L. Staiger. Kolmogorov complexity and Hausdorff dimension. Inf. Comput., 103(2):159--194, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....fractal sets from sets of a rather smooth geometrical nature. In the context of computability, originating from applications of Hausdorff dimension in information theory (see for example [4] a close connection between Hausdorff dimension and Kolmogorov complexity was established [6, 16, 17, 18, 20, 21], and the notion of effective dimension was introduced [14] Recently, LUTZ [13] has extended the theory of effective dimension to complexity theory by introducing resource bounded dimension. Like for classical Hausdorff dimension, LUTZ s approach yields a generalization of resource bounded ....

L. Staiger. Kolmogorov complexity and Hausdorff dimension. Inf. Comput., 103(2):159--194, 1993.

....fractal sets from sets of a rather smooth geometrical nature. In the context of computability, originating from applications of Hausdorff dimension in information theory (see for example [4] a close connection between Hausdorff dimension and Kolmogorov complexity was established [6, 16, 17, 18, 20, 21], and the notion of effective dimension was introduced [14] Recently, LUTZ [13] has extended the theory of effective dimension to complexity theory by introducing resource bounded dimension. Like for classical Hausdorff dimension, LUTZ s approach yields a generalization of resource bounded ....

L. Staiger. Kolmogorov complexity and Hausdorff dimension. Inf. Comput., 103(2):159--194, 1993.

....of constructive strong dimension and upper Kolmogorov complexity #(# ) The aim of this note is to show that, in fact, this coincidence holds for arbitrary sets of infinite strings, F . Here #(F) sup #(# ) F and #(F) sup #(# ) F . This elucidates the relations between the papers [CH, R2, R3, S1, S2] and the subject matter of [AH, L1, L2, L3] in a more precise manner than the mere remark in [L2] that Moreover, Ryabko, Staiger, and Cai and Hartmanis have all proven results establishing quantitative relationships between Hausdorff dimension and Kolmogorov complexity. As a consequence, several ....

....= inf s (w) 0) 11) Substituting, in the proof of Lemma 4, lim sup by lim inf and simultaneously infinitely often by almost all yields a proof of its counterpart. 1. liminf w## d 2. #(# ) s implies lim inf w## d (w) Here, using the notation of [S1], we refer to #(# ) limsup = limsup (12) as the upper Kolmogorov complexity, #(# ) of an # word # . If we set #(F) sup #(# ) # F we obtain likewise the analogue of Theorem 5 for strong constructive dimension and upper Kolmogorov complexity. cDim(F) #(F) ....

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L. Staiger, Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 102 (1993), 159 -- 194.

....in [10] Here, we show how to generalize the notion of entropy (of a language) in order to obtain new formulae to determine the Hausdorff dimension of fractal sets (also in Euclidean spaces) especially defined via regular (w )languages. By doing this, we can sharpen and generalize earlier results [1, 14, 15, 26, 38] in two ways: firstly, we treat the case where the underlying basic iterated function system contains non contractive mappings, and secondly, we obtain results valid for non regular languages as well. A preliminary version appeared in: Mathematical Foundations of Computer Science (L. Brim, J. ....

....of entropy. Moreover, we investigate special metrizations (induced by valuations) of spaces of w words; Hausdorff measure and dimension within these spaces are directly related to those entities within Euclidean spaces. Further material is contained in other works of the authors of this paper [12, 13, 14, 15, 27, 36, 37, 38, 39]. A preliminary version of this paper was presented at MFCS 98 [18] Our paper is structured as follows. In the Section 2, we introduce the notion of valuation b and the derived concept of b entropy which is central to the whole of this paper. Section 3 shows how one can compute the b entropy for ....

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L. Staiger. Kolmogorov complexity and Hausdorff dimension. Inf. & Comp., 103:159--194, 1993.

....F 1 of Example 1 which has a size of only. Consequently, we follow an different line, mentioning that several papers investigated the relationship between the Kolmogorov complexity of infinite strings and size measures known from information theory and fractal geometry. It turned out in [Ry86] [St93] and [St98] that the Hausdorff dimension of subsets F ae is closely related to sup k(x) Definition 3 The Hausdorff dimension of a set F f0;1g , dimF , is the smallest real number a 0 such that for all g a it holds 8e 0 : 9W f0;1g : F W Delta f0;1g w2W Gammag e ....

....5. For the converse inequality, observe that Theorem 7 and Lemma 2 prove that g k(x) whenever g dimF , x 2 F and F f0;1g is S 2 definable. Thus, dimF sup x2F k(x) o Concluding Remark Our Theorems 7 and 8 in connection with previous results of Ryabko ( Ry86, Ry93] and this author ([St93, St98]) give evidence that there is a strong coincidence between the concepts of Kolmogorov complexity, gambling strategies and Hausdorff dimension for a class of recursive (computable) sets of infinite zeroone sequences. The results of the last section show a borderline in the Arithmetical hierarchy ....

L. Staiger, Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 102 (1993), No. 2, 159 -- 194.

....DT(ran; B( Pi 1 ) B( Sigma 1 ) DT(inf ; B( Sigma 2 ) B( Pi 2 ) Figure 2 shows the relations between the six classes of Theorem 2 and Corollary 1, the classes Pi 3 , P and its related subclass S : fF X : F is closed pref (F ) is recursively enumerableg : Example 1. 15 of [St93] shows that S is incomparable to the classes Pi 1 , Sigma 1 and Sigma 2 . Thus, the inclusion properties presented in Figure 2 are proper and other inclusions than the ones represented do not exist. 6.2 Composition and decomposition theorems In this section we analyze the relation between ....

L. Staiger, Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 103 (1993) 2, 159--194.

....numbers. 7 Corollary 9 No Liouville number is a random real. Though Liouville numbers are not random, we show that the upper limit of complexity reaches its maximum value k#a#=1 also for certain Liouville numbers a. We consider the following set constructed similar to the one in Example 3. 18 of [St93]. Example 10 Define F : X r # i2IN X 2i##2i# r # 0 #2i 1###2i 1# : It is interesting to note that the set of finite prefixes of F , A#F# : fw : w 2 X # r 9x#x 2 F w x#g, is recursive. If we consider w words b = 0 # i2IN w i # 0 #2i 1###2i 1# where jw i j = 2i # #2i# andK r ....

....it is evident that Hausdorff Dimension is monotone with respect to set inclusion and that dimfxg = 0. We mention still that Hausdorff Dimension is also countably stable. dim # i2IN F i = sup i2IN dimF i (7) For further properties of the Hausdorff dimension see, e.g. Fa90] In the papers [St93, St98] several connections between Hausdorff dimension and Kolmogorov complexity are derived. We need here the following one proofs of which can be found in [Ry86] or [St93] Lemma 11 For every F # X w r the following bound is true. dimF # supfk#x# : x 2 Fg Now we obtain Theorem 2.4 in [Ox71] ....

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L. Staiger, Kolmogorov complexity and Hausdorff dimension. Inform. and Comput. 103 (1993) 2, 159 -- 194.

....Corollary 9 No Liouville number is a random real. Though Liouville numbers are not random, we show that the upper limit of complexity reaches its maximum value k(a) 1 also for certain Liouville numbers a. We consider the following set constructed in a similar way as the one in Example 3. 18 of [St93]. The Kolmogorov Complexity of Liouville Numbers 11 Example 10 Define F : X r Delta i2IN X 2i Delta(2i) r Delta 0 (2i 1) Delta(2i 1) It is interesting to note that the set of finite prefixes of F , A(F) fw : w 2 X r 9x(x 2 F w x)g, is recursive. If we consider w words b = ....

....of Liouville numbers First we consider the Hausdorff dimension of the set of Liouville numbers, L [0; 1] It was mentioned in [MS94] that the Hausdorff dimension of a subset M [0; 1] coincides with the one of fx : x 2 X w r n r (x) 2 Mg. The latter can be defined as follows (see, e.g. [St93, St98]) 12 L. Staiger Definition 3 The Hausdorff dimension of a set F X w r , dimF , is the smallest real number a 0 such that for all g a it holds 8e(e 0 9W (W X r F W Delta X w r w2W Gamma r Gammag Delta jwj e) From the definition it is evident that Hausdorff ....

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L. Staiger, Kolmogorov complexity and Hausdorff dimension. Inform. and Comput. 103 (1993) 2, 159 -- 194.

....to generalize the notion of entropy (of a language) in order to obtain new formulae to determine the Hausdorff dimension of fractal sets (especially in Euclidean spaces) especially defined via regular languages. In this way, we can sharpen and generalize theorems formulated in earlier papers [1,2,10,11,19,27]. 1 Introduction Fractal geometry is now a budding branch of mathematics with a variety of possible applications [7,17] One of the main problems encountered in that field is the determination of the Hausdorff dimension of fractals. We have shown earlier that, for certain types of fractals ....

....that fi(w) 1 for w 2 L and nevertheless fi s (L) 1 for all s 2 [0; 1) In the sequel, however, we are not interested in such pathological cases. If fi(a) 1 for any a 2 Sigma n , then there is a finite change over point ff of the function fi s (L) for any L Sigma n . It was shown in [20,26,27] that the entropy of languages introduced by Chomsky and Miller (cf. 15] is a useful tool for the calculation of the Hausdorff dimension of certain subsets of the Cantor space Sigma n or of the Euclidean space IR d . Here, we will see the usefulness of the generalized notion of ....

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L. Staiger. Kolmogorov complexity and Hausdorff dimension. Inf. Comp., 103:159-- 194, 1993.

....of Example 1 which has a size of 1 2 only. Consequently, we follow an different line, mentioning that several papers investigated the relationship between the Kolmogorov complexity of infinite strings and size measures known from information theory and fractal geometry. It turned out in [Ry86] [St93] and [St98] that the Hausdorff dimension of subsets F ae f0;1g w is closely related to sup x2F k(x) Definition 3 The Hausdorff dimension of a set F f0;1g w , dimF , is the smallest real number a 0 such that for all g a it holds 8e 0 : 9W f0;1g : F W Delta f0;1g w w2W ....

....5. For the converse inequality, observe that Theorem 7 and Lemma 2 prove that g k(x) whenever g dimF , x 2 F and F f0;1g w is S 2 definable. Thus, dimF sup x2F k(x) o Concluding Remark Our Theorems 7 and 8 in connection with previous results of Ryabko ( Ry86, Ry93] and this author ([St93, St98]) give evidence that there is a strong coincidence between the concepts of Kolmogorov complexity, gambling strategies and Hausdorff dimension for a class of recursive (computable) sets of infinite zeroone sequences. The results of the last section show a borderline in the Arithmetical hierarchy ....

L. Staiger, Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 102 (1993), No. 2, 159 -- 194.

.... Then as usual (cf. Ed90] Fa90] the Hausdorff dimension dim F is given by dim F : sup fff : L ff (F ) 1g = inffff : L ff (F ) 0g : Thus the Hausdorff dimension assigns to each subset F X a real number ff = dim F between 0 and 1 which defines in some sense a size of F (cf. Fa90] St93] MS94] For every w 2 X the ball w DeltaX is a disjoint union of the balls wx DeltaX (x 2 X) Thus (w Delta X ) P x2X (wx Delta X ) for every measure on X . As 3 In X and X ZZ every complement of a meager set (a comeager set) is of second Baire ....

....one counter automaton. The language E is a G ffi set dense in the space fa; bg , hence of second Baire category (cf. Ku66] but it has (E) 0, for the (equidistribution) measure defined by = w Delta fa; bg ) 2 Gammajwj (cf. St80] and dim E dimfa; bg (cf. St93, Example 6.3] ut As it was announced in the introduction we are going to prove our Theorem 3 as a special case of its relativized version presented in Theorem 4 below. This gives further evidence that the above mentioned notions density, measure or dimension are in some sense strongly ....

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L. Staiger, Kolmogorov Complexity and Hausdorff Dimension, Inform. and Comput. Vol. 102 (1993), No. 2, 159--194.

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