σ -gales d the cut point χ d coincides with σ and is necessarily a computable real. Theorem 4.7.
Let ( q i ) i ∈ N , < q i < , q i = q i +1 , be a computable sequence of rationals with lim inf i →∞ q i = α . If α is neither left- nor right-computable then there is an ω -language F ⊆ X ω such that pref ( F ) is a computable language, α = dim F = dim comp F , Eq. (4.2) issatisfied but there is no computable α -supergale with lim sup w → ξ d ( w ) = ∞ for all ξ ∈ F .Proof. In view of the preceding Theorems 4.2 and 4.6 it suffices to show that under theadditional assumption that α is not left-computable no computable α -supergale satisfies lim sup w → ξ d ( w ) = ∞ for all ξ ∈ F .Assume the contrary. Since α is not left-computable, the cut point χ d of the computable α -supergale d cannot coincide with α . Hence α > χ d , and we have some rational number q, α > q > χ d . Consequently, d is a q -supergale with lim sup w → ξ d ( w ) = ∞ for all ξ ∈ F .This contradicts q < α = dim comp F Since there are computably approximable reals which are neither right- not left-computableTheorem 4.7 shows that in some cases Schnorr’s [Sch71] combination of martingales with(exponential) order functions can be more precise than Lutz’s approach via supergales.5.
Concluding remark
As the constructive dimension of subsets of X ω is sandwiched between the computable andthe Hausdorff dimension ([Lut03a, Lut03b, Hit05]) the result of Theorem 4.5 holds likewisefor constructive dimension, too. Acknowledgement
I wish to thank the anonymous reviewers for their suggestions for improving the presentationof this article.
References [CST06] Cristian S. Calude, Ludwig Staiger, and Sebastiaan A. Terwijn. On partial randomness.
Ann. PureAppl. Logic , 138(1-3):20–30, 2006.[DH10] Rodney G. Downey and Denis R. Hirschfeldt.
Algorithmic Randomness and Complexity . Theoryand Applications of Computability. Springer-Verlag, New York, 2010.[Fal03] Kenneth Falconer.
Fractal geometry. Mathematical foundations and applications . John Wiley &Sons, Inc., Hoboken, N.J., second edition, 2003.[Fur70] Harry Furstenberg. Intersections of Cantor sets and transversality of semigroups. In Robert C.Gunning, editor,
Problems in Analysis , pages 41–59. Princeton University Press, Princeton, N.J.,1970.[Hau18] Felix Hausdorff. Dimension und äußeres Maß.
Math. Ann. , 79(1-2):157–179, 1918.[Hit05] John M. Hitchcock. Correspondence principles for effective dimensions.
Theory Comput. Syst. ,38(5):559–571, 2005. [Ko98] Ker-I. Ko. On the computability of fractal dimensions and Hausdorff measure.
Ann. Pure Appl.Logic , 93(1-3):195–216, 1998.[Lut03a] Jack H. Lutz. Dimension in complexity classes.
SIAM J. Comput. , 32(5):1236–1259, 2003.[Lut03b] Jack H. Lutz. The dimensions of individual strings and sequences.
Inform. and Comput. , 187(1):49–79, 2003.[MSS18] Birzhan Moldagaliyev, Ludwig Staiger, and Frank Stephan. On the values for factor complexity. InCezar Câmpeanu, editor,
Implementation and Application of Automata - 23rd International Con-ference, CIAA 2018, Charlottetown, PE, Canada, July 30 - August 2, 2018, Proceedings , volume10977 of
Lecture Notes in Computer Science , pages 274–285. Springer, 2018.[Sch71] Claus-Peter Schnorr.
Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung derWahrscheinlichkeitstheorie . Lecture Notes in Mathematics, Vol. 218. Springer-Verlag, Berlin, 1971.[Sta89] Ludwig Staiger. Combinatorial properties of the Hausdorff dimension.
J. Statist. Plann. Inference ,23(1):95–100, 1989.[Sta93] Ludwig Staiger. Kolmogorov complexity and Hausdorff dimension.
Inform. and Comput. ,103(2):159–194, 1993.[Sta07] Ludwig Staiger. The Kolmogorov complexity of infinite words.
Theoret. Comput. Sci. , 383(2-3):187–199, 2007.[Ter04] Sebastiaan A. Terwijn. Complexity and randomness.
Rend. Semin. Mat., Torino , 62(1):1–37, 2004.[ZW01] Xizhong Zheng and Klaus Weihrauch. The arithmetical hierarchy of real numbers.
MLQ Math. Log.Q. , 47(1):51–65, 2001.
This work is licensed under the Creative Commons Attribution License. To view a copy of thislicense, visit https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/