structured/0704.2386.structured.json

{"paper_meta":{"paper_id":"arxiv:0704.2386","title":"0704.2386","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0704.2386v1 [cs.CC] 18 Apr 2007\nBounded Pushdown dimension vs Lempel Ziv\ninformation density\nPilar Albert, Elvira Mayordomo, and Philippe Moser ∗\nAbstract\nIn this paper we introduce a variant of pushdown dimension called bounded push-\ndown (BPD) dimension, that measures the density of information contained in a\nsequence, relative to a BPD automata, i.e. a finite state machine equipped with an\nextra infinite memory stack, with the additional requirement that every input symbol\nonly allows a bounded number of stack movements. BPD automata are a natural\nreal-time restriction of pushdown automata. We show that BPD dimension is a ro-\nbust notion by giving an equivalent characterization of BPD dimension in terms of\nBPD compressors. We then study the relationships between BPD compression, and\nthe standard Lempel-Ziv (LZ) compression algorithm, and show that in contrast to\nthe finite-state compressor case, LZ is not universal for bounded pushdown compres-\nsors in a strong sense: we construct a sequence that LZ fails to compress significantly,\nbut that is compressed by at least a factor 2 by a BPD compressor. As a corollary\nwe obtain a strong separation between finite-state and BPD dimension.\nKeywords\nInformation lossless compressors, finite state (bounded pushdown) dimension, Lempel-Ziv\ncompression algorithm.\n1\nIntroduction\nEffective versions of fractal dimension have been developed since 2000 [9, 10] and used\nfor the quantitative study of complexity classes, information theory and data compression,\nand back in fractal geometry (see recent surveys in [11, 7, 12]). Here we are interested\nin information theory and data compression, where it is known that for several different\n∗Dept. de Inform ́atica e Ingenier ́ıa de Sistemas , Universidad de Zaragoza. Edificio Ada Byron, Mar ́ıa\nde Luna 1 - E-50018 Zaragoza (Spain). Email: {mpalbert, elvira}@unizar.es and mosersan@gmail.com.\nResearch supported in part by Spanish Government MEC Project TIN 2005-08832-C03-02, by Arag ́on\nGovernment Dept. Ciencia, Tecnolog ́ıa y Universidad, subvenci ́on destinada a la formaci ́on de personal\ninvestigador-B068/2006 and by Spanish Government MEC Program Juan de la Cierva.\n1\n\nbounds on the computing power, effective dimensions capture what can be considered the\ninherent information content of a sequence in the corresponding setting [12]. In the today\nrealistic context of massive data streams we need to consider very low resource-bounds,\nsuch as finite memory or finite-time per input symbol.\nThe finite state dimension of an infinite sequence [3], is a measure of the amount of ran-\ndomness contained in the sequence within a finite-memory setting. It is a robust quantity,\nthat has been shown to admit several characterizations in terms of finite-state information\nlossless compressors (introduced by Huffman [8], [3]), finite-state decompressors [4, 13],\nfinite-state predictors in the logloss model [1], and block entropy rates [2]. It is an effec-\ntivization of the general notion of Hausdorffdimension at the level of finite-state machines.\nInformally, the finite state dimension assigns every sequence a number s ∈[0, 1], that char-\nacterizes the randomness density in the sequence (or equivalently its compression ratio),\nwhere the larger the dimension the more randomness is contained in the sequence.\nIn a recent line of research, Doty and Nichols [5] investigated a variant of finite-state\ndimension, where the finite state machine comes equipped with an infinite memory stack\nand is called a pushdown automata, yielding the notion of pushdown dimension. Hence\nthe pushdown dimension of a sequence, is a measure of the density of randomness in the\nsequence as viewed by a pushdown automata. Since a finite-state automata is a special\ncase of a pushdown automata, the pushdown dimension of a sequence is a lower bound\nfor its finite state dimension. It was shown in [5], that there are sequences for which the\npushdown dimension is at most half its finite state dimension, hence yielding a strong\nseparation between the two notions. Unfortunately the notion of pushdown dimension is\nnot known to enjoy any of the equivalent characterizations that finite state dimension does.\nMoreover, the computation time per input symbol can be unbounded, which rules out this\nmodel for many real-time applications.\nIn this paper we introduce a variant of pushdown dimension called bounded pushdown\n(BPD) dimension: Whereas pushdown automata can choose not to read their input and\nonly work with their stack for as many steps as they wish (each such step is called a\nlambda transition), we add the additional real-time constraint that the sequences of lambda\ntransitions are bounded, i.e. we only allow a bounded number of stack movements per each\ninput symbol.\nWe define the notion of bounded pushdown dimension as the natural effectivitation of\nHausdorffdimension via Lutz’s gale characterization [9]. We provide evidence that bounded\npushdown dimension is a robust notion by giving a compression characterization; i.e. we\nintroduce BPD information-lossless compressors and show that the best compression ratio\nachievable on a sequence by BPD compressors is exactly its BPD dimension.\nIn the context of compression, we study the relationship between BPD compression and\nthe standard Lempel-Ziv (LZ) compression algorithm [14]. It is well known that the LZ\ncompression ratio of any sequence is a lower bound for its finite state compressibility [14],\ni.e. LZ compresses every sequence at least as well as any finite-state information lossless\ncompressor. We show that this fails dramatically in the context of BPD compressors, by\nconstructing a sequence that LZ fails to compress significantly, but is compressed by at least\na factor 2 by a BPD compressor, thus yielding a strong separation between LZ and BPD\n2\n\ndimension. This implies that we have the same separation between LZ and (unbounded)\npushdown dimension, and between finite state dimension [3] and BPD dimension.\nSection 2 contains the preliminaries, section 3 presents BPD dimension and its basic\nproperties, section 4 proves the equivalence of BPD compression and dimension and section\n5 contains the separation of BPD compression from Lempel Ziv compression. The proofs\nare postponed to the appendix.\n2\nPreliminaries\nWe write Z for the set of all integers, N for the set of all nonnegative integers and Z+ for\nthe set of all positive integers. Let Σ be a finite alphabet, with |Σ| ≥2. Σ∗denotes the\nset of finite strings, and Σ∞the set of infinite sequences. We write |w| for the length of\na string w in Σ∗. The empty string is denoted λ. For S ∈Σ∞and i, j ∈N, we write\nS[i..j] for the string consisting of the ith through jth symbols of S, with the convention\nthat S[i..j] = λ if i > j, and S[0] is the leftmost symbol of S. We write S[i] for S[i..i] (the\nith symbol of S). For w ∈Σ∗and S ∈Σ∞, we write w ⊑S if w is a prefix of S, i.e., if\nw = S[0..|w| −1]. All logarithms are taken in base |Σ|.\n3\nBounded Pushdown Dimension\nIn this section we first recall Lutz’s characterization of Hasudorffdimension in terms of\ngales that can be used to effectivize dimension. Then we introduce Bounded Pushdown\ndimension based on the concept of BPD gamblers and give its basic properties.\nDefinition. [9] Let s ∈[0, ∞).\n1. An s-gale is a function d : Σ∗→[0, ∞) that satisfies the condition\nd(w) =\nP\na∈Σ\nd(wa)\n|Σ|s\n(1)\nfor all w ∈Σ∗.\n2. A martingale is a 1-gale.\nIntuitively, an s-gale is a strategy for betting on the successive symbols of a sequence\nS ∈Σ∞. For each prefix w of S, d(w) is the capital (amount of money) that d has after\nhaving bet on S[0..|w| −1].\nWhen betting on the next symbol b of a prefix wb of S,\nassuming symbol b is equally likely to be any value in Σ, equation (1) guarantees that the\nexpected value of d(wb) is |Σ|−1 P\na∈Σ\nd(wa) = |Σ|s−1d(w). If s = 1, this expected value is\nexactly d(w), so the payoffs are “fair”.\nDefinition.\nLet d be an s-gale, where s ∈[0, ∞).\n1. We say that d succeeds on a sequence S ∈Σ∞if\n3\n\nlim sup\nn→∞d(S[0..n −1]) = ∞.\n2. The success set of d is\nS∞[d] = {S ∈Σ∞| d succeeds on S}.\nObservation 3.1 Let s, s′ ∈[0, ∞). For every s-gale d, the function d′ : Σ∗→[0, ∞)\ndefined by d′(w) = |Σ|(s′−s)|w|d(w) is an s′-gale. Moreover, if s ≤s′, then S∞[d] ⊆S∞[d′].\nLutz characterized Hausdorffdimension using gales as follows.\nTheorem 3.2 [9] Given a set X ⊆Σ∞, if dimH(X) is the Hausdorffdimension of X [6],\nthen\ndimH(X) = inf{s | there is an s −gale d such that X ⊆S∞[d]}\nThe idea for a Bounded Pushdown dimension is to consider only s-gales that are com-\nputable by a Bounded Pushdown (BPD) gambler. Bounded Pushdown gamblers are finite-\nstate gamblers [3] with an extra memory stack, that is used both by the transition and\nbetting functions. Additionally, BPDG’s are allowed to delay reading the next character\nof the input –they read λ from the input– in order to alter the content of their stack, but\nthey cannot do this more than a constant number of times per each input symbol. During\nsuch λ-transitions, the gambler’s capital remains unchanged.\nThe betting function returns a probability measure over the input alphabet.\nDefinition.\nLet Σ be a finite alphabet. ∆Q(Σ) is the set of all rational-valued probability\nmeasures over Σ, i.e., all functions π : Σ −→[0, 1] ∩Q such that P\na∈Σ\nπ(a) = 1.\nWe are ready to define BPD gamblers.\nDefinition.\nA bounded pushdown gambler (BPDG) is an 8-tuple G =(Q, Σ, Γ, δ, β, q0,\nz0, c) where\n• Q is a finite set of states,\n• Σ is the finite input alphabet,\n• Γ is the finite stack alphabet,\n• δ : Q×(Σ∪{λ})×Γ →Q×Γ∗is the transition function (for simplicity we use the nota-\ntion δ(q, b, a) = ⊥when undefined; and we write δ(q, b, a) = (δQ(q, b, a), δΓ∗(q, b, a))),\n• β : Q × Γ →∆Q(Σ) is the betting function,\n• q0 ∈Q is the start state,\n• z0 ∈Γ is the start stack symbol,\n• c ∈N is a constant such that the number of λ-transitions per input symbol is at most\nc,\n4\n\nwith the two additional restrictions:\n1. for each q ∈Q and a ∈Γ at least one of the following holds\n• δ(q, λ, a) =⊥\n• δ(q, b, a) =⊥for all b ∈Σ\n2. for every q ∈Q, b ∈Σ ∪{λ}, either δ(q, b, z0) =⊥, or δ(q, b, z0) = (q′, vz0), where q′\n∈Q and v ∈Γ∗.\nWe denote with BPDG the set of all bounded pushdown gamblers.\nThe transition function δ outputs a new state and a string z′ ∈Γ∗.\nInformally,\nδ(q, w, a) = (q′, z′) means that in state q, reading input w, and popping symbol a from the\nstack, δ enters state q′ and pushes z′ to the stack.\nNote that w can be λ (ie, a λ-transition: the input is ignored and δ only computes with\nthe stack) but this only happens at most c times per input symbol. Any pair (state, stack\nsymbol) can either be a λ-transition pair or a non λ-transition pair exclusively, because\nthe first additional restriction enforces determinism.\nMoreover, since z0 represents the bottom of the stack, we restrict δ so that z0 cannot\nbe removed from the bottom by the second additional restriction.\nWe can extend δ in the usual way to\nδ∗: Q × (Σ ∪{λ}) × Γ+ →Q × Γ∗,\nwhere for all q ∈Q, a ∈Γ, v ∈Γ∗, and b ∈Σ ∪{λ}\nδ∗(q, b, av) =\n (δQ(q, b, a), δΓ∗(q, b, a)v)\nif δ(q, b, a) ̸=⊥,\n⊥\notherwise.\nWe denote δ∗by δ.\nFor each i ≥2, we will use the notation\nδi(q, λ, v) = δ(δi−1\nQ (q, λ, v), λ, δi−1\nΓ∗(q, λ, v))\nwhere\nδ1(q, λ, v) = δ(q, λ, v).\nSince δ is c-bounded we have that for any q ∈Q, v ∈Γ∗,\nδc+1(q, λ, v) = ⊥\nWe also consider the extended transition function\nδ∗∗: Q × Σ∗× Γ+ →Q × Γ∗,\ndefined for all q ∈Q, a ∈Γ, v ∈Γ∗, w ∈Σ∗, and b ∈Σ by\nδ∗∗(q, λ, av) = (q, av)\n5\n\nδ∗∗(q, wb, av) = δ(δi\nQ(eq, λ, eaev), b, δi\nΓ∗(eq, λ, eaev))\nif δ∗∗(q, w, av) = (eq, eaev), δi(eq, λ, eaev) ̸=⊥and δi+1(eq, λ, eaev) =⊥, i ≤c.\nThat is, λ-transitions are inside the definition of δ∗∗(q, b, av), for b ∈Σ. Notice that\nδ∗∗is not defined on an empty stack string, therefore av needs to be long enough in order\nthat δ∗∗(q, b, av) ̸=⊥.\nWe denote δ∗∗by δ, and δ(q0, w, z0) by δ(w). We write δ = (δQ, δΓ∗) for simplicity.\nWe also consider the usual extension of β\nβ∗: Q × Γ+ →∆Q(Σ),\ndefined for all q ∈Q, a ∈Γ, and v ∈Γ∗by\nβ∗(q, av) = β(q, a),\nand denote β∗by β.\nWe use BPDG to compute martingales. Intuitively, suppose a BPDG G is to bet on\nsequence S has already bet on w ❁S, with current capital x ∈Q, current state q ∈Q and\ncurrent top stack symbol a. Then for b ∈Σ, G bets the quantity xβ(q, a)(b) of its capital\nthat the next symbol of S is b. If the bet is correct (that is, if wb ❁S) and since payoffs\nare fair, G has capital |Σ|xβ(q, a)(b). Formally,\nDefinition.\nLet G = (Q, Σ, Γ, δ, β, q0, z0, c) be a bounded pushdown gambler.\nThe\nmartingale of G is the function\ndG : Σ∗→[0, ∞)\ndefined by the recursion\ndG(λ) = 1\ndG(wb) = |Σ|dG(w)β(δ(w))(b)\nfor all w ∈Σ∗and b ∈Σ.\nBy Observation 3.1, a BPDG G actually yields an s-gale for every s ∈[0, ∞). We call\nit the s-gale of G, and denote it by\nds\nG(w) = |Σ|(s−1)|w|dG(w).\nA bounded pushdown s-gale is an s-gale d for which there exists a BPDG such that ds\nG = d.\nThe first two properties of BPD gamblers are that any number of λ-transitions can\nbe replaced by a single λ-transition and that the stack alphabet does not give additional\npower.\nProposition 3.3 Let G = (Q, Σ, Γ, δ, β, q0, z0, c) be a BPDG. Then there is a BPDG G′ =\n(Q′, Σ, Γ′, δ′, β′, q′\n0, z′\n0, 1) such that dG = dG′.\nFrom now on we shall assume that the maximum number of λ-transitions c is 1.\nProposition 3.4 Let G = (Q, Σ, Γ, δ, β, q0, z0, c) be a BPDG. Then there is a BPDG G′ =\n(Q′, Σ, {0, 1, z′\n0}, δ′, β′, q′\n0, z′\n0, c′) such that dG = dG′.\n6\n\nLet us define bounded pushdown dimension. Intuitively, the BPD dimension of a se-\nquence is the smallest s such that there is a BPD-s-gale that succeeds on the sequence.\nDefinition.\nThe bounded pushdown dimension of a set X ⊆Σ∞is\ndimBPD(X) = inf{s | there is a bounded pushdown s −gale d such that X ⊆S∞[d]}.\n4\nDimension and compression\nIn this section we characterize the bounded pushdown dimension of individual sequences\nin terms of bounded pushdown compressibility, therefore BPD dimension is a natural and\nrobust definition.\nDefinition. A bounded pushdown compressor (BPDC) is an 8-tuple\nC = (Q, Σ, Γ, δ, ν, q0, z0, c)\nwhere\n• Q is a finite set of states,\n• Σ is the finite input and output alphabet,\n• Γ is the finite stack alphabet,\n• δ : Q × (Σ ∪{λ}) × Γ →Q × Γ∗is the transition function,\n• ν : Q × Σ × Γ →Σ∗is the output function,\n• q0 ∈Q is the initial state,\n• z0 ∈Γ is the start stack symbol,\n• c ∈N is a constant such that the number of λ-transitions per input symbol is at most\nc,\nwith the two additional restrictions:\n1. for each q ∈Q and a ∈Γ at least one of the following holds\n• δ(q, λ, a) =⊥\n• δ(q, b, a) =⊥for all b ∈Σ\n2. for every q ∈Q, b ∈Σ ∪{λ}, either δ(q, b, z0) =⊥, or δ(q, b, z0) = (q′, vz0), where q′\n∈Q and v ∈Γ∗.\n7\n\nWe extend δ to δ∗∗: Q×Σ∗×Γ+ →Q×Γ∗as before, and denote δ∗∗by δ and δ(q0, w, z0)\nby δ(w).\nFor q ∈Q, w ∈Σ∗and z ∈Γ+, we define the output from state q on input w reading\nz on the top of the stack to be the string ν∗(q, w, z) (denoted by ν(q, w, z)) with\nν(q, λ, z) = λ\nν(q, wb, z) = ν(q, w, z)ν(δQ(q, w, z), b, δΓ∗(q, w, z))\nfor w ∈Σ∗and b ∈Σ. We then define the output of C on input w ∈Σ∗to be the string\nC(w) = ν(q0, w, z0).\nWe can restrict λ-transitions to a single one and the stack alphabet to three symbols.\nProposition 4.1 Let C = (Q, Σ, Γ, δ, ν, q0, z0, c) be a BPDC. Then there is a BPDC C′ =\n(Q′, Σ, Γ′, δ′, ν′, q′\n0, z′\n0, 1) such that C(w) = C′(w) for every w ∈Σ∗.\nProposition 4.2 Let C = (Q, Σ, Γ, δ, ν, q0, z0, c) be a BPDC. Then there is a BPDC C′ =\n(Q′, Σ, {0, 1, z′\n0}, δ′, ν′, q′\n0, z′\n0, c′) such that C(w) = C′(w) for every w ∈Σ∗.\nWe are interested in information lossless compressors, that is, w must be recoverable\nfrom C(w) and the final state.\nDefinition.\nA BPDC C = (Q, Σ, Γ, δ, ν, q0, z0) is information-lossless (IL) if the function\nΣ∗→Σ∗× Q\nw →(C(w), δQ(w))\nis one-to-one.\nAn information-lossless bounded pushdown compressor (ILBPDC) is a\nBPDC that is IL.\nIntuitively, a BPDC compresses a string w if |C(w)| is significantly less than |w|. Of\ncourse, if C is IL, then not all strings can be compressed. Our interest here is in the\ndegree (if any) to which the prefixes of a given sequence S ∈Σ∞can be compressed by an\nILBPDC.\nDefinition.\nIf C is a BPDC and S ∈Σ∞, then the compression ratio of C on S is\nρC(S) = lim inf\nn→∞\n|C(S[0..n −1])|\nn\n.\nThe BPD compression ratio of a sequence is the best compression ratio achievable by\nan ILBPDC, that is\nDefinition.\nThe bounded pushdown compression ratio of a sequence S ∈Σ∞is\nρBPD(S) = inf{ρC(S) | C is a ILBPDC}.\nThe main result in this section states that the BPD dimension of a sequence and its\nILBPD compression ratio are the same, therefore BPD dimension is the natural concept\nof density of information in the BPD setting.\n8\n\nTheorem 4.3 For all S ∈Σ∞,\ndimBPD(S) = ρBPD(S).\n5\nSeparating LZ from BPD\nIn this section we prove that BPD compression can be much better than the compression\nattained with the celebrated Lempel-Ziv algorithm.\nWe start with a brief description of the LZ algorithm [14].\nWe finish relating BPD dimension (and compression) with the Lempel-Ziv algorithm.\nGiven an input x ∈Σ∗, LZ parses x in different phrases xi, i.e., x = x1x2 . . . xn (xi ∈Σ∗)\nsuch that every prefix y ❁xi, appears before xi in the parsing (i.e. there exists j < i s.t.\nxj = y). Therefore for every i, xi = xl(i)bi for l(i) < i and bi ∈Σ. We sometimes denote\nthe number of phrases in the parsing of x as C(x).\nLZ encodes xi by a prefix free encoding of l(i) and the symbol bi, that is, if x =\nx1x2 . . . xn as before, the output of LZ on input x is\nLZ(x) = cl(1)b1cl(2)b2 . . . cl(n)bn\nwhere ci is a prefix-free coding of i (and x0 = λ).\nLZ is usually restricted to the binary alphabet, but the description above is valid for\nany Σ.\nFor a sequence S ∈Σ∞, the LZ compression ratio is given by\nρLZ(S) = lim inf\nn→∞\n|LZ(S[0 . . . n −1])|\nn\n.\nIt is well known that LZ [14] yields a lower bound on the finite-state dimension (or finite-\nstate compressibility) of a sequence [14], ie, LZ is universal for finite-state compressors.\nThe following result shows that this is not true for BPD (hence PD) dimension, in a\nstrong sense: we construct a sequence S that cannot be compressed by LZ, but that has\nBPD compression ratio less than 1\n2.\nTheorem 5.1 For every m ∈N, there is a sequence S ∈{0, 1}∞such that\nρLZ(S) > 1 −1\nm\nand\ndimBPD(S) ≤1\n2.\nAs a corollary we obtain a separation of finite-state dimension and bounded pushdown\ndimension. A similar result between finite-state dimension and pushdown dimension was\nproved in [5].\n9\n\nCorollary 5.2 For any m ∈N, there exists a sequence S ∈{0, 1}∞such that\ndimFS(S) > 1 −1\nm\nand\ndimBPD(S) ≤1\n2.\nConclusion\nWe have introduced Bounded Pushdown dimension, characterized it with compression and\ncompared it with Lempel-Ziv compression. It is open if there is a BPD compressor that\nis universal for Finite-State compressors, which is true for the Lempel-Ziv algorithm, and\nwhether Lempel-Ziv compression can surpass BPD-compression for some sequence.\nReferences\n[1] K. B. Athreya, J. M. Hitchcock, J. H. Lutz, and E. Mayordomo. Effective strong\ndimension in algorithmic information and computational complexity. SIAM Journal\non Computing. To appear.\n[2] Chris Bourke, John M. Hitchcock, and N. V. Vinodchandran.\nEntropy rates and\nfinite-state dimension. Theor. Comput. Sci., 349(3):392–406, 2005.\n[3] Jack J. Dai, James I. Lathrop, Jack H. Lutz, and Elvira Mayordomo. Finite-state\ndimension. Theoretical Computer Science, 310(1–3):1–33, January 2004.\n[4] D. Doty and P. Moser. Personal communication, based on [13]. 2006.\n[5] David Doty and Jared Nichols. Pushdown dimension. Theoretical Computer Science.\nTo appear.\n[6] K. Falconer. The Geometry of Fractal Sets. Cambridge University Press, 1985.\n[7] J. M. Hitchcock, J. H. Lutz, and E. Mayordomo. The fractal geometry of complexity\nclasses. SIGACT News Complexity Theory Column, 36:24–38, 2005.\n[8] D. A. Huffman. Canonical forms for information-lossless finite-state logical machines.\nTrans. Circuit Theory, pages 41–59, 1959.\n[9] J. H. Lutz. Dimension in complexity classes. SIAM Journal on Computing, 32:1236–\n1259, 2003.\n[10] J. H. Lutz. The dimensions of individual strings and sequences. Information and\nComputation, 187:49–79, 2003.\n10\n\n[11] J. H. Lutz. Effective fractal dimensions.\nMathematical Logic Quarterly, 51:62–72,\n2005.\n[12] E. Mayordomo. Effective fractal dimension in algorithmic information theory. In New\nComputational Paradigms: Changing Conceptions of What is Computable. Springer-\nVerlag, 2007. To appear.\n[13] D. Sheinwald, A. Lempel, and J. Ziv. On compression with two-way head machines.\nIn Data Compression Conference, pages 218–227, 1991.\n[14] Jacob Ziv and Abraham Lempel. Compression of individual sequences via variable-\nrate coding. IEEE Transactions on Information Theory, 24(5):530–536, 1978.\n11\n\nTechnical Appendix\nThis appendix is devoted to proving Theorem 4.3 and Theorem 5.1. For the first one,\nwe need the following:\nA\nProof of Theorem 4.3\nDefinition.\nA BPDG G = (Q, Σ, Γ, δ, β, q0, z0) is nonvanishing if 0 < β(q, z)(b) < 1 for\nall q ∈Q, b ∈Σ and z ∈Γ.\nLemma A.1 For every BPDG G and each ε > 0, there is a nonvanishing BPDG G′ such\nthat for all w ∈Σ∗, dG′(w) ≥|Σ|−ε|w|dG(w).\nProof of Lemma A.1 . Let G = (Q, Σ, δ, β, q0, Γ, z0) be a BPDG, and let ε > 0. For\neach q ∈Q, z ∈Γ, b ∈Σ,\n1 −|Σ|−ε X\nb∈Σ\nβ(q, z)(b) = 1 −|Σ|−ε > 0,\nso we can fix a rational β′(q, z)(b) such that\n|Σ|−εβ(q, z)(b) < β′(q, z)(b) < 1 −|Σ|−ε\nX\na∈Σ,a̸=b\nβ(q, z)(a)\nand\nX\nb∈Σ\nβ′(q, z)(b) = 1.\nThen, 0 < β′(q, z)(b) < 1 for each q ∈Q, b ∈Σ and z ∈Γ, therefore the BPDG G′ =\n(Q, Σ, δ, β′, q0, Γ, z0) is nonvanishing.\nAlso, for all q ∈Q, b ∈Σ, z ∈Γ,\nβ′(q, z)(b) ≥|Σ|−εβ(q, z)(b)\nso for all w ∈Σ∗, dG′(w) ≥|Σ|−ε|w|dG(w).\n✷\nProof of Theorem 4.3 Let S ∈Σ∞. For each n ∈N, let wn = S[0..n −1].\nTo see that dimBPD(S) ≤ρBPD(S), let s > s′ > ρBPD(S).\nIt suffices to show that\ndimBPD(S) ≤s. By our choice of s′, there is an 1-ILBPDC C = (Q, Σ, Γ, δ, ν, q0, z0) for\nwhich the set\nI = {n ∈N | |C(wn)| < s′n}\nis infinite.\n1\n\nCONSTRUCTION A.1 Given a 1-bounded pushdown compressor (BPDC)\nC = (Q, Σ, Γ, δ, ν, q0, z0), and k ∈Z+ , we construct the 1-bounded pushdown gambler\n(BPDG) G = G(C, k) = (Q′, Σ, Γ′, δ′, β′, q′\n0, z′\n0) as follows:\ni) Q′ = Q × {0, 1, . . . , k −1}\nii) q′\n0 = (q0, 0)\niii) Γ′ =\n4k−1\nS\ni=2k\nΓi\niv) z′\n0 = z2k\n0\nv) ∀(q, i) ∈Q′, b ∈Σ, a ∈Γ′,\nδ′((q, i), b, a) =\n \nδQ(q, b, a), (i + 1) mod k\n \n,\n\\\nδΓ∗(q, b, a)\n \nwhere for each z ∈(Γ′)+, z ∈Γ+ is the Γ-string obtained by concatenating the symbols of\nz, and for each y ∈Γ+, if y = y1y2 · · · y2kl+n with n < 2k, then by ∈(Γ′)+ is such that\nby1 = y1 · · · y2k+n, by2 = y2k+n+1 · · · y4k+n, . . . , byl = y2k(l−1)+n+1 · · · y2kl+n.\nvi) ∀(q, i) ∈Q′, a ∈Γ′, b ∈Σ\nβ′((q, i), a)(b) = σ(q, bΣk−i−1, a)\nσ(q, Σk−i, a)\nwhere σ(q, A, a) = P\nx∈A\n|Σ|−|ν(q,x,a)| .\nLemma A.2 In Construction A.1, if |w| is a multiple of k and u ∈Σ≤k, then\ndG(wu) = |Σ||u|−|ν(δQ(w),u,δΓ∗(w))| σ(δQ(wu), Σk−|u|, \\\nδΓ∗(wu))\nσ(δQ(w), Σk, \\\nδΓ∗(w))\ndG(w).\nProof of Lemma A.2. We use induction on the string u. If u = λ, the lemma is\nclear. Assume that it holds for u, where u ∈Σ<k, and let b ∈Σ. Then\ndG(wub) = |Σ|σ(δQ(wu), bΣk−|u|−1, \\\nδΓ∗(wu))\nσ(δQ(wu), Σk−|u|, \\\nδΓ∗(wu))\ndG(wu)\n= |Σ|1−|ν(δQ(wu),b,δΓ∗(wu))|σ(δQ(wub), Σk−|u|−1,\n\\\nδΓ∗(wub))\nσ(δQ(wu), Σk−|u|, \\\nδΓ∗(wu))\ndG(wu)\nso by the induction hypothesis the lemma holds for ub.\n2\n\n✷\nLemma A.3 In Construction A.1, if w = w0w1 · · ·wn−1, where each wi ∈Σk , then\ndG(w) =\n|Σ||w|−|C(w)|\nn−1\nQ\ni=0\nσ(δQ(w0 · · ·wi−1), Σk,\n\\\nδΓ∗(w0 · · · wi−1))\n.\nProof of Lemma A.3. We use induction on n. For n = 0, the identity is clear.\nAssume that it holds for w = w0w1 · · · wn−1, with each wi ∈Σk, and let w′ = w0w1 · · · wn.\nThen Lemma A.2 with u = wn tells us that\ndG(w′) = |Σ|k−|ν(δQ(w),wn,δΓ∗(w))|\nσ(δQ(w), Σk, \\\nδΓ∗(w))\ndG(w)\nwhence the identity holds for w′ by the induction hypothesis.\n✷\nLemma A.4 In Construction A.1, if C is IL and |w| is a multiple of k, then\ndG(w) ≥|Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1),\nwhere l = ⌈log |Q|⌉and m = max{|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}.\nProof of Lemma A.4. We prove that for each z ∈Σ∗,\nσ(δQ(z), Σk, \\\nδΓ∗(z)) ≤|Σ|l+log m+log k+1.\nTo see this, fix z ∈Σ∗and observe that at most |Q| strings w ∈Σk can have the same\noutput from state δQ(z) with stack content δΓ∗(z). Therefore, the number of w ∈Σk for\nwhich |ν(δQ(z), w, δΓ∗(z))| = j does not exceed |Q||Σ|j. Hence\nσ(δQ(z), Σk, \\\nδΓ∗(z)) =\nX\nw∈Σk\n|Σ|−|ν(δQ(z),w,δΓ∗(z))| ≤\nmk\nX\nj=0\n|Q||Σ|j|Σ|−j = |Q|(mk + 1)\n≤|Σ|l+log m+log k+1.\nIt follows by Lemma A.3 that\ndG(w) = |Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1).\n✷\nLemma A.5 In Construction A.1, if C is IL, then for all w ∈Σ∗,\ndG(w) ≥|Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1)−(km+l+log m+log k+1),\nwhere l = ⌈log |Q|⌉and m = max {|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}.\n3\n\nProof of Lemma A.5. Assume the hypothesis, let l and m be as given, and let w ∈\nΣ∗. Fix 0 ≤j < k such that |w| + j is divisible by k. By Lemma A.4 we have\ndG(w) ≥|Σ|−jdG(w0j)\n≥|Σ|−j+|w0j|−|C(w0j)|−|w0j |\nk\n(l+log m+log k+1)\n= |Σ||w|−|C(w0j)|−|w|\nk (l+log m+log k+1)−j\nk (l+log m+log k+1)\n≥|Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1)−(km+l+log m+log k+1)\n✷\nLet l = ⌈log |Q|⌉and m = max{|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}, and fix k ∈Z+ such\nthat l+log m+log k+1\nk\n< s −s′. Let G = G(C, k) be as in Construction A.1. Then, by Lemma\nA.5, for all n ∈I we have\nd(s)\nG (wn) ≥|Σ|sn−|C(wn)|−n\nk (l+log m+log k+1)−(km+l+log m+log k+1)\n≥|Σ|(s−s′−l+log m+log k+1\nk\n)n−(km+l+log m+log k+1)\nSince s −s′ −l+log m+log k+1\nk\n> 0, this implies that S ∈S∞[d(s)\nG ].\nThus, dimBPD(S) ≤s.\nTo see that ρBPD(S) ≤dimBPD(S), let s > s′ > s′′ > dimBPD(S). It suffices to show\nthat ρBPD(S) ≤s. By our choice of s′′, there is a 1-BPDG G such that the set\nJ = {n ∈N | ds′′\nG (wn) ≥1}\nis infinite. By Lemma A.1 there is a nonvanishing 1-BPDG eG such that\nd e\nG(w) ≥|Σ|(s′′−s′)|w|dG(w) for all w ∈Σ∗.\nCONSTRUCTION A.2 Let G = (Q, Σ, Γ, δ, β, q0, z0) be a nonvanishing 1-BPDG, and\nlet k ∈Z+. For each z ∈Γ∗(long enough for dGq,z(w) to be defined for all w ∈Σk)\nand q ∈Q, let Gq,z = (Q, Σ, Γ, δ, β, q, z), and define pq,z : Σk →[0, 1] by pq,z(w) =\n|Σ|−kdGq,z(w). Since G is nonvanishing and each dGq,z is a martingale with dGq,z(λ) = 1,\neach of the functions pq,z is a positive probability measure on Σk. For each z ∈Γ∗, q ∈Q, let\nΘq,z : Σk →Σ∗be the Shannon-Fano-Elias code given by the probability measure pq,z. Then\n|Θq,z(w)| = lq,z(w)\nlq,z(w) = 1 + ⌈log\n1\npq,z(w)⌉\nfor all q ∈Q and w ∈Σk, and each of the sets range(Θq,z) is an instantaneous code. We\ndefine the 1-BPDC C = C(G, k) = (Q′, Σ, Γ′, δ′, ν′, q′\n0, z′\n0) whose components are as follows:\ni) Q′ = Q × Σ<k\n4\n\nii) q′\n0 = (q0, λ)\niii) Γ′ =\n4k−1\nS\ni=2k\nΓi\niv) z′\n0 = z2k\n0\nv) ∀(q, w) ∈Q′, b ∈Σ, a ∈Γ′,\nδ′((q, w), b, a) =\n(\n(q, wb, a)\nif |w| < k −1,\n(δQ(q, wb, a), λ,\n\\\nδΓ∗(q, wb, a))\nif |w| = k −1.\nvi) ∀(q, w) ∈Q′, b ∈Σ, a ∈Γ′,\nν′((q, w), b, a) =\n λ\nif |w| < k −1,\nΘq,a(wb)\nif |w| = k −1.\nSince each range(Θq,z) is an instantaneous code, it is easy to see that the BPDC C =\nC(G, k) is IL.\nLemma A.6 In Construction A.2, if |w| is a multiple of k, then\n|C(w)| ≤\n \n1 + 2\nk\n \n|w| −log dG(w).\nProof of Lemma A.6.\nLet w = w0w1 · · · wn−1, where each wi ∈Σk.\nFor each\n0 ≤i < n, let qi = δQ(w0 · · · wi−1) and zi = δΓ∗(w0 · · · wi−1). Then,\n|C(w)| =\nn−1\nX\ni=0\nlqi,zi(wi)\n=\nn−1\nX\ni=0\n \n1 + ⌈log\n1\npqi,zi(wi)⌉\n \n≤\nn−1\nX\ni=0\n \n2 + log\n1\npqi,zi(wi)\n \n=\nn−1\nX\ni=0\n \n2 + log\n|Σ|k\ndGqi,zi(wi)\n \n= (k + 2)n −log\nn−1\nY\ni=0\ndGqi,zi(wi)\n= (k + 2)n −log dG(w) = (1 + 2\nk)|w| −log dG(w)\n✷\nLemma A.7 In Construction A.2, for all w ∈Σ∗,\n|C(w)| ≤\n \n1 + 2\nk\n \n|w| −log dG(w).\n5\n\nProof of Lemma A.7. If |w| is multiple of k, then we apply the Lemma A.6.\nOtherwise, let w = w′z, where |w′| is a multiple of k and |z| = j, 0 < j < k.\nThen, Lemma A.6 tell us that\n|C(w)| = |C(w′)|\n≤\n \n1 + 2\nk\n \n|w′| −log dG(w′)\n≤\n \n1 + 2\nk\n \n|w′| −log(|Σ|−jdG(w))\n=\n \n1 + 2\nk\n \n|w| −log dG(w) −2j\nk\n≤\n \n1 + 2\nk\n \n|w| −log dG(w).\n✷\nFix k >\n2\ns−s′, and let C = C( eG, k) be as in Construction A.2. Then Lemma A.7 tell us\nthat for all n ∈J,\n| C(wn) | ≤\n \n1 + 2\nk\n \nn −log d e\nG(wn)\n≤\n \n1 + 2\nk + s′ −s′′ \nn −log dG(wn)\n≤\n 2\nk + s′ \nn −log ds′′\nG (wn)\n≤\n 2\nk + s′ \nn\n< sn.\nThus, ρBPD(S) ≤s.\n✷\nB\nProof of Theorem 5.1\nFor a string x, x−1 denotes x written in reverse order.\nProof of Theorem 5.1 Let m ∈N, and let k = k(m) be an integer to be determined\nlater. For any integer n, let Tn denote the set of strings x of size n such that 1j does not\nappear in x, for every j ≥k. Since Tn contains {0, 1}k−1 × {0} × {0, 1}k−1 × {0} . . . (i.e.\nthe set of strings whose every kth bit is zero), it follows that |Tn| ≥2an, where a = 1−1/k.\nRemark B.1 For every string x ∈Tn there is a string y ∈Tn−1 and a bit b such that\nyb = x.\n6\n\nLet An = {a1, . . . au} be the set of palindromes in Tn. Since fixing the n/2 first bits\nof a palindrome (wlog n is even) completely determines it, it follows that |An| ≤2\nn\n2 .\nLet us separate the remaining strings in Tn −An into two sets Xn = {x1, . . . xt} and\nYn = {y1, . . . yt} with (xi)−1 = yi for every 1 ≤i ≤t. Let us choose X, Y such that x1 and\nyt start with a zero. We construct S in stages. For n ≤k −1, Sn is an enumeration of all\nstrings of size n in lexicographical order. For n ≥k,\nSn = a1 . . . au 12n x1 . . . xt 12n+1 yt . . . y1\ni.e.\na concatenation of all strings in An (the A zone of Sn) followed by a flag of 2n\nones, followed by the concatenations of all strings in X (the X-zone) and Y (the Y zone)\nseparated by a flag of 2n + 1 ones. Let\nS = S1S2 . . . Sk−1 1k 1k+1 . . . 12k−1 SkSk+1 . . .\ni.e. the concatenation of the Sj’s with some extra flags between Sk−1 and Sk. We claim\nthat the parsing of Sn (n ≥k) by LZ, is as follows:\nSn = a1, . . . , au, 12n, x1, . . . , xt, 12n+1, yt, . . . , y1.\nIndeed after S1, . . . Sk−1 1k 1k+1 . . . 12k−1, LZ has parsed every string of size ≤k −1 and\nthe flags 1k 1k+1 . . . 12k−1. Together with Remark B.1, this guarantees that LZ parses Sn\ninto phrases that are exactly all the strings in Tn and the two flags 12n, 12n+1.\nLet us compute the compression ratio ρLZ(S). Let n, i be integers. By construction of\nS, LZ encodes every phrase in Si (except the two flags), by a phrase in Si−1 (plus a bit).\nIndexing a phrase in Si−1 requires a codeword of length at least logarithmic in the number\nof phrase parsed before, i.e. log(C(S1S2 . . . Si−2)). Since C(Si) ≥|Ti| ≥2ai, it follows\nC(S1 . . . Si−2) ≥\ni−2\nX\nj=1\n2aj = 2a(i−1) −2a\n2a −1\n≥b2a(i−1)\nwhere b = b(a) is arbitrarily close to 1. Letting ti = |Ti|, the number of bits output by LZ\non Si is at least\nC(Si) log C(S1 . . . Si−2) ≥ti log b2a(i−1)\n≥cti(i −1)\nwhere c = c(b) is arbitrarily close to 1. Therefore\n|LZ(S1 . . . Sn)| ≥\nn\nX\nj=1\nctj(j −1)\nSince |S1 . . . Sn| ≤2k2 + Pn\nj=1(jtj + 4j), (the two flags plus the extra flags between Sk−1\nand Sk) the compression ratio is given by\nρLZ(S1 . . . Sn) ≥c\nPn\nj=1 tj(j −1)\n2k2 + Pn\nj=1 j(tj + 4)\n(2)\n= c −c\n2k2 + Pn\nj=1(tj + 4j)\n2k2 + Pn\nj=1 j(tj + 4)\n(3)\n7\n\nThe second term in Equation 3 can be made arbitrarily small for n large enough: Let\nM ≤n, we have\n2k2 +\nn\nX\nj=1\nj(tj + 4) ≥2k2 +\nM\nX\nj=1\njtj + (M + 1)\nn\nX\nj=M+1\ntj\n= 2k2 +\nM\nX\nj=1\njtj + M\nn\nX\nj=M+1\ntj +\nn\nX\nj=M+1\ntj\n≥2k2 +\nM\nX\nj=1\njtj + M\nn\nX\nj=M+1\ntj +\nn\nX\nj=M+1\n2aj\n≥2k2 +\nM\nX\nj=1\njtj + M\nn\nX\nj=M+1\ntj + 2an\n≥M\nn\nX\nj=M+1\ntj + M(2k2 + 2n(n + 1) +\nM\nX\nj=1\ntj)\nfor n big enough\n= M(2k2 +\nn\nX\nj=1\ntj + 4\nn\nX\nj=1\nj)\nHence\nρLZ(S1 . . . Sn) ≥c −c\nM\nwhich by definition of c, M can be made arbitrarily close to 1 by choosing k accordingly,\ni.e\nρLZ(S1 . . . Sn) ≥1 −1\nm.\nLet us show that dimBPD(S) ≤1\n2. Consider the following BPD martingale d. Informally,\nd on Sn goes through the An zone until the first flag, then starts pushing the whole X\nzone onto its stack until it hits the second flag. It then uses the stack to bet correctly on\nthe whole Y zone. Since the Y zone is exactly the X zone written in reverse order, d is\nable to double its capital on every bit of the Y zone. On the other zones, d does not bet.\nBefore giving a detailed construction of d, let us compute the upper bound it yields on\n8\n\ndimBPD(S).\ndimBPD(S) ≤1 −lim sup\nn→∞\nlog d(S1 . . . Sn)\n|S1 . . . Sn|\n≤1 −lim sup\nn→∞\nPn\nj=1 |Yj|\n2k2 + Pn\nj=1(j|Tj| + 4j)\n≤1 −lim sup\nn→∞\nPn\nj=1 j |Tj|−|Aj|\n2\n2k2 + Pn\nj=1(j|Tj| + 4j)\n≤1\n2 + 1\n2 lim sup\nn→∞\n2k2 + Pn\nj=1(j|Aj| + 4j)\n2k2 + Pn\nj=1(j|Tj| + 4j) .\nSince\nlim sup\nn→∞\n2k2 + Pn\nj=1(j|Aj| + 4j)\n2k2 + Pn\nj=1(j|Tj| + 4j) ≤lim sup\nn→∞\nPn\nj=1 j(|Aj| + 4 + 2k2)\nPn\nj=1 |Tj|\n≤lim sup\nn→∞\nPn\nj=1 j(2\nj\n2 + 2\nj\n4)\nPn\nj=1 2aj\n≤lim sup\nn→∞\nn2\n3n\n4\n2an\n= 0.\nIt follows that\ndimBPD(S) ≤1\n2.\nLet us give a detailed description of d. Let Q be the following set of states:\n• The start state q0, and q1, . . . qv the “early” states that will count up to\nv = |S1S2 . . . Sk−1 1k 1k+1 . . . 12k−1|.\n• qa\n0, . . . , qa\nk the A zone states that cruise through the A zone until the first flag.\n• q1f the first flag state.\n• qX\n0 , . . . , qX\nk the X zone states that cruise through the X zone, pushing every bit on\nthe stack, until the second flag is met.\n• qr\n0, . . . , qr\nk which after the second flag is detected, pop k symbols from the stack that\nwere erroneously pushed while reading the second flag.\n• q2f the second flag state.\n• qb the betting on zone Y state.\n9\n\nLet us describe the transition function δ : Q × {0, 1} × {0, 1} →Q × {0, 1}. First δ counts\nuntil v i.e. for i = 0, . . . v −1\nδ(qi, x, y) = (qi+1, y)\nfor any x, y\nand after reading v bits, it enters in the first A zone state, i.e. for any x, y\nδ(qv, x, y) = (qa\n0, y).\nThen δ skips through A until the string 1k is met, i.e. for i = 0, . . . k −1 and any x, y\nδ(qa\ni , x, y) =\n(\n(qa\ni+1, y)\nif x = 1\n(qa\n0, y)\nif x = 0\nand\nδ(qa\nk, x, y) = (q1f, y).\nOnce 1k has been seen, δ knows the first flag has started, so it skips through the flag until\na zero is met, i.e. for every x, y\nδ(q1f, x, y) =\n(\n(q1f, y)\nif x = 1\n(qX\n0 , 0y)\nif x = 0\nwhere state qX\n0 means that the first bit of the X zone (a zero bit) has been read, therefore\nδ pushes a zero. In the X zone, delta pushes every bit it sees until it reads a sequence of\nk ones, i.e until the start of the second flag, i.e for i = 0, . . . k −1 and any x, y\nδ(qX\ni , x, y) =\n(\n(qX\ni+1, xy)\nif x = 1\n(qX\n0 , xy)\nif x = 0\nand\nδ(qX\nk , x, y) = (qr\n0, y).\nAt this point, δ has pushed all the X zone on the stack, followed by k ones. The next step\nis to pop k ones, i.e for i = 0, . . . k −1 and any x, y\nδ(qr\ni , x, y) = (qr\ni+1, λ)\nand\nδ(qr\nk, x, y) = (q2f\n0 , y).\nAt this stage, δ is still in the second flag (the second flag is always bigger than 2k) therefore\nit keeps on reading ones until a zero (the first bit of the Y zone) is met. For any x, y\nδ(q2f, x, y) =\n(\n(q2f, y)\nif x = 1\n(qb, λ)\nif x = 0.\n10\n\nOn the last step δ has read the first bit of the Y zone, therefore it pops it. At this stage,\nthe stack exactly contains the Y zone (i.e. the X zone written in reverse order) except\nthe first bit; δ thus uses its stack to bet and double its capital on every bit in the Y zone.\nOnce the stack is empty, a new A zone begins. Thus, for any x, y\nδ(qb, x, y) = (qb, λ).\nand\nδ(qb, x, z0) =\n(\n(qa\n1, z0)\nif x = 1\n(qa\n0, z0)\nif x = 0.\nThe betting function is equal to 1/2 everywhere (i.e no bet) except on state qb, where\nβ(qb, y)(z) =\n(\n1\nif y = z\n0\nif y ̸= z.\nand β stops betting once start stack symbol is met, i.e.\nβ(qb, z0) = 1\n2.\n⊓⊔\n11","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0704.2386v1 [cs.CC] 18 Apr 2007\nBounded Pushdown dimension vs Lempel Ziv\ninformation density\nPilar Albert, Elvira Mayordomo, and Philippe Moser ∗\nAbstract\nIn this paper we introduce a variant of pushdown dimension called bounded push-\ndown (BPD) dimension, that measures the density of information contained in a\nsequence, relative to a BPD automata, i.e. a finite state machine equipped with an\nextra infinite memory stack, with the additional requirement that every input symbol\nonly allows a bounded number of stack movements. BPD automata are a natural\nreal-time restriction of pushdown automata. We show that BPD dimension is a ro-\nbust notion by giving an equivalent characterization of BPD dimension in terms of\nBPD compressors. We then study the relationships between BPD compression, and\nthe standard Lempel-Ziv (LZ) compression algorithm, and show that in contrast to\nthe finite-state compressor case, LZ is not universal for bounded pushdown compres-\nsors in a strong sense: we construct a sequence that LZ fails to compress significantly,\nbut that is compressed by at least a factor 2 by a BPD compressor. As a corollary\nwe obtain a strong separation between finite-state and BPD dimension.\nKeywords\nInformation lossless compressors, finite state (bounded pushdown) dimension, Lempel-Ziv\ncompression algorithm.\n1\nIntroduction\nEffective versions of fractal dimension have been developed since 2000 [9, 10] and used\nfor the quantitative study of complexity classes, information theory and data compression,\nand back in fractal geometry (see recent surveys in [11, 7, 12]). Here we are interested\nin information theory and data compression, where it is known that for several different\n∗Dept. de Inform ́atica e Ingenier ́ıa de Sistemas , Universidad de Zaragoza. Edificio Ada Byron, Mar ́ıa\nde Luna 1 - E-50018 Zaragoza (Spain). Email: {mpalbert, elvira}@unizar.es and mosersan@gmail.com.\nResearch supported in part by Spanish Government MEC Project TIN 2005-08832-C03-02, by Arag ́on\nGovernment Dept. Ciencia, Tecnolog ́ıa y Universidad, subvenci ́on destinada a la formaci ́on de personal\ninvestigador-B068/2006 and by Spanish Government MEC Program Juan de la Cierva.\n1"},{"paragraph_id":"p2","order":2,"text":"bounds on the computing power, effective dimensions capture what can be considered the\ninherent information content of a sequence in the corresponding setting [12]. In the today\nrealistic context of massive data streams we need to consider very low resource-bounds,\nsuch as finite memory or finite-time per input symbol.\nThe finite state dimension of an infinite sequence [3], is a measure of the amount of ran-\ndomness contained in the sequence within a finite-memory setting. It is a robust quantity,\nthat has been shown to admit several characterizations in terms of finite-state information\nlossless compressors (introduced by Huffman [8], [3]), finite-state decompressors [4, 13],\nfinite-state predictors in the logloss model [1], and block entropy rates [2]. It is an effec-\ntivization of the general notion of Hausdorffdimension at the level of finite-state machines.\nInformally, the finite state dimension assigns every sequence a number s ∈[0, 1], that char-\nacterizes the randomness density in the sequence (or equivalently its compression ratio),\nwhere the larger the dimension the more randomness is contained in the sequence.\nIn a recent line of research, Doty and Nichols [5] investigated a variant of finite-state\ndimension, where the finite state machine comes equipped with an infinite memory stack\nand is called a pushdown automata, yielding the notion of pushdown dimension. Hence\nthe pushdown dimension of a sequence, is a measure of the density of randomness in the\nsequence as viewed by a pushdown automata. Since a finite-state automata is a special\ncase of a pushdown automata, the pushdown dimension of a sequence is a lower bound\nfor its finite state dimension. It was shown in [5], that there are sequences for which the\npushdown dimension is at most half its finite state dimension, hence yielding a strong\nseparation between the two notions. Unfortunately the notion of pushdown dimension is\nnot known to enjoy any of the equivalent characterizations that finite state dimension does.\nMoreover, the computation time per input symbol can be unbounded, which rules out this\nmodel for many real-time applications.\nIn this paper we introduce a variant of pushdown dimension called bounded pushdown\n(BPD) dimension: Whereas pushdown automata can choose not to read their input and\nonly work with their stack for as many steps as they wish (each such step is called a\nlambda transition), we add the additional real-time constraint that the sequences of lambda\ntransitions are bounded, i.e. we only allow a bounded number of stack movements per each\ninput symbol.\nWe define the notion of bounded pushdown dimension as the natural effectivitation of\nHausdorffdimension via Lutz’s gale characterization [9]. We provide evidence that bounded\npushdown dimension is a robust notion by giving a compression characterization; i.e. we\nintroduce BPD information-lossless compressors and show that the best compression ratio\nachievable on a sequence by BPD compressors is exactly its BPD dimension.\nIn the context of compression, we study the relationship between BPD compression and\nthe standard Lempel-Ziv (LZ) compression algorithm [14]. It is well known that the LZ\ncompression ratio of any sequence is a lower bound for its finite state compressibility [14],\ni.e. LZ compresses every sequence at least as well as any finite-state information lossless\ncompressor. We show that this fails dramatically in the context of BPD compressors, by\nconstructing a sequence that LZ fails to compress significantly, but is compressed by at least\na factor 2 by a BPD compressor, thus yielding a strong separation between LZ and BPD\n2"},{"paragraph_id":"p3","order":3,"text":"dimension. This implies that we have the same separation between LZ and (unbounded)\npushdown dimension, and between finite state dimension [3] and BPD dimension.\nSection 2 contains the preliminaries, section 3 presents BPD dimension and its basic\nproperties, section 4 proves the equivalence of BPD compression and dimension and section\n5 contains the separation of BPD compression from Lempel Ziv compression. The proofs\nare postponed to the appendix.\n2\nPreliminaries\nWe write Z for the set of all integers, N for the set of all nonnegative integers and Z+ for\nthe set of all positive integers. Let Σ be a finite alphabet, with |Σ| ≥2. Σ∗denotes the\nset of finite strings, and Σ∞the set of infinite sequences. We write |w| for the length of\na string w in Σ∗. The empty string is denoted λ. For S ∈Σ∞and i, j ∈N, we write\nS[i..j] for the string consisting of the ith through jth symbols of S, with the convention\nthat S[i..j] = λ if i > j, and S[0] is the leftmost symbol of S. We write S[i] for S[i..i] (the\nith symbol of S). For w ∈Σ∗and S ∈Σ∞, we write w ⊑S if w is a prefix of S, i.e., if\nw = S[0..|w| −1]. All logarithms are taken in base |Σ|.\n3\nBounded Pushdown Dimension\nIn this section we first recall Lutz’s characterization of Hasudorffdimension in terms of\ngales that can be used to effectivize dimension. Then we introduce Bounded Pushdown\ndimension based on the concept of BPD gamblers and give its basic properties.\nDefinition. [9] Let s ∈[0, ∞).\n1. An s-gale is a function d : Σ∗→[0, ∞) that satisfies the condition\nd(w) =\nP\na∈Σ\nd(wa)\n|Σ|s\n(1)\nfor all w ∈Σ∗.\n2. A martingale is a 1-gale.\nIntuitively, an s-gale is a strategy for betting on the successive symbols of a sequence\nS ∈Σ∞. For each prefix w of S, d(w) is the capital (amount of money) that d has after\nhaving bet on S[0..|w| −1].\nWhen betting on the next symbol b of a prefix wb of S,\nassuming symbol b is equally likely to be any value in Σ, equation (1) guarantees that the\nexpected value of d(wb) is |Σ|−1 P\na∈Σ\nd(wa) = |Σ|s−1d(w). If s = 1, this expected value is\nexactly d(w), so the payoffs are “fair”.\nDefinition.\nLet d be an s-gale, where s ∈[0, ∞).\n1. We say that d succeeds on a sequence S ∈Σ∞if\n3"},{"paragraph_id":"p4","order":4,"text":"lim sup\nn→∞d(S[0..n −1]) = ∞.\n2. The success set of d is\nS∞[d] = {S ∈Σ∞| d succeeds on S}.\nObservation 3.1 Let s, s′ ∈[0, ∞). For every s-gale d, the function d′ : Σ∗→[0, ∞)\ndefined by d′(w) = |Σ|(s′−s)|w|d(w) is an s′-gale. Moreover, if s ≤s′, then S∞[d] ⊆S∞[d′].\nLutz characterized Hausdorffdimension using gales as follows.\nTheorem 3.2 [9] Given a set X ⊆Σ∞, if dimH(X) is the Hausdorffdimension of X [6],\nthen\ndimH(X) = inf{s | there is an s −gale d such that X ⊆S∞[d]}\nThe idea for a Bounded Pushdown dimension is to consider only s-gales that are com-\nputable by a Bounded Pushdown (BPD) gambler. Bounded Pushdown gamblers are finite-\nstate gamblers [3] with an extra memory stack, that is used both by the transition and\nbetting functions. Additionally, BPDG’s are allowed to delay reading the next character\nof the input –they read λ from the input– in order to alter the content of their stack, but\nthey cannot do this more than a constant number of times per each input symbol. During\nsuch λ-transitions, the gambler’s capital remains unchanged.\nThe betting function returns a probability measure over the input alphabet.\nDefinition.\nLet Σ be a finite alphabet. ∆Q(Σ) is the set of all rational-valued probability\nmeasures over Σ, i.e., all functions π : Σ −→[0, 1] ∩Q such that P\na∈Σ\nπ(a) = 1.\nWe are ready to define BPD gamblers.\nDefinition.\nA bounded pushdown gambler (BPDG) is an 8-tuple G =(Q, Σ, Γ, δ, β, q0,\nz0, c) where\n• Q is a finite set of states,\n• Σ is the finite input alphabet,\n• Γ is the finite stack alphabet,\n• δ : Q×(Σ∪{λ})×Γ →Q×Γ∗is the transition function (for simplicity we use the nota-\ntion δ(q, b, a) = ⊥when undefined; and we write δ(q, b, a) = (δQ(q, b, a), δΓ∗(q, b, a))),\n• β : Q × Γ →∆Q(Σ) is the betting function,\n• q0 ∈Q is the start state,\n• z0 ∈Γ is the start stack symbol,\n• c ∈N is a constant such that the number of λ-transitions per input symbol is at most\nc,\n4"},{"paragraph_id":"p5","order":5,"text":"with the two additional restrictions:\n1. for each q ∈Q and a ∈Γ at least one of the following holds\n• δ(q, λ, a) =⊥\n• δ(q, b, a) =⊥for all b ∈Σ\n2. for every q ∈Q, b ∈Σ ∪{λ}, either δ(q, b, z0) =⊥, or δ(q, b, z0) = (q′, vz0), where q′\n∈Q and v ∈Γ∗.\nWe denote with BPDG the set of all bounded pushdown gamblers.\nThe transition function δ outputs a new state and a string z′ ∈Γ∗.\nInformally,\nδ(q, w, a) = (q′, z′) means that in state q, reading input w, and popping symbol a from the\nstack, δ enters state q′ and pushes z′ to the stack.\nNote that w can be λ (ie, a λ-transition: the input is ignored and δ only computes with\nthe stack) but this only happens at most c times per input symbol. Any pair (state, stack\nsymbol) can either be a λ-transition pair or a non λ-transition pair exclusively, because\nthe first additional restriction enforces determinism.\nMoreover, since z0 represents the bottom of the stack, we restrict δ so that z0 cannot\nbe removed from the bottom by the second additional restriction.\nWe can extend δ in the usual way to\nδ∗: Q × (Σ ∪{λ}) × Γ+ →Q × Γ∗,\nwhere for all q ∈Q, a ∈Γ, v ∈Γ∗, and b ∈Σ ∪{λ}\nδ∗(q, b, av) =\n (δQ(q, b, a), δΓ∗(q, b, a)v)\nif δ(q, b, a) ̸=⊥,\n⊥\notherwise.\nWe denote δ∗by δ.\nFor each i ≥2, we will use the notation\nδi(q, λ, v) = δ(δi−1\nQ (q, λ, v), λ, δi−1\nΓ∗(q, λ, v))\nwhere\nδ1(q, λ, v) = δ(q, λ, v).\nSince δ is c-bounded we have that for any q ∈Q, v ∈Γ∗,\nδc+1(q, λ, v) = ⊥\nWe also consider the extended transition function\nδ∗∗: Q × Σ∗× Γ+ →Q × Γ∗,\ndefined for all q ∈Q, a ∈Γ, v ∈Γ∗, w ∈Σ∗, and b ∈Σ by\nδ∗∗(q, λ, av) = (q, av)\n5"},{"paragraph_id":"p6","order":6,"text":"δ∗∗(q, wb, av) = δ(δi\nQ(eq, λ, eaev), b, δi\nΓ∗(eq, λ, eaev))\nif δ∗∗(q, w, av) = (eq, eaev), δi(eq, λ, eaev) ̸=⊥and δi+1(eq, λ, eaev) =⊥, i ≤c.\nThat is, λ-transitions are inside the definition of δ∗∗(q, b, av), for b ∈Σ. Notice that\nδ∗∗is not defined on an empty stack string, therefore av needs to be long enough in order\nthat δ∗∗(q, b, av) ̸=⊥.\nWe denote δ∗∗by δ, and δ(q0, w, z0) by δ(w). We write δ = (δQ, δΓ∗) for simplicity.\nWe also consider the usual extension of β\nβ∗: Q × Γ+ →∆Q(Σ),\ndefined for all q ∈Q, a ∈Γ, and v ∈Γ∗by\nβ∗(q, av) = β(q, a),\nand denote β∗by β.\nWe use BPDG to compute martingales. Intuitively, suppose a BPDG G is to bet on\nsequence S has already bet on w ❁S, with current capital x ∈Q, current state q ∈Q and\ncurrent top stack symbol a. Then for b ∈Σ, G bets the quantity xβ(q, a)(b) of its capital\nthat the next symbol of S is b. If the bet is correct (that is, if wb ❁S) and since payoffs\nare fair, G has capital |Σ|xβ(q, a)(b). Formally,\nDefinition.\nLet G = (Q, Σ, Γ, δ, β, q0, z0, c) be a bounded pushdown gambler.\nThe\nmartingale of G is the function\ndG : Σ∗→[0, ∞)\ndefined by the recursion\ndG(λ) = 1\ndG(wb) = |Σ|dG(w)β(δ(w))(b)\nfor all w ∈Σ∗and b ∈Σ.\nBy Observation 3.1, a BPDG G actually yields an s-gale for every s ∈[0, ∞). We call\nit the s-gale of G, and denote it by\nds\nG(w) = |Σ|(s−1)|w|dG(w).\nA bounded pushdown s-gale is an s-gale d for which there exists a BPDG such that ds\nG = d.\nThe first two properties of BPD gamblers are that any number of λ-transitions can\nbe replaced by a single λ-transition and that the stack alphabet does not give additional\npower.\nProposition 3.3 Let G = (Q, Σ, Γ, δ, β, q0, z0, c) be a BPDG. Then there is a BPDG G′ =\n(Q′, Σ, Γ′, δ′, β′, q′\n0, z′\n0, 1) such that dG = dG′.\nFrom now on we shall assume that the maximum number of λ-transitions c is 1.\nProposition 3.4 Let G = (Q, Σ, Γ, δ, β, q0, z0, c) be a BPDG. Then there is a BPDG G′ =\n(Q′, Σ, {0, 1, z′\n0}, δ′, β′, q′\n0, z′\n0, c′) such that dG = dG′.\n6"},{"paragraph_id":"p7","order":7,"text":"Let us define bounded pushdown dimension. Intuitively, the BPD dimension of a se-\nquence is the smallest s such that there is a BPD-s-gale that succeeds on the sequence.\nDefinition.\nThe bounded pushdown dimension of a set X ⊆Σ∞is\ndimBPD(X) = inf{s | there is a bounded pushdown s −gale d such that X ⊆S∞[d]}.\n4\nDimension and compression\nIn this section we characterize the bounded pushdown dimension of individual sequences\nin terms of bounded pushdown compressibility, therefore BPD dimension is a natural and\nrobust definition.\nDefinition. A bounded pushdown compressor (BPDC) is an 8-tuple\nC = (Q, Σ, Γ, δ, ν, q0, z0, c)\nwhere\n• Q is a finite set of states,\n• Σ is the finite input and output alphabet,\n• Γ is the finite stack alphabet,\n• δ : Q × (Σ ∪{λ}) × Γ →Q × Γ∗is the transition function,\n• ν : Q × Σ × Γ →Σ∗is the output function,\n• q0 ∈Q is the initial state,\n• z0 ∈Γ is the start stack symbol,\n• c ∈N is a constant such that the number of λ-transitions per input symbol is at most\nc,\nwith the two additional restrictions:\n1. for each q ∈Q and a ∈Γ at least one of the following holds\n• δ(q, λ, a) =⊥\n• δ(q, b, a) =⊥for all b ∈Σ\n2. for every q ∈Q, b ∈Σ ∪{λ}, either δ(q, b, z0) =⊥, or δ(q, b, z0) = (q′, vz0), where q′\n∈Q and v ∈Γ∗.\n7"},{"paragraph_id":"p8","order":8,"text":"We extend δ to δ∗∗: Q×Σ∗×Γ+ →Q×Γ∗as before, and denote δ∗∗by δ and δ(q0, w, z0)\nby δ(w).\nFor q ∈Q, w ∈Σ∗and z ∈Γ+, we define the output from state q on input w reading\nz on the top of the stack to be the string ν∗(q, w, z) (denoted by ν(q, w, z)) with\nν(q, λ, z) = λ\nν(q, wb, z) = ν(q, w, z)ν(δQ(q, w, z), b, δΓ∗(q, w, z))\nfor w ∈Σ∗and b ∈Σ. We then define the output of C on input w ∈Σ∗to be the string\nC(w) = ν(q0, w, z0).\nWe can restrict λ-transitions to a single one and the stack alphabet to three symbols.\nProposition 4.1 Let C = (Q, Σ, Γ, δ, ν, q0, z0, c) be a BPDC. Then there is a BPDC C′ =\n(Q′, Σ, Γ′, δ′, ν′, q′\n0, z′\n0, 1) such that C(w) = C′(w) for every w ∈Σ∗.\nProposition 4.2 Let C = (Q, Σ, Γ, δ, ν, q0, z0, c) be a BPDC. Then there is a BPDC C′ =\n(Q′, Σ, {0, 1, z′\n0}, δ′, ν′, q′\n0, z′\n0, c′) such that C(w) = C′(w) for every w ∈Σ∗.\nWe are interested in information lossless compressors, that is, w must be recoverable\nfrom C(w) and the final state.\nDefinition.\nA BPDC C = (Q, Σ, Γ, δ, ν, q0, z0) is information-lossless (IL) if the function\nΣ∗→Σ∗× Q\nw →(C(w), δQ(w))\nis one-to-one.\nAn information-lossless bounded pushdown compressor (ILBPDC) is a\nBPDC that is IL.\nIntuitively, a BPDC compresses a string w if |C(w)| is significantly less than |w|. Of\ncourse, if C is IL, then not all strings can be compressed. Our interest here is in the\ndegree (if any) to which the prefixes of a given sequence S ∈Σ∞can be compressed by an\nILBPDC.\nDefinition.\nIf C is a BPDC and S ∈Σ∞, then the compression ratio of C on S is\nρC(S) = lim inf\nn→∞\n|C(S[0..n −1])|\nn\n.\nThe BPD compression ratio of a sequence is the best compression ratio achievable by\nan ILBPDC, that is\nDefinition.\nThe bounded pushdown compression ratio of a sequence S ∈Σ∞is\nρBPD(S) = inf{ρC(S) | C is a ILBPDC}.\nThe main result in this section states that the BPD dimension of a sequence and its\nILBPD compression ratio are the same, therefore BPD dimension is the natural concept\nof density of information in the BPD setting.\n8"},{"paragraph_id":"p9","order":9,"text":"Theorem 4.3 For all S ∈Σ∞,\ndimBPD(S) = ρBPD(S).\n5\nSeparating LZ from BPD\nIn this section we prove that BPD compression can be much better than the compression\nattained with the celebrated Lempel-Ziv algorithm.\nWe start with a brief description of the LZ algorithm [14].\nWe finish relating BPD dimension (and compression) with the Lempel-Ziv algorithm.\nGiven an input x ∈Σ∗, LZ parses x in different phrases xi, i.e., x = x1x2 . . . xn (xi ∈Σ∗)\nsuch that every prefix y ❁xi, appears before xi in the parsing (i.e. there exists j < i s.t.\nxj = y). Therefore for every i, xi = xl(i)bi for l(i) < i and bi ∈Σ. We sometimes denote\nthe number of phrases in the parsing of x as C(x).\nLZ encodes xi by a prefix free encoding of l(i) and the symbol bi, that is, if x =\nx1x2 . . . xn as before, the output of LZ on input x is\nLZ(x) = cl(1)b1cl(2)b2 . . . cl(n)bn\nwhere ci is a prefix-free coding of i (and x0 = λ).\nLZ is usually restricted to the binary alphabet, but the description above is valid for\nany Σ.\nFor a sequence S ∈Σ∞, the LZ compression ratio is given by\nρLZ(S) = lim inf\nn→∞\n|LZ(S[0 . . . n −1])|\nn\n.\nIt is well known that LZ [14] yields a lower bound on the finite-state dimension (or finite-\nstate compressibility) of a sequence [14], ie, LZ is universal for finite-state compressors.\nThe following result shows that this is not true for BPD (hence PD) dimension, in a\nstrong sense: we construct a sequence S that cannot be compressed by LZ, but that has\nBPD compression ratio less than 1\n2.\nTheorem 5.1 For every m ∈N, there is a sequence S ∈{0, 1}∞such that\nρLZ(S) > 1 −1\nm\nand\ndimBPD(S) ≤1\n2.\nAs a corollary we obtain a separation of finite-state dimension and bounded pushdown\ndimension. A similar result between finite-state dimension and pushdown dimension was\nproved in [5].\n9"},{"paragraph_id":"p10","order":10,"text":"Corollary 5.2 For any m ∈N, there exists a sequence S ∈{0, 1}∞such that\ndimFS(S) > 1 −1\nm\nand\ndimBPD(S) ≤1\n2.\nConclusion\nWe have introduced Bounded Pushdown dimension, characterized it with compression and\ncompared it with Lempel-Ziv compression. It is open if there is a BPD compressor that\nis universal for Finite-State compressors, which is true for the Lempel-Ziv algorithm, and\nwhether Lempel-Ziv compression can surpass BPD-compression for some sequence.\nReferences\n[1] K. B. Athreya, J. M. Hitchcock, J. H. Lutz, and E. Mayordomo. Effective strong\ndimension in algorithmic information and computational complexity. SIAM Journal\non Computing. To appear.\n[2] Chris Bourke, John M. Hitchcock, and N. V. Vinodchandran.\nEntropy rates and\nfinite-state dimension. Theor. Comput. Sci., 349(3):392–406, 2005.\n[3] Jack J. Dai, James I. Lathrop, Jack H. Lutz, and Elvira Mayordomo. Finite-state\ndimension. Theoretical Computer Science, 310(1–3):1–33, January 2004.\n[4] D. Doty and P. Moser. Personal communication, based on [13]. 2006.\n[5] David Doty and Jared Nichols. Pushdown dimension. Theoretical Computer Science.\nTo appear.\n[6] K. Falconer. The Geometry of Fractal Sets. Cambridge University Press, 1985.\n[7] J. M. Hitchcock, J. H. Lutz, and E. Mayordomo. The fractal geometry of complexity\nclasses. SIGACT News Complexity Theory Column, 36:24–38, 2005.\n[8] D. A. Huffman. Canonical forms for information-lossless finite-state logical machines.\nTrans. Circuit Theory, pages 41–59, 1959.\n[9] J. H. Lutz. Dimension in complexity classes. SIAM Journal on Computing, 32:1236–\n1259, 2003.\n[10] J. H. Lutz. The dimensions of individual strings and sequences. Information and\nComputation, 187:49–79, 2003.\n10"},{"paragraph_id":"p11","order":11,"text":"[11] J. H. Lutz. Effective fractal dimensions.\nMathematical Logic Quarterly, 51:62–72,\n2005.\n[12] E. Mayordomo. Effective fractal dimension in algorithmic information theory. In New\nComputational Paradigms: Changing Conceptions of What is Computable. Springer-\nVerlag, 2007. To appear.\n[13] D. Sheinwald, A. Lempel, and J. Ziv. On compression with two-way head machines.\nIn Data Compression Conference, pages 218–227, 1991.\n[14] Jacob Ziv and Abraham Lempel. Compression of individual sequences via variable-\nrate coding. IEEE Transactions on Information Theory, 24(5):530–536, 1978.\n11"},{"paragraph_id":"p12","order":12,"text":"Technical Appendix\nThis appendix is devoted to proving Theorem 4.3 and Theorem 5.1. For the first one,\nwe need the following:\nA\nProof of Theorem 4.3\nDefinition.\nA BPDG G = (Q, Σ, Γ, δ, β, q0, z0) is nonvanishing if 0 < β(q, z)(b) < 1 for\nall q ∈Q, b ∈Σ and z ∈Γ.\nLemma A.1 For every BPDG G and each ε > 0, there is a nonvanishing BPDG G′ such\nthat for all w ∈Σ∗, dG′(w) ≥|Σ|−ε|w|dG(w).\nProof of Lemma A.1 . Let G = (Q, Σ, δ, β, q0, Γ, z0) be a BPDG, and let ε > 0. For\neach q ∈Q, z ∈Γ, b ∈Σ,\n1 −|Σ|−ε X\nb∈Σ\nβ(q, z)(b) = 1 −|Σ|−ε > 0,\nso we can fix a rational β′(q, z)(b) such that\n|Σ|−εβ(q, z)(b) < β′(q, z)(b) < 1 −|Σ|−ε\nX\na∈Σ,a̸=b\nβ(q, z)(a)\nand\nX\nb∈Σ\nβ′(q, z)(b) = 1.\nThen, 0 < β′(q, z)(b) < 1 for each q ∈Q, b ∈Σ and z ∈Γ, therefore the BPDG G′ =\n(Q, Σ, δ, β′, q0, Γ, z0) is nonvanishing.\nAlso, for all q ∈Q, b ∈Σ, z ∈Γ,\nβ′(q, z)(b) ≥|Σ|−εβ(q, z)(b)\nso for all w ∈Σ∗, dG′(w) ≥|Σ|−ε|w|dG(w).\n✷\nProof of Theorem 4.3 Let S ∈Σ∞. For each n ∈N, let wn = S[0..n −1].\nTo see that dimBPD(S) ≤ρBPD(S), let s > s′ > ρBPD(S).\nIt suffices to show that\ndimBPD(S) ≤s. By our choice of s′, there is an 1-ILBPDC C = (Q, Σ, Γ, δ, ν, q0, z0) for\nwhich the set\nI = {n ∈N | |C(wn)| < s′n}\nis infinite.\n1"},{"paragraph_id":"p13","order":13,"text":"CONSTRUCTION A.1 Given a 1-bounded pushdown compressor (BPDC)\nC = (Q, Σ, Γ, δ, ν, q0, z0), and k ∈Z+ , we construct the 1-bounded pushdown gambler\n(BPDG) G = G(C, k) = (Q′, Σ, Γ′, δ′, β′, q′\n0, z′\n0) as follows:\ni) Q′ = Q × {0, 1, . . . , k −1}\nii) q′\n0 = (q0, 0)\niii) Γ′ =\n4k−1\nS\ni=2k\nΓi\niv) z′\n0 = z2k\n0\nv) ∀(q, i) ∈Q′, b ∈Σ, a ∈Γ′,\nδ′((q, i), b, a) ="},{"paragraph_id":"p14","order":14,"text":"δQ(q, b, a), (i + 1) mod k"},{"paragraph_id":"p15","order":15,"text":",\n\\\nδΓ∗(q, b, a)"},{"paragraph_id":"p16","order":16,"text":"where for each z ∈(Γ′)+, z ∈Γ+ is the Γ-string obtained by concatenating the symbols of\nz, and for each y ∈Γ+, if y = y1y2 · · · y2kl+n with n < 2k, then by ∈(Γ′)+ is such that\nby1 = y1 · · · y2k+n, by2 = y2k+n+1 · · · y4k+n, . . . , byl = y2k(l−1)+n+1 · · · y2kl+n.\nvi) ∀(q, i) ∈Q′, a ∈Γ′, b ∈Σ\nβ′((q, i), a)(b) = σ(q, bΣk−i−1, a)\nσ(q, Σk−i, a)\nwhere σ(q, A, a) = P\nx∈A\n|Σ|−|ν(q,x,a)| .\nLemma A.2 In Construction A.1, if |w| is a multiple of k and u ∈Σ≤k, then\ndG(wu) = |Σ||u|−|ν(δQ(w),u,δΓ∗(w))| σ(δQ(wu), Σk−|u|, \\\nδΓ∗(wu))\nσ(δQ(w), Σk, \\\nδΓ∗(w))\ndG(w).\nProof of Lemma A.2. We use induction on the string u. If u = λ, the lemma is\nclear. Assume that it holds for u, where u ∈Σ<k, and let b ∈Σ. Then\ndG(wub) = |Σ|σ(δQ(wu), bΣk−|u|−1, \\\nδΓ∗(wu))\nσ(δQ(wu), Σk−|u|, \\\nδΓ∗(wu))\ndG(wu)\n= |Σ|1−|ν(δQ(wu),b,δΓ∗(wu))|σ(δQ(wub), Σk−|u|−1,\n\\\nδΓ∗(wub))\nσ(δQ(wu), Σk−|u|, \\\nδΓ∗(wu))\ndG(wu)\nso by the induction hypothesis the lemma holds for ub.\n2"},{"paragraph_id":"p17","order":17,"text":"✷\nLemma A.3 In Construction A.1, if w = w0w1 · · ·wn−1, where each wi ∈Σk , then\ndG(w) =\n|Σ||w|−|C(w)|\nn−1\nQ\ni=0\nσ(δQ(w0 · · ·wi−1), Σk,\n\\\nδΓ∗(w0 · · · wi−1))\n.\nProof of Lemma A.3. We use induction on n. For n = 0, the identity is clear.\nAssume that it holds for w = w0w1 · · · wn−1, with each wi ∈Σk, and let w′ = w0w1 · · · wn.\nThen Lemma A.2 with u = wn tells us that\ndG(w′) = |Σ|k−|ν(δQ(w),wn,δΓ∗(w))|\nσ(δQ(w), Σk, \\\nδΓ∗(w))\ndG(w)\nwhence the identity holds for w′ by the induction hypothesis.\n✷\nLemma A.4 In Construction A.1, if C is IL and |w| is a multiple of k, then\ndG(w) ≥|Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1),\nwhere l = ⌈log |Q|⌉and m = max{|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}.\nProof of Lemma A.4. We prove that for each z ∈Σ∗,\nσ(δQ(z), Σk, \\\nδΓ∗(z)) ≤|Σ|l+log m+log k+1.\nTo see this, fix z ∈Σ∗and observe that at most |Q| strings w ∈Σk can have the same\noutput from state δQ(z) with stack content δΓ∗(z). Therefore, the number of w ∈Σk for\nwhich |ν(δQ(z), w, δΓ∗(z))| = j does not exceed |Q||Σ|j. Hence\nσ(δQ(z), Σk, \\\nδΓ∗(z)) =\nX\nw∈Σk\n|Σ|−|ν(δQ(z),w,δΓ∗(z))| ≤\nmk\nX\nj=0\n|Q||Σ|j|Σ|−j = |Q|(mk + 1)\n≤|Σ|l+log m+log k+1.\nIt follows by Lemma A.3 that\ndG(w) = |Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1).\n✷\nLemma A.5 In Construction A.1, if C is IL, then for all w ∈Σ∗,\ndG(w) ≥|Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1)−(km+l+log m+log k+1),\nwhere l = ⌈log |Q|⌉and m = max {|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}.\n3"},{"paragraph_id":"p18","order":18,"text":"Proof of Lemma A.5. Assume the hypothesis, let l and m be as given, and let w ∈\nΣ∗. Fix 0 ≤j < k such that |w| + j is divisible by k. By Lemma A.4 we have\ndG(w) ≥|Σ|−jdG(w0j)\n≥|Σ|−j+|w0j|−|C(w0j)|−|w0j |\nk\n(l+log m+log k+1)\n= |Σ||w|−|C(w0j)|−|w|\nk (l+log m+log k+1)−j\nk (l+log m+log k+1)\n≥|Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1)−(km+l+log m+log k+1)\n✷\nLet l = ⌈log |Q|⌉and m = max{|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}, and fix k ∈Z+ such\nthat l+log m+log k+1\nk\n< s −s′. Let G = G(C, k) be as in Construction A.1. Then, by Lemma\nA.5, for all n ∈I we have\nd(s)\nG (wn) ≥|Σ|sn−|C(wn)|−n\nk (l+log m+log k+1)−(km+l+log m+log k+1)\n≥|Σ|(s−s′−l+log m+log k+1\nk\n)n−(km+l+log m+log k+1)\nSince s −s′ −l+log m+log k+1\nk\n> 0, this implies that S ∈S∞[d(s)\nG ].\nThus, dimBPD(S) ≤s.\nTo see that ρBPD(S) ≤dimBPD(S), let s > s′ > s′′ > dimBPD(S). It suffices to show\nthat ρBPD(S) ≤s. By our choice of s′′, there is a 1-BPDG G such that the set\nJ = {n ∈N | ds′′\nG (wn) ≥1}\nis infinite. By Lemma A.1 there is a nonvanishing 1-BPDG eG such that\nd e\nG(w) ≥|Σ|(s′′−s′)|w|dG(w) for all w ∈Σ∗.\nCONSTRUCTION A.2 Let G = (Q, Σ, Γ, δ, β, q0, z0) be a nonvanishing 1-BPDG, and\nlet k ∈Z+. For each z ∈Γ∗(long enough for dGq,z(w) to be defined for all w ∈Σk)\nand q ∈Q, let Gq,z = (Q, Σ, Γ, δ, β, q, z), and define pq,z : Σk →[0, 1] by pq,z(w) =\n|Σ|−kdGq,z(w). Since G is nonvanishing and each dGq,z is a martingale with dGq,z(λ) = 1,\neach of the functions pq,z is a positive probability measure on Σk. For each z ∈Γ∗, q ∈Q, let\nΘq,z : Σk →Σ∗be the Shannon-Fano-Elias code given by the probability measure pq,z. Then\n|Θq,z(w)| = lq,z(w)\nlq,z(w) = 1 + ⌈log\n1\npq,z(w)⌉\nfor all q ∈Q and w ∈Σk, and each of the sets range(Θq,z) is an instantaneous code. We\ndefine the 1-BPDC C = C(G, k) = (Q′, Σ, Γ′, δ′, ν′, q′\n0, z′\n0) whose components are as follows:\ni) Q′ = Q × Σ<k\n4"},{"paragraph_id":"p19","order":19,"text":"ii) q′\n0 = (q0, λ)\niii) Γ′ =\n4k−1\nS\ni=2k\nΓi\niv) z′\n0 = z2k\n0\nv) ∀(q, w) ∈Q′, b ∈Σ, a ∈Γ′,\nδ′((q, w), b, a) =\n(\n(q, wb, a)\nif |w| < k −1,\n(δQ(q, wb, a), λ,\n\\\nδΓ∗(q, wb, a))\nif |w| = k −1.\nvi) ∀(q, w) ∈Q′, b ∈Σ, a ∈Γ′,\nν′((q, w), b, a) =\n λ\nif |w| < k −1,\nΘq,a(wb)\nif |w| = k −1.\nSince each range(Θq,z) is an instantaneous code, it is easy to see that the BPDC C =\nC(G, k) is IL.\nLemma A.6 In Construction A.2, if |w| is a multiple of k, then\n|C(w)| ≤"},{"paragraph_id":"p20","order":20,"text":"1 + 2\nk"},{"paragraph_id":"p21","order":21,"text":"|w| −log dG(w).\nProof of Lemma A.6.\nLet w = w0w1 · · · wn−1, where each wi ∈Σk.\nFor each\n0 ≤i < n, let qi = δQ(w0 · · · wi−1) and zi = δΓ∗(w0 · · · wi−1). Then,\n|C(w)| =\nn−1\nX\ni=0\nlqi,zi(wi)\n=\nn−1\nX\ni=0"},{"paragraph_id":"p22","order":22,"text":"1 + ⌈log\n1\npqi,zi(wi)⌉"},{"paragraph_id":"p23","order":23,"text":"≤\nn−1\nX\ni=0"},{"paragraph_id":"p24","order":24,"text":"2 + log\n1\npqi,zi(wi)"},{"paragraph_id":"p25","order":25,"text":"=\nn−1\nX\ni=0"},{"paragraph_id":"p26","order":26,"text":"2 + log\n|Σ|k\ndGqi,zi(wi)"},{"paragraph_id":"p27","order":27,"text":"= (k + 2)n −log\nn−1\nY\ni=0\ndGqi,zi(wi)\n= (k + 2)n −log dG(w) = (1 + 2\nk)|w| −log dG(w)\n✷\nLemma A.7 In Construction A.2, for all w ∈Σ∗,\n|C(w)| ≤"},{"paragraph_id":"p28","order":28,"text":"1 + 2\nk"},{"paragraph_id":"p29","order":29,"text":"|w| −log dG(w).\n5"},{"paragraph_id":"p30","order":30,"text":"Proof of Lemma A.7. If |w| is multiple of k, then we apply the Lemma A.6.\nOtherwise, let w = w′z, where |w′| is a multiple of k and |z| = j, 0 < j < k.\nThen, Lemma A.6 tell us that\n|C(w)| = |C(w′)|\n≤"},{"paragraph_id":"p31","order":31,"text":"1 + 2\nk"},{"paragraph_id":"p32","order":32,"text":"|w′| −log dG(w′)\n≤"},{"paragraph_id":"p33","order":33,"text":"1 + 2\nk"},{"paragraph_id":"p34","order":34,"text":"|w′| −log(|Σ|−jdG(w))\n="},{"paragraph_id":"p35","order":35,"text":"1 + 2\nk"},{"paragraph_id":"p36","order":36,"text":"|w| −log dG(w) −2j\nk\n≤"},{"paragraph_id":"p37","order":37,"text":"1 + 2\nk"},{"paragraph_id":"p38","order":38,"text":"|w| −log dG(w).\n✷\nFix k >\n2\ns−s′, and let C = C( eG, k) be as in Construction A.2. Then Lemma A.7 tell us\nthat for all n ∈J,\n| C(wn) | ≤"},{"paragraph_id":"p39","order":39,"text":"1 + 2\nk"},{"paragraph_id":"p40","order":40,"text":"n −log d e\nG(wn)\n≤"},{"paragraph_id":"p41","order":41,"text":"1 + 2\nk + s′ −s′′ \nn −log dG(wn)\n≤\n 2\nk + s′ \nn −log ds′′\nG (wn)\n≤\n 2\nk + s′ \nn\n< sn.\nThus, ρBPD(S) ≤s.\n✷\nB\nProof of Theorem 5.1\nFor a string x, x−1 denotes x written in reverse order.\nProof of Theorem 5.1 Let m ∈N, and let k = k(m) be an integer to be determined\nlater. For any integer n, let Tn denote the set of strings x of size n such that 1j does not\nappear in x, for every j ≥k. Since Tn contains {0, 1}k−1 × {0} × {0, 1}k−1 × {0} . . . (i.e.\nthe set of strings whose every kth bit is zero), it follows that |Tn| ≥2an, where a = 1−1/k.\nRemark B.1 For every string x ∈Tn there is a string y ∈Tn−1 and a bit b such that\nyb = x.\n6"},{"paragraph_id":"p42","order":42,"text":"Let An = {a1, . . . au} be the set of palindromes in Tn. Since fixing the n/2 first bits\nof a palindrome (wlog n is even) completely determines it, it follows that |An| ≤2\nn\n2 .\nLet us separate the remaining strings in Tn −An into two sets Xn = {x1, . . . xt} and\nYn = {y1, . . . yt} with (xi)−1 = yi for every 1 ≤i ≤t. Let us choose X, Y such that x1 and\nyt start with a zero. We construct S in stages. For n ≤k −1, Sn is an enumeration of all\nstrings of size n in lexicographical order. For n ≥k,\nSn = a1 . . . au 12n x1 . . . xt 12n+1 yt . . . y1\ni.e.\na concatenation of all strings in An (the A zone of Sn) followed by a flag of 2n\nones, followed by the concatenations of all strings in X (the X-zone) and Y (the Y zone)\nseparated by a flag of 2n + 1 ones. Let\nS = S1S2 . . . Sk−1 1k 1k+1 . . . 12k−1 SkSk+1 . . .\ni.e. the concatenation of the Sj’s with some extra flags between Sk−1 and Sk. We claim\nthat the parsing of Sn (n ≥k) by LZ, is as follows:\nSn = a1, . . . , au, 12n, x1, . . . , xt, 12n+1, yt, . . . , y1.\nIndeed after S1, . . . Sk−1 1k 1k+1 . . . 12k−1, LZ has parsed every string of size ≤k −1 and\nthe flags 1k 1k+1 . . . 12k−1. Together with Remark B.1, this guarantees that LZ parses Sn\ninto phrases that are exactly all the strings in Tn and the two flags 12n, 12n+1.\nLet us compute the compression ratio ρLZ(S). Let n, i be integers. By construction of\nS, LZ encodes every phrase in Si (except the two flags), by a phrase in Si−1 (plus a bit).\nIndexing a phrase in Si−1 requires a codeword of length at least logarithmic in the number\nof phrase parsed before, i.e. log(C(S1S2 . . . Si−2)). Since C(Si) ≥|Ti| ≥2ai, it follows\nC(S1 . . . Si−2) ≥\ni−2\nX\nj=1\n2aj = 2a(i−1) −2a\n2a −1\n≥b2a(i−1)\nwhere b = b(a) is arbitrarily close to 1. Letting ti = |Ti|, the number of bits output by LZ\non Si is at least\nC(Si) log C(S1 . . . Si−2) ≥ti log b2a(i−1)\n≥cti(i −1)\nwhere c = c(b) is arbitrarily close to 1. Therefore\n|LZ(S1 . . . Sn)| ≥\nn\nX\nj=1\nctj(j −1)\nSince |S1 . . . Sn| ≤2k2 + Pn\nj=1(jtj + 4j), (the two flags plus the extra flags between Sk−1\nand Sk) the compression ratio is given by\nρLZ(S1 . . . Sn) ≥c\nPn\nj=1 tj(j −1)\n2k2 + Pn\nj=1 j(tj + 4)\n(2)\n= c −c\n2k2 + Pn\nj=1(tj + 4j)\n2k2 + Pn\nj=1 j(tj + 4)\n(3)\n7"},{"paragraph_id":"p43","order":43,"text":"The second term in Equation 3 can be made arbitrarily small for n large enough: Let\nM ≤n, we have\n2k2 +\nn\nX\nj=1\nj(tj + 4) ≥2k2 +\nM\nX\nj=1\njtj + (M + 1)\nn\nX\nj=M+1\ntj\n= 2k2 +\nM\nX\nj=1\njtj + M\nn\nX\nj=M+1\ntj +\nn\nX\nj=M+1\ntj\n≥2k2 +\nM\nX\nj=1\njtj + M\nn\nX\nj=M+1\ntj +\nn\nX\nj=M+1\n2aj\n≥2k2 +\nM\nX\nj=1\njtj + M\nn\nX\nj=M+1\ntj + 2an\n≥M\nn\nX\nj=M+1\ntj + M(2k2 + 2n(n + 1) +\nM\nX\nj=1\ntj)\nfor n big enough\n= M(2k2 +\nn\nX\nj=1\ntj + 4\nn\nX\nj=1\nj)\nHence\nρLZ(S1 . . . Sn) ≥c −c\nM\nwhich by definition of c, M can be made arbitrarily close to 1 by choosing k accordingly,\ni.e\nρLZ(S1 . . . Sn) ≥1 −1\nm.\nLet us show that dimBPD(S) ≤1\n2. Consider the following BPD martingale d. Informally,\nd on Sn goes through the An zone until the first flag, then starts pushing the whole X\nzone onto its stack until it hits the second flag. It then uses the stack to bet correctly on\nthe whole Y zone. Since the Y zone is exactly the X zone written in reverse order, d is\nable to double its capital on every bit of the Y zone. On the other zones, d does not bet.\nBefore giving a detailed construction of d, let us compute the upper bound it yields on\n8"},{"paragraph_id":"p44","order":44,"text":"dimBPD(S).\ndimBPD(S) ≤1 −lim sup\nn→∞\nlog d(S1 . . . Sn)\n|S1 . . . Sn|\n≤1 −lim sup\nn→∞\nPn\nj=1 |Yj|\n2k2 + Pn\nj=1(j|Tj| + 4j)\n≤1 −lim sup\nn→∞\nPn\nj=1 j |Tj|−|Aj|\n2\n2k2 + Pn\nj=1(j|Tj| + 4j)\n≤1\n2 + 1\n2 lim sup\nn→∞\n2k2 + Pn\nj=1(j|Aj| + 4j)\n2k2 + Pn\nj=1(j|Tj| + 4j) .\nSince\nlim sup\nn→∞\n2k2 + Pn\nj=1(j|Aj| + 4j)\n2k2 + Pn\nj=1(j|Tj| + 4j) ≤lim sup\nn→∞\nPn\nj=1 j(|Aj| + 4 + 2k2)\nPn\nj=1 |Tj|\n≤lim sup\nn→∞\nPn\nj=1 j(2\nj\n2 + 2\nj\n4)\nPn\nj=1 2aj\n≤lim sup\nn→∞\nn2\n3n\n4\n2an\n= 0.\nIt follows that\ndimBPD(S) ≤1\n2.\nLet us give a detailed description of d. Let Q be the following set of states:\n• The start state q0, and q1, . . . qv the “early” states that will count up to\nv = |S1S2 . . . Sk−1 1k 1k+1 . . . 12k−1|.\n• qa\n0, . . . , qa\nk the A zone states that cruise through the A zone until the first flag.\n• q1f the first flag state.\n• qX\n0 , . . . , qX\nk the X zone states that cruise through the X zone, pushing every bit on\nthe stack, until the second flag is met.\n• qr\n0, . . . , qr\nk which after the second flag is detected, pop k symbols from the stack that\nwere erroneously pushed while reading the second flag.\n• q2f the second flag state.\n• qb the betting on zone Y state.\n9"},{"paragraph_id":"p45","order":45,"text":"Let us describe the transition function δ : Q × {0, 1} × {0, 1} →Q × {0, 1}. First δ counts\nuntil v i.e. for i = 0, . . . v −1\nδ(qi, x, y) = (qi+1, y)\nfor any x, y\nand after reading v bits, it enters in the first A zone state, i.e. for any x, y\nδ(qv, x, y) = (qa\n0, y).\nThen δ skips through A until the string 1k is met, i.e. for i = 0, . . . k −1 and any x, y\nδ(qa\ni , x, y) =\n(\n(qa\ni+1, y)\nif x = 1\n(qa\n0, y)\nif x = 0\nand\nδ(qa\nk, x, y) = (q1f, y).\nOnce 1k has been seen, δ knows the first flag has started, so it skips through the flag until\na zero is met, i.e. for every x, y\nδ(q1f, x, y) =\n(\n(q1f, y)\nif x = 1\n(qX\n0 , 0y)\nif x = 0\nwhere state qX\n0 means that the first bit of the X zone (a zero bit) has been read, therefore\nδ pushes a zero. In the X zone, delta pushes every bit it sees until it reads a sequence of\nk ones, i.e until the start of the second flag, i.e for i = 0, . . . k −1 and any x, y\nδ(qX\ni , x, y) =\n(\n(qX\ni+1, xy)\nif x = 1\n(qX\n0 , xy)\nif x = 0\nand\nδ(qX\nk , x, y) = (qr\n0, y).\nAt this point, δ has pushed all the X zone on the stack, followed by k ones. The next step\nis to pop k ones, i.e for i = 0, . . . k −1 and any x, y\nδ(qr\ni , x, y) = (qr\ni+1, λ)\nand\nδ(qr\nk, x, y) = (q2f\n0 , y).\nAt this stage, δ is still in the second flag (the second flag is always bigger than 2k) therefore\nit keeps on reading ones until a zero (the first bit of the Y zone) is met. For any x, y\nδ(q2f, x, y) =\n(\n(q2f, y)\nif x = 1\n(qb, λ)\nif x = 0.\n10"},{"paragraph_id":"p46","order":46,"text":"On the last step δ has read the first bit of the Y zone, therefore it pops it. At this stage,\nthe stack exactly contains the Y zone (i.e. the X zone written in reverse order) except\nthe first bit; δ thus uses its stack to bet and double its capital on every bit in the Y zone.\nOnce the stack is empty, a new A zone begins. Thus, for any x, y\nδ(qb, x, y) = (qb, λ).\nand\nδ(qb, x, z0) =\n(\n(qa\n1, z0)\nif x = 1\n(qa\n0, z0)\nif x = 0.\nThe betting function is equal to 1/2 everywhere (i.e no bet) except on state qb, where\nβ(qb, y)(z) =\n(\n1\nif y = z\n0\nif y ̸= z.\nand β stops betting once start stack symbol is met, i.e.\nβ(qb, z0) = 1\n2.\n⊓⊔\n11"}],"pages":[{"page":1,"text":"arXiv:0704.2386v1 [cs.CC] 18 Apr 2007\nBounded Pushdown dimension vs Lempel Ziv\ninformation density\nPilar Albert, Elvira Mayordomo, and Philippe Moser ∗\nAbstract\nIn this paper we introduce a variant of pushdown dimension called bounded push-\ndown (BPD) dimension, that measures the density of information contained in a\nsequence, relative to a BPD automata, i.e. a finite state machine equipped with an\nextra infinite memory stack, with the additional requirement that every input symbol\nonly allows a bounded number of stack movements. BPD automata are a natural\nreal-time restriction of pushdown automata. We show that BPD dimension is a ro-\nbust notion by giving an equivalent characterization of BPD dimension in terms of\nBPD compressors. We then study the relationships between BPD compression, and\nthe standard Lempel-Ziv (LZ) compression algorithm, and show that in contrast to\nthe finite-state compressor case, LZ is not universal for bounded pushdown compres-\nsors in a strong sense: we construct a sequence that LZ fails to compress significantly,\nbut that is compressed by at least a factor 2 by a BPD compressor. As a corollary\nwe obtain a strong separation between finite-state and BPD dimension.\nKeywords\nInformation lossless compressors, finite state (bounded pushdown) dimension, Lempel-Ziv\ncompression algorithm.\n1\nIntroduction\nEffective versions of fractal dimension have been developed since 2000 [9, 10] and used\nfor the quantitative study of complexity classes, information theory and data compression,\nand back in fractal geometry (see recent surveys in [11, 7, 12]). Here we are interested\nin information theory and data compression, where it is known that for several different\n∗Dept. de Inform ́atica e Ingenier ́ıa de Sistemas , Universidad de Zaragoza. Edificio Ada Byron, Mar ́ıa\nde Luna 1 - E-50018 Zaragoza (Spain). Email: {mpalbert, elvira}@unizar.es and mosersan@gmail.com.\nResearch supported in part by Spanish Government MEC Project TIN 2005-08832-C03-02, by Arag ́on\nGovernment Dept. Ciencia, Tecnolog ́ıa y Universidad, subvenci ́on destinada a la formaci ́on de personal\ninvestigador-B068/2006 and by Spanish Government MEC Program Juan de la Cierva.\n1"},{"page":2,"text":"bounds on the computing power, effective dimensions capture what can be considered the\ninherent information content of a sequence in the corresponding setting [12]. In the today\nrealistic context of massive data streams we need to consider very low resource-bounds,\nsuch as finite memory or finite-time per input symbol.\nThe finite state dimension of an infinite sequence [3], is a measure of the amount of ran-\ndomness contained in the sequence within a finite-memory setting. It is a robust quantity,\nthat has been shown to admit several characterizations in terms of finite-state information\nlossless compressors (introduced by Huffman [8], [3]), finite-state decompressors [4, 13],\nfinite-state predictors in the logloss model [1], and block entropy rates [2]. It is an effec-\ntivization of the general notion of Hausdorffdimension at the level of finite-state machines.\nInformally, the finite state dimension assigns every sequence a number s ∈[0, 1], that char-\nacterizes the randomness density in the sequence (or equivalently its compression ratio),\nwhere the larger the dimension the more randomness is contained in the sequence.\nIn a recent line of research, Doty and Nichols [5] investigated a variant of finite-state\ndimension, where the finite state machine comes equipped with an infinite memory stack\nand is called a pushdown automata, yielding the notion of pushdown dimension. Hence\nthe pushdown dimension of a sequence, is a measure of the density of randomness in the\nsequence as viewed by a pushdown automata. Since a finite-state automata is a special\ncase of a pushdown automata, the pushdown dimension of a sequence is a lower bound\nfor its finite state dimension. It was shown in [5], that there are sequences for which the\npushdown dimension is at most half its finite state dimension, hence yielding a strong\nseparation between the two notions. Unfortunately the notion of pushdown dimension is\nnot known to enjoy any of the equivalent characterizations that finite state dimension does.\nMoreover, the computation time per input symbol can be unbounded, which rules out this\nmodel for many real-time applications.\nIn this paper we introduce a variant of pushdown dimension called bounded pushdown\n(BPD) dimension: Whereas pushdown automata can choose not to read their input and\nonly work with their stack for as many steps as they wish (each such step is called a\nlambda transition), we add the additional real-time constraint that the sequences of lambda\ntransitions are bounded, i.e. we only allow a bounded number of stack movements per each\ninput symbol.\nWe define the notion of bounded pushdown dimension as the natural effectivitation of\nHausdorffdimension via Lutz’s gale characterization [9]. We provide evidence that bounded\npushdown dimension is a robust notion by giving a compression characterization; i.e. we\nintroduce BPD information-lossless compressors and show that the best compression ratio\nachievable on a sequence by BPD compressors is exactly its BPD dimension.\nIn the context of compression, we study the relationship between BPD compression and\nthe standard Lempel-Ziv (LZ) compression algorithm [14]. It is well known that the LZ\ncompression ratio of any sequence is a lower bound for its finite state compressibility [14],\ni.e. LZ compresses every sequence at least as well as any finite-state information lossless\ncompressor. We show that this fails dramatically in the context of BPD compressors, by\nconstructing a sequence that LZ fails to compress significantly, but is compressed by at least\na factor 2 by a BPD compressor, thus yielding a strong separation between LZ and BPD\n2"},{"page":3,"text":"dimension. This implies that we have the same separation between LZ and (unbounded)\npushdown dimension, and between finite state dimension [3] and BPD dimension.\nSection 2 contains the preliminaries, section 3 presents BPD dimension and its basic\nproperties, section 4 proves the equivalence of BPD compression and dimension and section\n5 contains the separation of BPD compression from Lempel Ziv compression. The proofs\nare postponed to the appendix.\n2\nPreliminaries\nWe write Z for the set of all integers, N for the set of all nonnegative integers and Z+ for\nthe set of all positive integers. Let Σ be a finite alphabet, with |Σ| ≥2. Σ∗denotes the\nset of finite strings, and Σ∞the set of infinite sequences. We write |w| for the length of\na string w in Σ∗. The empty string is denoted λ. For S ∈Σ∞and i, j ∈N, we write\nS[i..j] for the string consisting of the ith through jth symbols of S, with the convention\nthat S[i..j] = λ if i > j, and S[0] is the leftmost symbol of S. We write S[i] for S[i..i] (the\nith symbol of S). For w ∈Σ∗and S ∈Σ∞, we write w ⊑S if w is a prefix of S, i.e., if\nw = S[0..|w| −1]. All logarithms are taken in base |Σ|.\n3\nBounded Pushdown Dimension\nIn this section we first recall Lutz’s characterization of Hasudorffdimension in terms of\ngales that can be used to effectivize dimension. Then we introduce Bounded Pushdown\ndimension based on the concept of BPD gamblers and give its basic properties.\nDefinition. [9] Let s ∈[0, ∞).\n1. An s-gale is a function d : Σ∗→[0, ∞) that satisfies the condition\nd(w) =\nP\na∈Σ\nd(wa)\n|Σ|s\n(1)\nfor all w ∈Σ∗.\n2. A martingale is a 1-gale.\nIntuitively, an s-gale is a strategy for betting on the successive symbols of a sequence\nS ∈Σ∞. For each prefix w of S, d(w) is the capital (amount of money) that d has after\nhaving bet on S[0..|w| −1].\nWhen betting on the next symbol b of a prefix wb of S,\nassuming symbol b is equally likely to be any value in Σ, equation (1) guarantees that the\nexpected value of d(wb) is |Σ|−1 P\na∈Σ\nd(wa) = |Σ|s−1d(w). If s = 1, this expected value is\nexactly d(w), so the payoffs are “fair”.\nDefinition.\nLet d be an s-gale, where s ∈[0, ∞).\n1. We say that d succeeds on a sequence S ∈Σ∞if\n3"},{"page":4,"text":"lim sup\nn→∞d(S[0..n −1]) = ∞.\n2. The success set of d is\nS∞[d] = {S ∈Σ∞| d succeeds on S}.\nObservation 3.1 Let s, s′ ∈[0, ∞). For every s-gale d, the function d′ : Σ∗→[0, ∞)\ndefined by d′(w) = |Σ|(s′−s)|w|d(w) is an s′-gale. Moreover, if s ≤s′, then S∞[d] ⊆S∞[d′].\nLutz characterized Hausdorffdimension using gales as follows.\nTheorem 3.2 [9] Given a set X ⊆Σ∞, if dimH(X) is the Hausdorffdimension of X [6],\nthen\ndimH(X) = inf{s | there is an s −gale d such that X ⊆S∞[d]}\nThe idea for a Bounded Pushdown dimension is to consider only s-gales that are com-\nputable by a Bounded Pushdown (BPD) gambler. Bounded Pushdown gamblers are finite-\nstate gamblers [3] with an extra memory stack, that is used both by the transition and\nbetting functions. Additionally, BPDG’s are allowed to delay reading the next character\nof the input –they read λ from the input– in order to alter the content of their stack, but\nthey cannot do this more than a constant number of times per each input symbol. During\nsuch λ-transitions, the gambler’s capital remains unchanged.\nThe betting function returns a probability measure over the input alphabet.\nDefinition.\nLet Σ be a finite alphabet. ∆Q(Σ) is the set of all rational-valued probability\nmeasures over Σ, i.e., all functions π : Σ −→[0, 1] ∩Q such that P\na∈Σ\nπ(a) = 1.\nWe are ready to define BPD gamblers.\nDefinition.\nA bounded pushdown gambler (BPDG) is an 8-tuple G =(Q, Σ, Γ, δ, β, q0,\nz0, c) where\n• Q is a finite set of states,\n• Σ is the finite input alphabet,\n• Γ is the finite stack alphabet,\n• δ : Q×(Σ∪{λ})×Γ →Q×Γ∗is the transition function (for simplicity we use the nota-\ntion δ(q, b, a) = ⊥when undefined; and we write δ(q, b, a) = (δQ(q, b, a), δΓ∗(q, b, a))),\n• β : Q × Γ →∆Q(Σ) is the betting function,\n• q0 ∈Q is the start state,\n• z0 ∈Γ is the start stack symbol,\n• c ∈N is a constant such that the number of λ-transitions per input symbol is at most\nc,\n4"},{"page":5,"text":"with the two additional restrictions:\n1. for each q ∈Q and a ∈Γ at least one of the following holds\n• δ(q, λ, a) =⊥\n• δ(q, b, a) =⊥for all b ∈Σ\n2. for every q ∈Q, b ∈Σ ∪{λ}, either δ(q, b, z0) =⊥, or δ(q, b, z0) = (q′, vz0), where q′\n∈Q and v ∈Γ∗.\nWe denote with BPDG the set of all bounded pushdown gamblers.\nThe transition function δ outputs a new state and a string z′ ∈Γ∗.\nInformally,\nδ(q, w, a) = (q′, z′) means that in state q, reading input w, and popping symbol a from the\nstack, δ enters state q′ and pushes z′ to the stack.\nNote that w can be λ (ie, a λ-transition: the input is ignored and δ only computes with\nthe stack) but this only happens at most c times per input symbol. Any pair (state, stack\nsymbol) can either be a λ-transition pair or a non λ-transition pair exclusively, because\nthe first additional restriction enforces determinism.\nMoreover, since z0 represents the bottom of the stack, we restrict δ so that z0 cannot\nbe removed from the bottom by the second additional restriction.\nWe can extend δ in the usual way to\nδ∗: Q × (Σ ∪{λ}) × Γ+ →Q × Γ∗,\nwhere for all q ∈Q, a ∈Γ, v ∈Γ∗, and b ∈Σ ∪{λ}\nδ∗(q, b, av) =\n (δQ(q, b, a), δΓ∗(q, b, a)v)\nif δ(q, b, a) ̸=⊥,\n⊥\notherwise.\nWe denote δ∗by δ.\nFor each i ≥2, we will use the notation\nδi(q, λ, v) = δ(δi−1\nQ (q, λ, v), λ, δi−1\nΓ∗(q, λ, v))\nwhere\nδ1(q, λ, v) = δ(q, λ, v).\nSince δ is c-bounded we have that for any q ∈Q, v ∈Γ∗,\nδc+1(q, λ, v) = ⊥\nWe also consider the extended transition function\nδ∗∗: Q × Σ∗× Γ+ →Q × Γ∗,\ndefined for all q ∈Q, a ∈Γ, v ∈Γ∗, w ∈Σ∗, and b ∈Σ by\nδ∗∗(q, λ, av) = (q, av)\n5"},{"page":6,"text":"δ∗∗(q, wb, av) = δ(δi\nQ(eq, λ, eaev), b, δi\nΓ∗(eq, λ, eaev))\nif δ∗∗(q, w, av) = (eq, eaev), δi(eq, λ, eaev) ̸=⊥and δi+1(eq, λ, eaev) =⊥, i ≤c.\nThat is, λ-transitions are inside the definition of δ∗∗(q, b, av), for b ∈Σ. Notice that\nδ∗∗is not defined on an empty stack string, therefore av needs to be long enough in order\nthat δ∗∗(q, b, av) ̸=⊥.\nWe denote δ∗∗by δ, and δ(q0, w, z0) by δ(w). We write δ = (δQ, δΓ∗) for simplicity.\nWe also consider the usual extension of β\nβ∗: Q × Γ+ →∆Q(Σ),\ndefined for all q ∈Q, a ∈Γ, and v ∈Γ∗by\nβ∗(q, av) = β(q, a),\nand denote β∗by β.\nWe use BPDG to compute martingales. Intuitively, suppose a BPDG G is to bet on\nsequence S has already bet on w ❁S, with current capital x ∈Q, current state q ∈Q and\ncurrent top stack symbol a. Then for b ∈Σ, G bets the quantity xβ(q, a)(b) of its capital\nthat the next symbol of S is b. If the bet is correct (that is, if wb ❁S) and since payoffs\nare fair, G has capital |Σ|xβ(q, a)(b). Formally,\nDefinition.\nLet G = (Q, Σ, Γ, δ, β, q0, z0, c) be a bounded pushdown gambler.\nThe\nmartingale of G is the function\ndG : Σ∗→[0, ∞)\ndefined by the recursion\ndG(λ) = 1\ndG(wb) = |Σ|dG(w)β(δ(w))(b)\nfor all w ∈Σ∗and b ∈Σ.\nBy Observation 3.1, a BPDG G actually yields an s-gale for every s ∈[0, ∞). We call\nit the s-gale of G, and denote it by\nds\nG(w) = |Σ|(s−1)|w|dG(w).\nA bounded pushdown s-gale is an s-gale d for which there exists a BPDG such that ds\nG = d.\nThe first two properties of BPD gamblers are that any number of λ-transitions can\nbe replaced by a single λ-transition and that the stack alphabet does not give additional\npower.\nProposition 3.3 Let G = (Q, Σ, Γ, δ, β, q0, z0, c) be a BPDG. Then there is a BPDG G′ =\n(Q′, Σ, Γ′, δ′, β′, q′\n0, z′\n0, 1) such that dG = dG′.\nFrom now on we shall assume that the maximum number of λ-transitions c is 1.\nProposition 3.4 Let G = (Q, Σ, Γ, δ, β, q0, z0, c) be a BPDG. Then there is a BPDG G′ =\n(Q′, Σ, {0, 1, z′\n0}, δ′, β′, q′\n0, z′\n0, c′) such that dG = dG′.\n6"},{"page":7,"text":"Let us define bounded pushdown dimension. Intuitively, the BPD dimension of a se-\nquence is the smallest s such that there is a BPD-s-gale that succeeds on the sequence.\nDefinition.\nThe bounded pushdown dimension of a set X ⊆Σ∞is\ndimBPD(X) = inf{s | there is a bounded pushdown s −gale d such that X ⊆S∞[d]}.\n4\nDimension and compression\nIn this section we characterize the bounded pushdown dimension of individual sequences\nin terms of bounded pushdown compressibility, therefore BPD dimension is a natural and\nrobust definition.\nDefinition. A bounded pushdown compressor (BPDC) is an 8-tuple\nC = (Q, Σ, Γ, δ, ν, q0, z0, c)\nwhere\n• Q is a finite set of states,\n• Σ is the finite input and output alphabet,\n• Γ is the finite stack alphabet,\n• δ : Q × (Σ ∪{λ}) × Γ →Q × Γ∗is the transition function,\n• ν : Q × Σ × Γ →Σ∗is the output function,\n• q0 ∈Q is the initial state,\n• z0 ∈Γ is the start stack symbol,\n• c ∈N is a constant such that the number of λ-transitions per input symbol is at most\nc,\nwith the two additional restrictions:\n1. for each q ∈Q and a ∈Γ at least one of the following holds\n• δ(q, λ, a) =⊥\n• δ(q, b, a) =⊥for all b ∈Σ\n2. for every q ∈Q, b ∈Σ ∪{λ}, either δ(q, b, z0) =⊥, or δ(q, b, z0) = (q′, vz0), where q′\n∈Q and v ∈Γ∗.\n7"},{"page":8,"text":"We extend δ to δ∗∗: Q×Σ∗×Γ+ →Q×Γ∗as before, and denote δ∗∗by δ and δ(q0, w, z0)\nby δ(w).\nFor q ∈Q, w ∈Σ∗and z ∈Γ+, we define the output from state q on input w reading\nz on the top of the stack to be the string ν∗(q, w, z) (denoted by ν(q, w, z)) with\nν(q, λ, z) = λ\nν(q, wb, z) = ν(q, w, z)ν(δQ(q, w, z), b, δΓ∗(q, w, z))\nfor w ∈Σ∗and b ∈Σ. We then define the output of C on input w ∈Σ∗to be the string\nC(w) = ν(q0, w, z0).\nWe can restrict λ-transitions to a single one and the stack alphabet to three symbols.\nProposition 4.1 Let C = (Q, Σ, Γ, δ, ν, q0, z0, c) be a BPDC. Then there is a BPDC C′ =\n(Q′, Σ, Γ′, δ′, ν′, q′\n0, z′\n0, 1) such that C(w) = C′(w) for every w ∈Σ∗.\nProposition 4.2 Let C = (Q, Σ, Γ, δ, ν, q0, z0, c) be a BPDC. Then there is a BPDC C′ =\n(Q′, Σ, {0, 1, z′\n0}, δ′, ν′, q′\n0, z′\n0, c′) such that C(w) = C′(w) for every w ∈Σ∗.\nWe are interested in information lossless compressors, that is, w must be recoverable\nfrom C(w) and the final state.\nDefinition.\nA BPDC C = (Q, Σ, Γ, δ, ν, q0, z0) is information-lossless (IL) if the function\nΣ∗→Σ∗× Q\nw →(C(w), δQ(w))\nis one-to-one.\nAn information-lossless bounded pushdown compressor (ILBPDC) is a\nBPDC that is IL.\nIntuitively, a BPDC compresses a string w if |C(w)| is significantly less than |w|. Of\ncourse, if C is IL, then not all strings can be compressed. Our interest here is in the\ndegree (if any) to which the prefixes of a given sequence S ∈Σ∞can be compressed by an\nILBPDC.\nDefinition.\nIf C is a BPDC and S ∈Σ∞, then the compression ratio of C on S is\nρC(S) = lim inf\nn→∞\n|C(S[0..n −1])|\nn\n.\nThe BPD compression ratio of a sequence is the best compression ratio achievable by\nan ILBPDC, that is\nDefinition.\nThe bounded pushdown compression ratio of a sequence S ∈Σ∞is\nρBPD(S) = inf{ρC(S) | C is a ILBPDC}.\nThe main result in this section states that the BPD dimension of a sequence and its\nILBPD compression ratio are the same, therefore BPD dimension is the natural concept\nof density of information in the BPD setting.\n8"},{"page":9,"text":"Theorem 4.3 For all S ∈Σ∞,\ndimBPD(S) = ρBPD(S).\n5\nSeparating LZ from BPD\nIn this section we prove that BPD compression can be much better than the compression\nattained with the celebrated Lempel-Ziv algorithm.\nWe start with a brief description of the LZ algorithm [14].\nWe finish relating BPD dimension (and compression) with the Lempel-Ziv algorithm.\nGiven an input x ∈Σ∗, LZ parses x in different phrases xi, i.e., x = x1x2 . . . xn (xi ∈Σ∗)\nsuch that every prefix y ❁xi, appears before xi in the parsing (i.e. there exists j < i s.t.\nxj = y). Therefore for every i, xi = xl(i)bi for l(i) < i and bi ∈Σ. We sometimes denote\nthe number of phrases in the parsing of x as C(x).\nLZ encodes xi by a prefix free encoding of l(i) and the symbol bi, that is, if x =\nx1x2 . . . xn as before, the output of LZ on input x is\nLZ(x) = cl(1)b1cl(2)b2 . . . cl(n)bn\nwhere ci is a prefix-free coding of i (and x0 = λ).\nLZ is usually restricted to the binary alphabet, but the description above is valid for\nany Σ.\nFor a sequence S ∈Σ∞, the LZ compression ratio is given by\nρLZ(S) = lim inf\nn→∞\n|LZ(S[0 . . . n −1])|\nn\n.\nIt is well known that LZ [14] yields a lower bound on the finite-state dimension (or finite-\nstate compressibility) of a sequence [14], ie, LZ is universal for finite-state compressors.\nThe following result shows that this is not true for BPD (hence PD) dimension, in a\nstrong sense: we construct a sequence S that cannot be compressed by LZ, but that has\nBPD compression ratio less than 1\n2.\nTheorem 5.1 For every m ∈N, there is a sequence S ∈{0, 1}∞such that\nρLZ(S) > 1 −1\nm\nand\ndimBPD(S) ≤1\n2.\nAs a corollary we obtain a separation of finite-state dimension and bounded pushdown\ndimension. A similar result between finite-state dimension and pushdown dimension was\nproved in [5].\n9"},{"page":10,"text":"Corollary 5.2 For any m ∈N, there exists a sequence S ∈{0, 1}∞such that\ndimFS(S) > 1 −1\nm\nand\ndimBPD(S) ≤1\n2.\nConclusion\nWe have introduced Bounded Pushdown dimension, characterized it with compression and\ncompared it with Lempel-Ziv compression. It is open if there is a BPD compressor that\nis universal for Finite-State compressors, which is true for the Lempel-Ziv algorithm, and\nwhether Lempel-Ziv compression can surpass BPD-compression for some sequence.\nReferences\n[1] K. B. Athreya, J. M. Hitchcock, J. H. Lutz, and E. Mayordomo. Effective strong\ndimension in algorithmic information and computational complexity. SIAM Journal\non Computing. To appear.\n[2] Chris Bourke, John M. Hitchcock, and N. V. Vinodchandran.\nEntropy rates and\nfinite-state dimension. Theor. Comput. Sci., 349(3):392–406, 2005.\n[3] Jack J. Dai, James I. Lathrop, Jack H. Lutz, and Elvira Mayordomo. Finite-state\ndimension. Theoretical Computer Science, 310(1–3):1–33, January 2004.\n[4] D. Doty and P. Moser. Personal communication, based on [13]. 2006.\n[5] David Doty and Jared Nichols. Pushdown dimension. Theoretical Computer Science.\nTo appear.\n[6] K. Falconer. The Geometry of Fractal Sets. Cambridge University Press, 1985.\n[7] J. M. Hitchcock, J. H. Lutz, and E. Mayordomo. The fractal geometry of complexity\nclasses. SIGACT News Complexity Theory Column, 36:24–38, 2005.\n[8] D. A. Huffman. Canonical forms for information-lossless finite-state logical machines.\nTrans. Circuit Theory, pages 41–59, 1959.\n[9] J. H. Lutz. Dimension in complexity classes. SIAM Journal on Computing, 32:1236–\n1259, 2003.\n[10] J. H. Lutz. The dimensions of individual strings and sequences. Information and\nComputation, 187:49–79, 2003.\n10"},{"page":11,"text":"[11] J. H. Lutz. Effective fractal dimensions.\nMathematical Logic Quarterly, 51:62–72,\n2005.\n[12] E. Mayordomo. Effective fractal dimension in algorithmic information theory. In New\nComputational Paradigms: Changing Conceptions of What is Computable. Springer-\nVerlag, 2007. To appear.\n[13] D. Sheinwald, A. Lempel, and J. Ziv. On compression with two-way head machines.\nIn Data Compression Conference, pages 218–227, 1991.\n[14] Jacob Ziv and Abraham Lempel. Compression of individual sequences via variable-\nrate coding. IEEE Transactions on Information Theory, 24(5):530–536, 1978.\n11"},{"page":12,"text":"Technical Appendix\nThis appendix is devoted to proving Theorem 4.3 and Theorem 5.1. For the first one,\nwe need the following:\nA\nProof of Theorem 4.3\nDefinition.\nA BPDG G = (Q, Σ, Γ, δ, β, q0, z0) is nonvanishing if 0 < β(q, z)(b) < 1 for\nall q ∈Q, b ∈Σ and z ∈Γ.\nLemma A.1 For every BPDG G and each ε > 0, there is a nonvanishing BPDG G′ such\nthat for all w ∈Σ∗, dG′(w) ≥|Σ|−ε|w|dG(w).\nProof of Lemma A.1 . Let G = (Q, Σ, δ, β, q0, Γ, z0) be a BPDG, and let ε > 0. For\neach q ∈Q, z ∈Γ, b ∈Σ,\n1 −|Σ|−ε X\nb∈Σ\nβ(q, z)(b) = 1 −|Σ|−ε > 0,\nso we can fix a rational β′(q, z)(b) such that\n|Σ|−εβ(q, z)(b) < β′(q, z)(b) < 1 −|Σ|−ε\nX\na∈Σ,a̸=b\nβ(q, z)(a)\nand\nX\nb∈Σ\nβ′(q, z)(b) = 1.\nThen, 0 < β′(q, z)(b) < 1 for each q ∈Q, b ∈Σ and z ∈Γ, therefore the BPDG G′ =\n(Q, Σ, δ, β′, q0, Γ, z0) is nonvanishing.\nAlso, for all q ∈Q, b ∈Σ, z ∈Γ,\nβ′(q, z)(b) ≥|Σ|−εβ(q, z)(b)\nso for all w ∈Σ∗, dG′(w) ≥|Σ|−ε|w|dG(w).\n✷\nProof of Theorem 4.3 Let S ∈Σ∞. For each n ∈N, let wn = S[0..n −1].\nTo see that dimBPD(S) ≤ρBPD(S), let s > s′ > ρBPD(S).\nIt suffices to show that\ndimBPD(S) ≤s. By our choice of s′, there is an 1-ILBPDC C = (Q, Σ, Γ, δ, ν, q0, z0) for\nwhich the set\nI = {n ∈N | |C(wn)| < s′n}\nis infinite.\n1"},{"page":13,"text":"CONSTRUCTION A.1 Given a 1-bounded pushdown compressor (BPDC)\nC = (Q, Σ, Γ, δ, ν, q0, z0), and k ∈Z+ , we construct the 1-bounded pushdown gambler\n(BPDG) G = G(C, k) = (Q′, Σ, Γ′, δ′, β′, q′\n0, z′\n0) as follows:\ni) Q′ = Q × {0, 1, . . . , k −1}\nii) q′\n0 = (q0, 0)\niii) Γ′ =\n4k−1\nS\ni=2k\nΓi\niv) z′\n0 = z2k\n0\nv) ∀(q, i) ∈Q′, b ∈Σ, a ∈Γ′,\nδ′((q, i), b, a) =\n \nδQ(q, b, a), (i + 1) mod k\n \n,\n\\\nδΓ∗(q, b, a)\n \nwhere for each z ∈(Γ′)+, z ∈Γ+ is the Γ-string obtained by concatenating the symbols of\nz, and for each y ∈Γ+, if y = y1y2 · · · y2kl+n with n < 2k, then by ∈(Γ′)+ is such that\nby1 = y1 · · · y2k+n, by2 = y2k+n+1 · · · y4k+n, . . . , byl = y2k(l−1)+n+1 · · · y2kl+n.\nvi) ∀(q, i) ∈Q′, a ∈Γ′, b ∈Σ\nβ′((q, i), a)(b) = σ(q, bΣk−i−1, a)\nσ(q, Σk−i, a)\nwhere σ(q, A, a) = P\nx∈A\n|Σ|−|ν(q,x,a)| .\nLemma A.2 In Construction A.1, if |w| is a multiple of k and u ∈Σ≤k, then\ndG(wu) = |Σ||u|−|ν(δQ(w),u,δΓ∗(w))| σ(δQ(wu), Σk−|u|, \\\nδΓ∗(wu))\nσ(δQ(w), Σk, \\\nδΓ∗(w))\ndG(w).\nProof of Lemma A.2. We use induction on the string u. If u = λ, the lemma is\nclear. Assume that it holds for u, where u ∈Σ<k, and let b ∈Σ. Then\ndG(wub) = |Σ|σ(δQ(wu), bΣk−|u|−1, \\\nδΓ∗(wu))\nσ(δQ(wu), Σk−|u|, \\\nδΓ∗(wu))\ndG(wu)\n= |Σ|1−|ν(δQ(wu),b,δΓ∗(wu))|σ(δQ(wub), Σk−|u|−1,\n\\\nδΓ∗(wub))\nσ(δQ(wu), Σk−|u|, \\\nδΓ∗(wu))\ndG(wu)\nso by the induction hypothesis the lemma holds for ub.\n2"},{"page":14,"text":"✷\nLemma A.3 In Construction A.1, if w = w0w1 · · ·wn−1, where each wi ∈Σk , then\ndG(w) =\n|Σ||w|−|C(w)|\nn−1\nQ\ni=0\nσ(δQ(w0 · · ·wi−1), Σk,\n\\\nδΓ∗(w0 · · · wi−1))\n.\nProof of Lemma A.3. We use induction on n. For n = 0, the identity is clear.\nAssume that it holds for w = w0w1 · · · wn−1, with each wi ∈Σk, and let w′ = w0w1 · · · wn.\nThen Lemma A.2 with u = wn tells us that\ndG(w′) = |Σ|k−|ν(δQ(w),wn,δΓ∗(w))|\nσ(δQ(w), Σk, \\\nδΓ∗(w))\ndG(w)\nwhence the identity holds for w′ by the induction hypothesis.\n✷\nLemma A.4 In Construction A.1, if C is IL and |w| is a multiple of k, then\ndG(w) ≥|Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1),\nwhere l = ⌈log |Q|⌉and m = max{|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}.\nProof of Lemma A.4. We prove that for each z ∈Σ∗,\nσ(δQ(z), Σk, \\\nδΓ∗(z)) ≤|Σ|l+log m+log k+1.\nTo see this, fix z ∈Σ∗and observe that at most |Q| strings w ∈Σk can have the same\noutput from state δQ(z) with stack content δΓ∗(z). Therefore, the number of w ∈Σk for\nwhich |ν(δQ(z), w, δΓ∗(z))| = j does not exceed |Q||Σ|j. Hence\nσ(δQ(z), Σk, \\\nδΓ∗(z)) =\nX\nw∈Σk\n|Σ|−|ν(δQ(z),w,δΓ∗(z))| ≤\nmk\nX\nj=0\n|Q||Σ|j|Σ|−j = |Q|(mk + 1)\n≤|Σ|l+log m+log k+1.\nIt follows by Lemma A.3 that\ndG(w) = |Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1).\n✷\nLemma A.5 In Construction A.1, if C is IL, then for all w ∈Σ∗,\ndG(w) ≥|Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1)−(km+l+log m+log k+1),\nwhere l = ⌈log |Q|⌉and m = max {|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}.\n3"},{"page":15,"text":"Proof of Lemma A.5. Assume the hypothesis, let l and m be as given, and let w ∈\nΣ∗. Fix 0 ≤j < k such that |w| + j is divisible by k. By Lemma A.4 we have\ndG(w) ≥|Σ|−jdG(w0j)\n≥|Σ|−j+|w0j|−|C(w0j)|−|w0j |\nk\n(l+log m+log k+1)\n= |Σ||w|−|C(w0j)|−|w|\nk (l+log m+log k+1)−j\nk (l+log m+log k+1)\n≥|Σ||w|−|C(w)|−|w|\nk (l+log m+log k+1)−(km+l+log m+log k+1)\n✷\nLet l = ⌈log |Q|⌉and m = max{|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}, and fix k ∈Z+ such\nthat l+log m+log k+1\nk\n< s −s′. Let G = G(C, k) be as in Construction A.1. Then, by Lemma\nA.5, for all n ∈I we have\nd(s)\nG (wn) ≥|Σ|sn−|C(wn)|−n\nk (l+log m+log k+1)−(km+l+log m+log k+1)\n≥|Σ|(s−s′−l+log m+log k+1\nk\n)n−(km+l+log m+log k+1)\nSince s −s′ −l+log m+log k+1\nk\n> 0, this implies that S ∈S∞[d(s)\nG ].\nThus, dimBPD(S) ≤s.\nTo see that ρBPD(S) ≤dimBPD(S), let s > s′ > s′′ > dimBPD(S). It suffices to show\nthat ρBPD(S) ≤s. By our choice of s′′, there is a 1-BPDG G such that the set\nJ = {n ∈N | ds′′\nG (wn) ≥1}\nis infinite. By Lemma A.1 there is a nonvanishing 1-BPDG eG such that\nd e\nG(w) ≥|Σ|(s′′−s′)|w|dG(w) for all w ∈Σ∗.\nCONSTRUCTION A.2 Let G = (Q, Σ, Γ, δ, β, q0, z0) be a nonvanishing 1-BPDG, and\nlet k ∈Z+. For each z ∈Γ∗(long enough for dGq,z(w) to be defined for all w ∈Σk)\nand q ∈Q, let Gq,z = (Q, Σ, Γ, δ, β, q, z), and define pq,z : Σk →[0, 1] by pq,z(w) =\n|Σ|−kdGq,z(w). Since G is nonvanishing and each dGq,z is a martingale with dGq,z(λ) = 1,\neach of the functions pq,z is a positive probability measure on Σk. For each z ∈Γ∗, q ∈Q, let\nΘq,z : Σk →Σ∗be the Shannon-Fano-Elias code given by the probability measure pq,z. Then\n|Θq,z(w)| = lq,z(w)\nlq,z(w) = 1 + ⌈log\n1\npq,z(w)⌉\nfor all q ∈Q and w ∈Σk, and each of the sets range(Θq,z) is an instantaneous code. We\ndefine the 1-BPDC C = C(G, k) = (Q′, Σ, Γ′, δ′, ν′, q′\n0, z′\n0) whose components are as follows:\ni) Q′ = Q × Σ<k\n4"},{"page":16,"text":"ii) q′\n0 = (q0, λ)\niii) Γ′ =\n4k−1\nS\ni=2k\nΓi\niv) z′\n0 = z2k\n0\nv) ∀(q, w) ∈Q′, b ∈Σ, a ∈Γ′,\nδ′((q, w), b, a) =\n(\n(q, wb, a)\nif |w| < k −1,\n(δQ(q, wb, a), λ,\n\\\nδΓ∗(q, wb, a))\nif |w| = k −1.\nvi) ∀(q, w) ∈Q′, b ∈Σ, a ∈Γ′,\nν′((q, w), b, a) =\n λ\nif |w| < k −1,\nΘq,a(wb)\nif |w| = k −1.\nSince each range(Θq,z) is an instantaneous code, it is easy to see that the BPDC C =\nC(G, k) is IL.\nLemma A.6 In Construction A.2, if |w| is a multiple of k, then\n|C(w)| ≤\n \n1 + 2\nk\n \n|w| −log dG(w).\nProof of Lemma A.6.\nLet w = w0w1 · · · wn−1, where each wi ∈Σk.\nFor each\n0 ≤i < n, let qi = δQ(w0 · · · wi−1) and zi = δΓ∗(w0 · · · wi−1). Then,\n|C(w)| =\nn−1\nX\ni=0\nlqi,zi(wi)\n=\nn−1\nX\ni=0\n \n1 + ⌈log\n1\npqi,zi(wi)⌉\n \n≤\nn−1\nX\ni=0\n \n2 + log\n1\npqi,zi(wi)\n \n=\nn−1\nX\ni=0\n \n2 + log\n|Σ|k\ndGqi,zi(wi)\n \n= (k + 2)n −log\nn−1\nY\ni=0\ndGqi,zi(wi)\n= (k + 2)n −log dG(w) = (1 + 2\nk)|w| −log dG(w)\n✷\nLemma A.7 In Construction A.2, for all w ∈Σ∗,\n|C(w)| ≤\n \n1 + 2\nk\n \n|w| −log dG(w).\n5"},{"page":17,"text":"Proof of Lemma A.7. If |w| is multiple of k, then we apply the Lemma A.6.\nOtherwise, let w = w′z, where |w′| is a multiple of k and |z| = j, 0 < j < k.\nThen, Lemma A.6 tell us that\n|C(w)| = |C(w′)|\n≤\n \n1 + 2\nk\n \n|w′| −log dG(w′)\n≤\n \n1 + 2\nk\n \n|w′| −log(|Σ|−jdG(w))\n=\n \n1 + 2\nk\n \n|w| −log dG(w) −2j\nk\n≤\n \n1 + 2\nk\n \n|w| −log dG(w).\n✷\nFix k >\n2\ns−s′, and let C = C( eG, k) be as in Construction A.2. Then Lemma A.7 tell us\nthat for all n ∈J,\n| C(wn) | ≤\n \n1 + 2\nk\n \nn −log d e\nG(wn)\n≤\n \n1 + 2\nk + s′ −s′′ \nn −log dG(wn)\n≤\n 2\nk + s′ \nn −log ds′′\nG (wn)\n≤\n 2\nk + s′ \nn\n< sn.\nThus, ρBPD(S) ≤s.\n✷\nB\nProof of Theorem 5.1\nFor a string x, x−1 denotes x written in reverse order.\nProof of Theorem 5.1 Let m ∈N, and let k = k(m) be an integer to be determined\nlater. For any integer n, let Tn denote the set of strings x of size n such that 1j does not\nappear in x, for every j ≥k. Since Tn contains {0, 1}k−1 × {0} × {0, 1}k−1 × {0} . . . (i.e.\nthe set of strings whose every kth bit is zero), it follows that |Tn| ≥2an, where a = 1−1/k.\nRemark B.1 For every string x ∈Tn there is a string y ∈Tn−1 and a bit b such that\nyb = x.\n6"},{"page":18,"text":"Let An = {a1, . . . au} be the set of palindromes in Tn. Since fixing the n/2 first bits\nof a palindrome (wlog n is even) completely determines it, it follows that |An| ≤2\nn\n2 .\nLet us separate the remaining strings in Tn −An into two sets Xn = {x1, . . . xt} and\nYn = {y1, . . . yt} with (xi)−1 = yi for every 1 ≤i ≤t. Let us choose X, Y such that x1 and\nyt start with a zero. We construct S in stages. For n ≤k −1, Sn is an enumeration of all\nstrings of size n in lexicographical order. For n ≥k,\nSn = a1 . . . au 12n x1 . . . xt 12n+1 yt . . . y1\ni.e.\na concatenation of all strings in An (the A zone of Sn) followed by a flag of 2n\nones, followed by the concatenations of all strings in X (the X-zone) and Y (the Y zone)\nseparated by a flag of 2n + 1 ones. Let\nS = S1S2 . . . Sk−1 1k 1k+1 . . . 12k−1 SkSk+1 . . .\ni.e. the concatenation of the Sj’s with some extra flags between Sk−1 and Sk. We claim\nthat the parsing of Sn (n ≥k) by LZ, is as follows:\nSn = a1, . . . , au, 12n, x1, . . . , xt, 12n+1, yt, . . . , y1.\nIndeed after S1, . . . Sk−1 1k 1k+1 . . . 12k−1, LZ has parsed every string of size ≤k −1 and\nthe flags 1k 1k+1 . . . 12k−1. Together with Remark B.1, this guarantees that LZ parses Sn\ninto phrases that are exactly all the strings in Tn and the two flags 12n, 12n+1.\nLet us compute the compression ratio ρLZ(S). Let n, i be integers. By construction of\nS, LZ encodes every phrase in Si (except the two flags), by a phrase in Si−1 (plus a bit).\nIndexing a phrase in Si−1 requires a codeword of length at least logarithmic in the number\nof phrase parsed before, i.e. log(C(S1S2 . . . Si−2)). Since C(Si) ≥|Ti| ≥2ai, it follows\nC(S1 . . . Si−2) ≥\ni−2\nX\nj=1\n2aj = 2a(i−1) −2a\n2a −1\n≥b2a(i−1)\nwhere b = b(a) is arbitrarily close to 1. Letting ti = |Ti|, the number of bits output by LZ\non Si is at least\nC(Si) log C(S1 . . . Si−2) ≥ti log b2a(i−1)\n≥cti(i −1)\nwhere c = c(b) is arbitrarily close to 1. Therefore\n|LZ(S1 . . . Sn)| ≥\nn\nX\nj=1\nctj(j −1)\nSince |S1 . . . Sn| ≤2k2 + Pn\nj=1(jtj + 4j), (the two flags plus the extra flags between Sk−1\nand Sk) the compression ratio is given by\nρLZ(S1 . . . Sn) ≥c\nPn\nj=1 tj(j −1)\n2k2 + Pn\nj=1 j(tj + 4)\n(2)\n= c −c\n2k2 + Pn\nj=1(tj + 4j)\n2k2 + Pn\nj=1 j(tj + 4)\n(3)\n7"},{"page":19,"text":"The second term in Equation 3 can be made arbitrarily small for n large enough: Let\nM ≤n, we have\n2k2 +\nn\nX\nj=1\nj(tj + 4) ≥2k2 +\nM\nX\nj=1\njtj + (M + 1)\nn\nX\nj=M+1\ntj\n= 2k2 +\nM\nX\nj=1\njtj + M\nn\nX\nj=M+1\ntj +\nn\nX\nj=M+1\ntj\n≥2k2 +\nM\nX\nj=1\njtj + M\nn\nX\nj=M+1\ntj +\nn\nX\nj=M+1\n2aj\n≥2k2 +\nM\nX\nj=1\njtj + M\nn\nX\nj=M+1\ntj + 2an\n≥M\nn\nX\nj=M+1\ntj + M(2k2 + 2n(n + 1) +\nM\nX\nj=1\ntj)\nfor n big enough\n= M(2k2 +\nn\nX\nj=1\ntj + 4\nn\nX\nj=1\nj)\nHence\nρLZ(S1 . . . Sn) ≥c −c\nM\nwhich by definition of c, M can be made arbitrarily close to 1 by choosing k accordingly,\ni.e\nρLZ(S1 . . . Sn) ≥1 −1\nm.\nLet us show that dimBPD(S) ≤1\n2. Consider the following BPD martingale d. Informally,\nd on Sn goes through the An zone until the first flag, then starts pushing the whole X\nzone onto its stack until it hits the second flag. It then uses the stack to bet correctly on\nthe whole Y zone. Since the Y zone is exactly the X zone written in reverse order, d is\nable to double its capital on every bit of the Y zone. On the other zones, d does not bet.\nBefore giving a detailed construction of d, let us compute the upper bound it yields on\n8"},{"page":20,"text":"dimBPD(S).\ndimBPD(S) ≤1 −lim sup\nn→∞\nlog d(S1 . . . Sn)\n|S1 . . . Sn|\n≤1 −lim sup\nn→∞\nPn\nj=1 |Yj|\n2k2 + Pn\nj=1(j|Tj| + 4j)\n≤1 −lim sup\nn→∞\nPn\nj=1 j |Tj|−|Aj|\n2\n2k2 + Pn\nj=1(j|Tj| + 4j)\n≤1\n2 + 1\n2 lim sup\nn→∞\n2k2 + Pn\nj=1(j|Aj| + 4j)\n2k2 + Pn\nj=1(j|Tj| + 4j) .\nSince\nlim sup\nn→∞\n2k2 + Pn\nj=1(j|Aj| + 4j)\n2k2 + Pn\nj=1(j|Tj| + 4j) ≤lim sup\nn→∞\nPn\nj=1 j(|Aj| + 4 + 2k2)\nPn\nj=1 |Tj|\n≤lim sup\nn→∞\nPn\nj=1 j(2\nj\n2 + 2\nj\n4)\nPn\nj=1 2aj\n≤lim sup\nn→∞\nn2\n3n\n4\n2an\n= 0.\nIt follows that\ndimBPD(S) ≤1\n2.\nLet us give a detailed description of d. Let Q be the following set of states:\n• The start state q0, and q1, . . . qv the “early” states that will count up to\nv = |S1S2 . . . Sk−1 1k 1k+1 . . . 12k−1|.\n• qa\n0, . . . , qa\nk the A zone states that cruise through the A zone until the first flag.\n• q1f the first flag state.\n• qX\n0 , . . . , qX\nk the X zone states that cruise through the X zone, pushing every bit on\nthe stack, until the second flag is met.\n• qr\n0, . . . , qr\nk which after the second flag is detected, pop k symbols from the stack that\nwere erroneously pushed while reading the second flag.\n• q2f the second flag state.\n• qb the betting on zone Y state.\n9"},{"page":21,"text":"Let us describe the transition function δ : Q × {0, 1} × {0, 1} →Q × {0, 1}. First δ counts\nuntil v i.e. for i = 0, . . . v −1\nδ(qi, x, y) = (qi+1, y)\nfor any x, y\nand after reading v bits, it enters in the first A zone state, i.e. for any x, y\nδ(qv, x, y) = (qa\n0, y).\nThen δ skips through A until the string 1k is met, i.e. for i = 0, . . . k −1 and any x, y\nδ(qa\ni , x, y) =\n(\n(qa\ni+1, y)\nif x = 1\n(qa\n0, y)\nif x = 0\nand\nδ(qa\nk, x, y) = (q1f, y).\nOnce 1k has been seen, δ knows the first flag has started, so it skips through the flag until\na zero is met, i.e. for every x, y\nδ(q1f, x, y) =\n(\n(q1f, y)\nif x = 1\n(qX\n0 , 0y)\nif x = 0\nwhere state qX\n0 means that the first bit of the X zone (a zero bit) has been read, therefore\nδ pushes a zero. In the X zone, delta pushes every bit it sees until it reads a sequence of\nk ones, i.e until the start of the second flag, i.e for i = 0, . . . k −1 and any x, y\nδ(qX\ni , x, y) =\n(\n(qX\ni+1, xy)\nif x = 1\n(qX\n0 , xy)\nif x = 0\nand\nδ(qX\nk , x, y) = (qr\n0, y).\nAt this point, δ has pushed all the X zone on the stack, followed by k ones. The next step\nis to pop k ones, i.e for i = 0, . . . k −1 and any x, y\nδ(qr\ni , x, y) = (qr\ni+1, λ)\nand\nδ(qr\nk, x, y) = (q2f\n0 , y).\nAt this stage, δ is still in the second flag (the second flag is always bigger than 2k) therefore\nit keeps on reading ones until a zero (the first bit of the Y zone) is met. For any x, y\nδ(q2f, x, y) =\n(\n(q2f, y)\nif x = 1\n(qb, λ)\nif x = 0.\n10"},{"page":22,"text":"On the last step δ has read the first bit of the Y zone, therefore it pops it. At this stage,\nthe stack exactly contains the Y zone (i.e. the X zone written in reverse order) except\nthe first bit; δ thus uses its stack to bet and double its capital on every bit in the Y zone.\nOnce the stack is empty, a new A zone begins. Thus, for any x, y\nδ(qb, x, y) = (qb, λ).\nand\nδ(qb, x, z0) =\n(\n(qa\n1, z0)\nif x = 1\n(qa\n0, z0)\nif x = 0.\nThe betting function is equal to 1/2 everywhere (i.e no bet) except on state qb, where\nβ(qb, y)(z) =\n(\n1\nif y = z\n0\nif y ̸= z.\nand β stops betting once start stack symbol is met, i.e.\nβ(qb, z0) = 1\n2.\n⊓⊔\n11"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"that S[i..j] = λ if i > j, and S[0] is the leftmost symbol of S. We write S[i] for S[i..i] (the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"w = S[0..|w| −1]. All logarithms are taken in base |Σ|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"d(w) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"d(wa) = |Σ|s−1d(w). If s = 1, this expected value is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"n→∞d(S[0..n −1]) = ∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"S∞[d] = {S ∈Σ∞| d succeeds on S}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"defined by d′(w) = |Σ|(s′−s)|w|d(w) is an s′-gale. Moreover, if s ≤s′, then S∞[d] ⊆S∞[d′].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"dimH(X) = inf{s | there is an s −gale d such that X ⊆S∞[d]}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"π(a) = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"A bounded pushdown gambler (BPDG) is an 8-tuple G =(Q, Σ, Γ, δ, β, q0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"tion δ(q, b, a) = ⊥when undefined; and we write δ(q, b, a) = (δQ(q, b, a), δΓ∗(q, b, a))),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"• δ(q, λ, a) =⊥","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"• δ(q, b, a) =⊥for all b ∈Σ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"2. for every q ∈Q, b ∈Σ ∪{λ}, either δ(q, b, z0) =⊥, or δ(q, b, z0) = (q′, vz0), where q′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"δ(q, w, a) = (q′, z′) means that in state q, reading input w, and popping symbol a from the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"δ∗(q, b, av) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"if δ(q, b, a) ̸=⊥,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"δi(q, λ, v) = δ(δi−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"δ1(q, λ, v) = δ(q, λ, v).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"δc+1(q, λ, v) = ⊥","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"δ∗∗(q, λ, av) = (q, av)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"δ∗∗(q, wb, av) = δ(δi","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"if δ∗∗(q, w, av) = (eq, eaev), δi(eq, λ, eaev) ̸=⊥and δi+1(eq, λ, eaev) =⊥, i ≤c.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"that δ∗∗(q, b, av) ̸=⊥.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"We denote δ∗∗by δ, and δ(q0, w, z0) by δ(w). We write δ = (δQ, δΓ∗) for simplicity.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"β∗(q, av) = β(q, a),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"Let G = (Q, Σ, Γ, δ, β, q0, z0, c) be a bounded pushdown gambler.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"dG(λ) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"dG(wb) = |Σ|dG(w)β(δ(w))(b)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"G(w) = |Σ|(s−1)|w|dG(w).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"G = d.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"Proposition 3.3 Let G = (Q, Σ, Γ, δ, β, q0, z0, c) be a BPDG. Then there is a BPDG G′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"0, 1) such that dG = dG′.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"Proposition 3.4 Let G = (Q, Σ, Γ, δ, β, q0, z0, c) be a BPDG. Then there is a BPDG G′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"0, c′) such that dG = dG′.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"dimBPD(X) = inf{s | there is a bounded pushdown s −gale d such that X ⊆S∞[d]}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"C = (Q, Σ, Γ, δ, ν, q0, z0, c)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"• δ(q, λ, a) =⊥","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"• δ(q, b, a) =⊥for all b ∈Σ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"2. for every q ∈Q, b ∈Σ ∪{λ}, either δ(q, b, z0) =⊥, or δ(q, b, z0) = (q′, vz0), where q′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"ν(q, λ, z) = λ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"ν(q, wb, z) = ν(q, w, z)ν(δQ(q, w, z), b, δΓ∗(q, w, z))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"C(w) = ν(q0, w, z0).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"Proposition 4.1 Let C = (Q, Σ, Γ, δ, ν, q0, z0, c) be a BPDC. Then there is a BPDC C′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"0, 1) such that C(w) = C′(w) for every w ∈Σ∗.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"Proposition 4.2 Let C = (Q, Σ, Γ, δ, ν, q0, z0, c) be a BPDC. Then there is a BPDC C′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"0, c′) such that C(w) = C′(w) for every w ∈Σ∗.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"A BPDC C = (Q, Σ, Γ, δ, ν, q0, z0) is information-lossless (IL) if the function","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"ρC(S) = lim inf","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"ρBPD(S) = inf{ρC(S) | C is a ILBPDC}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"dimBPD(S) = ρBPD(S).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"Given an input x ∈Σ∗, LZ parses x in different phrases xi, i.e., x = x1x2 . . . xn (xi ∈Σ∗)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"xj = y). Therefore for every i, xi = xl(i)bi for l(i) < i and bi ∈Σ. We sometimes denote","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"LZ encodes xi by a prefix free encoding of l(i) and the symbol bi, that is, if x =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"LZ(x) = cl(1)b1cl(2)b2 . . . cl(n)bn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"where ci is a prefix-free coding of i (and x0 = λ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"ρLZ(S) = lim inf","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"A BPDG G = (Q, Σ, Γ, δ, β, q0, z0) is nonvanishing if 0 < β(q, z)(b) < 1 for","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"Proof of Lemma A.1 . Let G = (Q, Σ, δ, β, q0, Γ, z0) be a BPDG, and let ε > 0. For","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"β(q, z)(b) = 1 −|Σ|−ε > 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"a∈Σ,a̸=b","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"β′(q, z)(b) = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"Then, 0 < β′(q, z)(b) < 1 for each q ∈Q, b ∈Σ and z ∈Γ, therefore the BPDG G′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"Proof of Theorem 4.3 Let S ∈Σ∞. For each n ∈N, let wn = S[0..n −1].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"dimBPD(S) ≤s. By our choice of s′, there is an 1-ILBPDC C = (Q, Σ, Γ, δ, ν, q0, z0) for","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"I = {n ∈N | |C(wn)| < s′n}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"C = (Q, Σ, Γ, δ, ν, q0, z0), and k ∈Z+ , we construct the 1-bounded pushdown gambler","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"(BPDG) G = G(C, k) = (Q′, Σ, Γ′, δ′, β′, q′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"i) Q′ = Q × {0, 1, . . . , k −1}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"0 = (q0, 0)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"iii) Γ′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"i=2k","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"0 = z2k","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"δ′((q, i), b, a) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"z, and for each y ∈Γ+, if y = y1y2 · · · y2kl+n with n < 2k, then by ∈(Γ′)+ is such that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"by1 = y1 · · · y2k+n, by2 = y2k+n+1 · · · y4k+n, . . . , byl = y2k(l−1)+n+1 · · · y2kl+n.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"β′((q, i), a)(b) = σ(q, bΣk−i−1, a)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"where σ(q, A, a) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"dG(wu) = |Σ||u|−|ν(δQ(w),u,δΓ∗(w))| σ(δQ(wu), Σk−|u|, \\","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"Proof of Lemma A.2. We use induction on the string u. If u = λ, the lemma is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"dG(wub) = |Σ|σ(δQ(wu), bΣk−|u|−1, \\","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"= |Σ|1−|ν(δQ(wu),b,δΓ∗(wu))|σ(δQ(wub), Σk−|u|−1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"Lemma A.3 In Construction A.1, if w = w0w1 · · ·wn−1, where each wi ∈Σk , then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"dG(w) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"Proof of Lemma A.3. We use induction on n. For n = 0, the identity is clear.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"Assume that it holds for w = w0w1 · · · wn−1, with each wi ∈Σk, and let w′ = w0w1 · · · wn.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"Then Lemma A.2 with u = wn tells us that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"dG(w′) = |Σ|k−|ν(δQ(w),wn,δΓ∗(w))|","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"where l = ⌈log |Q|⌉and m = max{|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"which |ν(δQ(z), w, δΓ∗(z))| = j does not exceed |Q||Σ|j. Hence","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"δΓ∗(z)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"j=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"|Q||Σ|j|Σ|−j = |Q|(mk + 1)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"dG(w) = |Σ||w|−|C(w)|−|w|","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"where l = ⌈log |Q|⌉and m = max {|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"= |Σ||w|−|C(w0j)|−|w|","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"Let l = ⌈log |Q|⌉and m = max{|ν(q, b, a)| | q ∈Q, b ∈Σ, a ∈Γ2}, and fix k ∈Z+ such","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"< s −s′. Let G = G(C, k) be as in Construction A.1. Then, by Lemma","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"J = {n ∈N | ds′′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"CONSTRUCTION A.2 Let G = (Q, Σ, Γ, δ, β, q0, z0) be a nonvanishing 1-BPDG, and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"and q ∈Q, let Gq,z = (Q, Σ, Γ, δ, β, q, z), and define pq,z : Σk →[0, 1] by pq,z(w) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq103","equation_number":null,"raw_text":"|Σ|−kdGq,z(w). Since G is nonvanishing and each dGq,z is a martingale with dGq,z(λ) = 1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq104","equation_number":null,"raw_text":"|Θq,z(w)| = lq,z(w)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq105","equation_number":null,"raw_text":"lq,z(w) = 1 + ⌈log","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq106","equation_number":null,"raw_text":"define the 1-BPDC C = C(G, k) = (Q′, Σ, Γ′, δ′, ν′, q′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq107","equation_number":null,"raw_text":"i) Q′ = Q × Σ<k","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq108","equation_number":null,"raw_text":"0 = (q0, λ)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq109","equation_number":null,"raw_text":"iii) Γ′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq110","equation_number":null,"raw_text":"i=2k","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq111","equation_number":null,"raw_text":"0 = z2k","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq112","equation_number":null,"raw_text":"δ′((q, w), b, a) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq113","equation_number":null,"raw_text":"if |w| = k −1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq114","equation_number":null,"raw_text":"ν′((q, w), b, a) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq115","equation_number":null,"raw_text":"if |w| = k −1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq116","equation_number":null,"raw_text":"Since each range(Θq,z) is an instantaneous code, it is easy to see that the BPDC C =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq117","equation_number":null,"raw_text":"Let w = w0w1 · · · wn−1, where each wi ∈Σk.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq118","equation_number":null,"raw_text":"0 ≤i < n, let qi = δQ(w0 · · · wi−1) and zi = δΓ∗(w0 · · · wi−1). Then,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq119","equation_number":null,"raw_text":"|C(w)| =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq120","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq121","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq122","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq123","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq124","equation_number":null,"raw_text":"= (k + 2)n −log","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq125","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq126","equation_number":null,"raw_text":"= (k + 2)n −log dG(w) = (1 + 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq127","equation_number":null,"raw_text":"Otherwise, let w = w′z, where |w′| is a multiple of k and |z| = j, 0 < j < k.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq128","equation_number":null,"raw_text":"|C(w)| = |C(w′)|","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq129","equation_number":null,"raw_text":"s−s′, and let C = C( eG, k) be as in Construction A.2. Then Lemma A.7 tell us","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq130","equation_number":null,"raw_text":"Proof of Theorem 5.1 Let m ∈N, and let k = k(m) be an integer to be determined","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq131","equation_number":null,"raw_text":"the set of strings whose every kth bit is zero), it follows that |Tn| ≥2an, where a = 1−1/k.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq132","equation_number":null,"raw_text":"yb = x.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq133","equation_number":null,"raw_text":"Let An = {a1, . . . au} be the set of palindromes in Tn. Since fixing the n/2 first bits","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq134","equation_number":null,"raw_text":"Let us separate the remaining strings in Tn −An into two sets Xn = {x1, . . . xt} and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq135","equation_number":null,"raw_text":"Yn = {y1, . . . yt} with (xi)−1 = yi for every 1 ≤i ≤t. Let us choose X, Y such that x1 and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq136","equation_number":null,"raw_text":"Sn = a1 . . . au 12n x1 . . . xt 12n+1 yt . . . y1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq137","equation_number":null,"raw_text":"S = S1S2 . . . Sk−1 1k 1k+1 . . . 12k−1 SkSk+1 . . .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq138","equation_number":null,"raw_text":"Sn = a1, . . . , au, 12n, x1, . . . , xt, 12n+1, yt, . . . , y1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq139","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq140","equation_number":null,"raw_text":"2aj = 2a(i−1) −2a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq141","equation_number":null,"raw_text":"where b = b(a) is arbitrarily close to 1. Letting ti = |Ti|, the number of bits output by LZ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq142","equation_number":null,"raw_text":"where c = c(b) is arbitrarily close to 1. Therefore","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq143","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq144","equation_number":null,"raw_text":"j=1(jtj + 4j), (the two flags plus the extra flags between Sk−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq145","equation_number":null,"raw_text":"j=1 tj(j −1)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq146","equation_number":null,"raw_text":"j=1 j(tj + 4)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq147","equation_number":null,"raw_text":"= c −c","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq148","equation_number":null,"raw_text":"j=1(tj + 4j)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq149","equation_number":null,"raw_text":"j=1 j(tj + 4)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq150","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq151","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq152","equation_number":null,"raw_text":"j=M+1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq153","equation_number":null,"raw_text":"= 2k2 +","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq154","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq155","equation_number":null,"raw_text":"j=M+1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq156","equation_number":null,"raw_text":"j=M+1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq157","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq158","equation_number":null,"raw_text":"j=M+1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq159","equation_number":null,"raw_text":"j=M+1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq160","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq161","equation_number":null,"raw_text":"j=M+1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq162","equation_number":null,"raw_text":"j=M+1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq163","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq164","equation_number":null,"raw_text":"= M(2k2 +","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq165","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq166","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq167","equation_number":null,"raw_text":"j=1 |Yj|","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq168","equation_number":null,"raw_text":"j=1(j|Tj| + 4j)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq169","equation_number":null,"raw_text":"j=1 j |Tj|−|Aj|","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq170","equation_number":null,"raw_text":"j=1(j|Tj| + 4j)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq171","equation_number":null,"raw_text":"j=1(j|Aj| + 4j)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq172","equation_number":null,"raw_text":"j=1(j|Tj| + 4j) .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq173","equation_number":null,"raw_text":"j=1(j|Aj| + 4j)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq174","equation_number":null,"raw_text":"j=1(j|Tj| + 4j) ≤lim sup","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq175","equation_number":null,"raw_text":"j=1 j(|Aj| + 4 + 2k2)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq176","equation_number":null,"raw_text":"j=1 |Tj|","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq177","equation_number":null,"raw_text":"j=1 j(2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq178","equation_number":null,"raw_text":"j=1 2aj","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq179","equation_number":null,"raw_text":"v = |S1S2 . . . Sk−1 1k 1k+1 . . . 12k−1|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq180","equation_number":null,"raw_text":"until v i.e. for i = 0, . . . v −1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq181","equation_number":null,"raw_text":"δ(qi, x, y) = (qi+1, y)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq182","equation_number":null,"raw_text":"δ(qv, x, y) = (qa","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq183","equation_number":null,"raw_text":"Then δ skips through A until the string 1k is met, i.e. for i = 0, . . . k −1 and any x, y","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq184","equation_number":null,"raw_text":"i , x, y) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq185","equation_number":null,"raw_text":"if x = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq186","equation_number":null,"raw_text":"if x = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq187","equation_number":null,"raw_text":"k, x, y) = (q1f, y).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq188","equation_number":null,"raw_text":"δ(q1f, x, y) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq189","equation_number":null,"raw_text":"if x = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq190","equation_number":null,"raw_text":"if x = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq191","equation_number":null,"raw_text":"k ones, i.e until the start of the second flag, i.e for i = 0, . . . k −1 and any x, y","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq192","equation_number":null,"raw_text":"i , x, y) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq193","equation_number":null,"raw_text":"if x = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq194","equation_number":null,"raw_text":"if x = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq195","equation_number":null,"raw_text":"k , x, y) = (qr","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq196","equation_number":null,"raw_text":"is to pop k ones, i.e for i = 0, . . . k −1 and any x, y","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq197","equation_number":null,"raw_text":"i , x, y) = (qr","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq198","equation_number":null,"raw_text":"k, x, y) = (q2f","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq199","equation_number":null,"raw_text":"δ(q2f, x, y) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq200","equation_number":null,"raw_text":"if x = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq201","equation_number":null,"raw_text":"if x = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq202","equation_number":null,"raw_text":"δ(qb, x, y) = (qb, λ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq203","equation_number":null,"raw_text":"δ(qb, x, z0) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq204","equation_number":null,"raw_text":"if x = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq205","equation_number":null,"raw_text":"if x = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq206","equation_number":null,"raw_text":"β(qb, y)(z) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq207","equation_number":null,"raw_text":"if y = z","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq208","equation_number":null,"raw_text":"if y ̸= z.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq209","equation_number":null,"raw_text":"β(qb, z0) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":35218,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}