## Dimension is compression (2005)

Venue: | In Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science |

Citations: | 15 - 9 self |

### BibTeX

@INPROCEEDINGS{López-valdés05dimensionis,

author = {María López-valdés and Elvira Mayordomo},

title = {Dimension is compression},

booktitle = {In Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science},

year = {2005},

pages = {676--685},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. Effective fractal dimension was defined by Lutz (2003) in order to quantitatively analyze the structure of complexity classes. Interesting connections of effective dimension with information theory were also found, in fact the cases of polynomial-space and constructive dimension can be precisely characterized in terms of Kolmogorov complexity, while analogous results for polynomial-time dimension haven’t been found. In this paper we remedy the situation by using the natural concept of reversible time-bounded compression for finite strings. We completely characterize polynomial-time dimension in terms of polynomial-time compressors. 1

### Citations

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Citation Context ...emes. Dually, strong polynomial-time-dimension [3] corresponds to the worst case asymptotic compression ratio. The proof of these results uses and interesting generalization of arithmetic coding (see =-=[7]-=- for an introduction to arithmetic coding and its history). Several results on the polynomial-time dimension of complexity classes can be now interpreted as compressibility results. For example, the (... |

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Citation Context ...on (1) holds 1. An extender, that is, ∀ w, w ′ ∈ {0, 1} ∗ w ⊑ w ′ ⇒ C(w) ⊑ C(w ′ ). 2. Lempel-Ziv data compression algorithm for its three most common variants (notice that it is not an extender. See =-=[10, 11]-=- for details). In fact, Lempel-Ziv compression algorithm verifies the common-prefix condition lemma 1. Let w ∈ {0, 1} ∗ . If w = w1w2 . . . wnv where w1, w2, . . . , wn are the phrases obtained by the... |

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Citation Context ...gue measure [18]. Important applications in Computational Complexity have been found including circuit-size complexity, polynomial-time degrees, the size of NP, zero-one laws, and oracle classes (see =-=[21, 13, 9]-=- for a summary of the main results). In parallel, the connections of this effective dimension with algorithmic information started being patent, as it could be suspected from earlier results by Ryabko... |

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Citation Context ...le time-bounded compression for finite strings. We completely characterize polynomial-time dimension in terms of polynomial-time compressors. 1 Introduction Effective fractal dimension was defined in =-=[13]-=- in order to quantitatively analyze the structure of complexity classes. See [12, 16] for a summary of the main results. In parallel, the connections of this effective dimension with algorithmic infor... |

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Citation Context ...e and polynomialspace dimension were characterized precisely as the best case asymptotic compression rate when using plain, recursive, and polynomial-space-bounded Kolmogorov complexity, respectively =-=[15, 14, 6]-=-. But the case of polynomial-time bounds remained elusive [8]. This is not strange since computing even approximately the value of time-bounded Kolmogorov complexity seems to require an exponential se... |

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Citation Context ...r to quantitatively analyze the structure of complexity classes with the immediate motivation of overcoming the limitations of resource-bounded measure, a generalization of classical Lebesgue measure =-=[18]-=-. Important applications in Computational Complexity have been found including circuit-size complexity, polynomial-time degrees, the size of NP, zero-one laws, and oracle classes (see [21, 13, 9] for ... |

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Citation Context ...ion ratio attained by these polynomial-time ⋆ This research was supported in part by Spanish Government MEC project TIC 2002-04019-C03-03scompression schemes. Dually, strong polynomial-time-dimension =-=[2]-=- corresponds to the worst case asymptotic compression ratio. Several results on the polynomial-time dimension of complexity classes can be now interpreted as compressibility results. For example, the ... |

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Citation Context ...e and polynomialspace dimension were characterized precisely as the best case asymptotic compression rate when using plain, recursive, and polynomial-space-bounded Kolmogorov complexity, respectively =-=[15, 14, 6]-=-. But the case of polynomial-time bounds remained elusive [8]. This is not strange since computing even approximately the value of time-bounded Kolmogorov complexity seems to require an exponential se... |

66 | Kolmogorov complexity and Hausdorff dimension
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Citation Context ...he main results). In parallel, the connections of this effective dimension with algorithmic information started being patent, as it could be suspected from earlier results by Ryabko [25, 26], Staiger =-=[27, 28]-=-, and Cai and Hartmanis [6]. The cases of constructive, recursive and polynomial-space dimension were characterized precisely as the best case asymptotic compression rate when using plain, recursive, ... |

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Citation Context ...for a summary of the main results). In parallel, the connections of this effective dimension with algorithmic information started being patent, as it could be suspected from earlier results by Ryabko =-=[25, 26]-=-, Staiger [27, 28], and Cai and Hartmanis [6]. The cases of constructive, recursive and polynomial-space dimension were characterized precisely as the best case asymptotic compression rate when using ... |

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Citation Context ...he main results). In parallel, the connections of this effective dimension with algorithmic information started being patent, as it could be suspected from earlier results by Ryabko [25, 26], Staiger =-=[27, 28]-=-, and Cai and Hartmanis [6]. The cases of constructive, recursive and polynomial-space dimension were characterized precisely as the best case asymptotic compression rate when using plain, recursive, ... |

40 |
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Citation Context ... each polynomial-time gale into a simple version that requires little accuracy. Then we apply a generalization of arithmetic coding [7] to this new gale. We will use the following result. 7Lemma 5.2 =-=[22]-=- Let d1 be a martingale. Let c : {0, 1} ∗ → [0, +∞) be an exactly p-computable function such that for each w ∈ {0, 1} ∗ , |c(w) − d1(w)| ≤ 2 −|w| . Let d2 be recursively defined as follows d2(λ) = c(λ... |

37 | Finite-state dimension
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Citation Context ...nomial-space-bounded Kolmogorov complexity, respectively [23, 20, 11], and the low resource-bounds of finite-state and pushdown devices have been connected to the corresponding compression algorithms =-=[8, 1]-=-. See [24] for an overall survey. ∗ Dept. de Informática e Ingeniería de Sistemas, Instituto de Investigación en Ingeniería de Aragón (I3A), Universidad de Zaragoza, Zaragoza, SPAIN. {marlopez, elvira... |

36 |
On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line
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Citation Context ...he connections of this effective dimension with algorithmic information started being patent, as it could be suspected from earlier results by Ryabko [25, 26], Staiger [27, 28], and Cai and Hartmanis =-=[6]-=-. The cases of constructive, recursive and polynomial-space dimension were characterized precisely as the best case asymptotic compression rate when using plain, recursive, and polynomial-space-bounde... |

35 | Fractal dimension and logarithmic loss unpredictability
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(Show Context)
Citation Context ... In section 5 we transform each gale into a compressor that requires only a time increase of a linear factor. In section 6 we show that compression is an upper bound on dimension. Hitchcock showed in =-=[7]-=- that p-dimension can be characterized in terms of on-line prediction algorithms, using the well-studied log-loss prediction ratio. Our result can thus be interpreted as a joining bridge between the p... |

34 | Hausdorff dimension in exponential time
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(Show Context)
Citation Context ... the notion of a.e. (almost everywhere) and i.o. (infinitely often) compressibility for sets of infinite sequences as the asymptotic best (respectively worse) compression ratio. Definition 6. For α ∈ =-=[0, 1]-=- and X ⊆ C, 1. X is α-i.o. polynomial-time compressible if there is a polynomial-time compressor (C, D) that does not start from scratch and such that for every A ∈ X lim inf n |C(A[0 . . . n − 1])| n... |

30 | MAX3SAT is exponentially hard to approximate if NP has positive dimension - Hitchcock - 2002 |

27 | Effective Fractal Dimension: Foundations and Applications
- Hitchcock
- 2003
(Show Context)
Citation Context ...e and polynomialspace dimension were characterized precisely as the best case asymptotic compression rate when using plain, recursive, and polynomial-space-bounded Kolmogorov complexity, respectively =-=[15, 14, 6]-=-. But the case of polynomial-time bounds remained elusive [8]. This is not strange since computing even approximately the value of time-bounded Kolmogorov complexity seems to require an exponential se... |

23 |
The fractal geometry of complexity classes
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(Show Context)
Citation Context ...gue measure [18]. Important applications in Computational Complexity have been found including circuit-size complexity, polynomial-time degrees, the size of NP, zero-one laws, and oracle classes (see =-=[21, 13, 9]-=- for a summary of the main results). In parallel, the connections of this effective dimension with algorithmic information started being patent, as it could be suspected from earlier results by Ryabko... |

15 |
Compressibility and resource bounded measure
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Citation Context ... sublinear amount. Here we obtain results on the compressibility of complete and autoreducible languages. Buhrman and Longprè have given a characterization of p-measure in terms of compressibility in =-=[4]-=-, but in that case the compressors are restricted to extenders and the encoder is required to give several alternatives, one of them being the correct output. In the light of our present results we ca... |

15 | Effective fractal dimensions
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(Show Context)
Citation Context ...gue measure [18]. Important applications in Computational Complexity have been found including circuit-size complexity, polynomial-time degrees, the size of NP, zero-one laws, and oracle classes (see =-=[21, 13, 9]-=- for a summary of the main results). In parallel, the connections of this effective dimension with algorithmic information started being patent, as it could be suspected from earlier results by Ryabko... |

15 |
Noiseless coding of combinatorial sources
- Ryabko
- 1986
(Show Context)
Citation Context ...for a summary of the main results). In parallel, the connections of this effective dimension with algorithmic information started being patent, as it could be suspected from earlier results by Ryabko =-=[25, 26]-=-, Staiger [27, 28], and Cai and Hartmanis [6]. The cases of constructive, recursive and polynomial-space dimension were characterized precisely as the best case asymptotic compression rate when using ... |

12 |
entropy rates, and compression
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- 2006
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Citation Context ...est case asymptotic compression rate when using plain, recursive, and polynomial-space-bounded Kolmogorov complexity, respectively [15, 14, 6]. But the case of polynomial-time bounds remained elusive =-=[8]-=-. This is not strange since computing even approximately the value of time-bounded Kolmogorov complexity seems to require an exponential search. The main difference with space-bounded Kolmogorov compl... |

7 |
Effective fractal dimensions. Mathematical Logic Quarterly, 51:62–72
- Lutz
- 2005
(Show Context)
Citation Context ...mial-time dimension in terms of polynomial-time compressors. 1 Introduction Effective fractal dimension was defined in [13] in order to quantitatively analyze the structure of complexity classes. See =-=[12, 16]-=- for a summary of the main results. In parallel, the connections of this effective dimension with algorithmic information started being patent. The cases of constructive, recursive and polynomialspace... |

7 | Effective fractal dimension in algorithmic information theory
- Mayordomo
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(Show Context)
Citation Context ...-bounded Kolmogorov complexity, respectively [23, 20, 11], and the low resource-bounds of finite-state and pushdown devices have been connected to the corresponding compression algorithms [8, 1]. See =-=[24]-=- for an overall survey. ∗ Dept. de Informática e Ingeniería de Sistemas, Instituto de Investigación en Ingeniería de Aragón (I3A), Universidad de Zaragoza, Zaragoza, SPAIN. {marlopez, elvira}@unizar.e... |

6 | Infinitely-often autoreducible sets
- Beigel, Fortnow, et al.
- 2003
(Show Context)
Citation Context ...ove in [1] that the class has p-dimension 1. Theorem 5. The class of sets that are NOT i.o. polynomial-time Turing autoreducible are i.o. polynomial-time incompressible. Proof. Beigel et al. prove in =-=[3]-=- that the class has p-dimension 1. We next show that there exist polynomial-time many-one degrees with every possible value for both a.e. and i.o. compressibility. Theorem 6. Let x, y be computable re... |

4 | Bounded pushdown dimension vs lempel ziv information density
- Albert, Mayordomo, et al.
- 2007
(Show Context)
Citation Context ...nomial-space-bounded Kolmogorov complexity, respectively [23, 20, 11], and the low resource-bounds of finite-state and pushdown devices have been connected to the corresponding compression algorithms =-=[8, 1]-=-. See [24] for an overall survey. ∗ Dept. de Informática e Ingeniería de Sistemas, Instituto de Investigación en Ingeniería de Aragón (I3A), Universidad de Zaragoza, Zaragoza, SPAIN. {marlopez, elvira... |

2 |
and En hui Yang, “Grammar based codes: A new class of universal lossless source codes
- Kieffer
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(Show Context)
Citation Context ...on prefix of length at least |LZ(w)|−log n ≥ |LZ(w)|−log(|w|). We leave for the complete version of this paper an analysis of the case of Grammar-based compressors, that generalize Lempel-Ziv methods =-=[9]-=-. Polynomial-time compressors (C, D) that are length increasing in the encoder C and for which we can control, for all w and all i ≥ 0, the number of strings u satisfying |C(wu)| = |C(w)| + i, don’t s... |

2 |
Effective Hausdorff dimension. In Classical and New Paradigms of Computation and their Complexity Hierarchies
- Mayordomo
- 2004
(Show Context)
Citation Context ...mial-time dimension in terms of polynomial-time compressors. 1 Introduction Effective fractal dimension was defined in [13] in order to quantitatively analyze the structure of complexity classes. See =-=[12, 16]-=- for a summary of the main results. In parallel, the connections of this effective dimension with algorithmic information started being patent. The cases of constructive, recursive and polynomialspace... |

1 |
A universal algortihm for sequential data compression
- Lempel, Ziv
- 1977
(Show Context)
Citation Context ...on (1) holds 1. An extender, that is, ∀ w, w ′ ∈ {0, 1} ∗ w ⊑ w ′ ⇒ C(w) ⊑ C(w ′ ). 2. Lempel-Ziv data compression algorithm for its three most common variants (notice that it is not an extender. See =-=[10, 11]-=- for details). In fact, Lempel-Ziv compression algorithm verifies the common-prefix condition lemma 1. Let w ∈ {0, 1} ∗ . If w = w1w2 . . . wnv where w1, w2, . . . , wn are the phrases obtained by the... |