## Dimension in Complexity Classes (2000)

Venue: | SIAM Journal on Computing |

Citations: | 114 - 17 self |

### BibTeX

@ARTICLE{Lutz00dimensionin,

author = {Jack H. Lutz},

title = {Dimension in Complexity Classes},

journal = {SIAM Journal on Computing},

year = {2000},

volume = {32},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

A theory of resource-bounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension"). Other choices of the parameter yield internal dimension theories in E, E 2 , ESPACE, and other complexity classes, and in the class of all decidable problems. In general, if C is such a class, then every set X of languages has a dimension in C, which is a real number dim(X j C) 2 [0; 1]. Along with the elements of this theory, two preliminary applications are presented: 1. For every real number 0 1 2 , the set FREQ( ), consisting of all languages that asymptotically contain at most of all strings, has dimension H() | the binary entropy of | in E and in E 2 . 2. For every real number 0 1, the set SIZE( 2 n n ), consisting of all languages decidable by Boolean circuits of at most 2 n n gates, has dimension in ESPACE.

### Citations

274 |
The geometry of fractal sets
- Falconer
- 1985
(Show Context)
Citation Context ...ly only the beginning. Classical Hausdor dimension is a sophisticated mathematical theory that has emerged as one of the most important tools for the investigation of fractal sets. (See, for example [=-=6] for -=-a good introduction and overview.) Many sets of interest in computational complexity seem to have \fractal-like" structures. Resource-bounded dimension will be a useful tool for the study of such... |

245 |
Ergodic theory and information
- Billingsley
- 1965
(Show Context)
Citation Context ...ws that dim(FREQ( ) j R()) H() : The case = all of Theorem 5.3 says simply that the classical Hausdor dimensions of FREQ() and FREQ( ) are both H(). This was proven in 1949 by Eggleston[5, 2]. The proof here is a new proof, using gales, of this classical result. However, it is complexity-theoretic results of the following kind that are of interest in this paper. Corollary 5.4. 1. For all ... |

172 | Almost everywhere high nonuniform complexity
- Lutz
- 1992
(Show Context)
Citation Context ...SIZE( 2 n n ), consisting of all languages decidable by Boolean circuits of at most 2 n n gates, has dimension in ESPACE. 1 Introduction Since the development of resource-bounded measure in 1991 [9]=-=-=-, the investigation of the internal, measure-theoretic structure of complexity classes has produced a rapidly growing body of new insights and results. As indicated by the survey papers [1, 3, 10, 12]... |

149 |
Zufälligkeit und Wahrscheinlichkeit
- Schnorr
- 1971
(Show Context)
Citation Context ...(δ)[0... n − 1]) ≤ 2 l ⌋ d(λ) 15whence R(δ) ∈ S ∞ [d]. ✷ We conclude this section by mentioning some relevant earlier work relating martingales and supermartingales to computable dimension. Schnorr =-=[14, 15]-=- defined a martingale d to have exponential order on a sequence (equivalently, language) S if lim sup n→∞ log d(S[0..n − 1]) n > 0 (4.1) and proved that no computable martingale can have exponential o... |

141 |
The synthesis of two-terminal switching circuits
- Shannon
- 1949
(Show Context)
Citation Context ...f gates in the smallest n-input Boolean circuit that decides A \ f0; 1g n . For each function f : N ! N, we dene the circuit-size complexity class SIZE(f) = fA 2 C j (8 1 n)CSA (n) f(n)g : Shannon [13] showed (essentially) that SIZE( 2 n n ) has measure 0 in C for all s1, and Lutz [9] showed that SIZE( 2 n n ) also has measure 0 in ESPACE for all s1. We now use resource-bounded dimension to giv... |

106 |
Etude Critique de la Notion de Collectif
- Ville
- 1939
(Show Context)
Citation Context ...ralization of the martingales that are the basis of resource-bounded measure. (Our characterization can be regarded as an analog of Ville's martingale characterization of the Lesbesgue measure 0 sets =-=[1-=-4].) We then generalize classical dimension by introducing a resource bound (a parameter of the theory) and requiring the gales to be -computable. We show that this induces a well-behaved notion of d... |

91 | The quantitative structure of exponential time - Lutz - 1997 |

43 |
The fractional dimension of a set defined by decimal properties
- Eggleston
- 1949
(Show Context)
Citation Context ...ows that dim(FREQ(≤ α) | R(∆)) ≥ H(α) . The case ∆ = all of Theorem 5.3 says simply that the classical Hausdorff dimensions of FREQ(α) and FREQ(≤ α) are both H(α). This was proven in 1949 by Eggleston=-=[5, 2]-=-. The proof here yields a new proof, using gales, of this classical result. However, it is complexity-theoretic results of the following kind that are of interest in this paper. Corollary 5.4. 1. For ... |

43 | A tight upper bound on Kolmogorov complexity and uniformly optimal prediction
- Staiger
- 1998
(Show Context)
Citation Context ... existence of an s < 1 for which the s-gale d (s) (w) = 2 (s−1)|w| d(w) succeeds on S. Thus Schnorr’s result says that dimcomp({S}) = 1 for every Church-stochastic sequence S. Ryabko [13] and Staiger =-=[17]-=- defined the exponent of increase λd(S) of a martingale d on a sequence S to be the left-hand side of (4.1). (We are using Staiger’s notation here.) Both papers paid particular attention to the quanti... |

40 | Resource-bounded measure and randomness
- Ambos-Spies, Mayordomo
- 1997
(Show Context)
Citation Context ...CE, and other complexity classes, and in the class of all decidable problems. In general, if C is such a class, then every set X of languages has a dimension in C, which is a real number dim(X j C) 2 =-=[-=-0; 1]. Along with the elements of this theory, two preliminary applications are presented: 1. For every real number 0 1 2 , the set FREQ( ), consisting of all languages that asymptotically contain ... |

33 |
Resource-bounded measure
- Lutz
- 1998
(Show Context)
Citation Context ...ntitative information that resource-bounded measure can provide in computational complexity. One of these limitation arises from the resource-bounded Kolmogorov zero-one law, which was proven by Lutz =-=[11-=-] and has recently been strengthened by Dai [4]. For any class C in which resourcebounded measure is dened, and for any set X of languages that is | like most sets of interest in computational complex... |

31 |
Twelve problems in resource-bounded measure
- Lutz, Mayordomo
- 1999
(Show Context)
Citation Context ...re in 1991 [9], the investigation of the internal, measure-theoretic structure of complexity classes has produced a rapidly growing body of new insights and results. As indicated by the survey papers =-=[1, 3, 10, 12]-=-, this line of inquiry has shed light on a wide variety of topics in computational complexity. The ongoing fruitfulness of this research is not surprising because resource-bounded measure is a complex... |

31 |
Dimension und äusseres Mass
- Hausdorff
- 1919
(Show Context)
Citation Context ... both these limitations were already present in classical Lesbesgue measure theory. ∗ This research was supported in part by National Science Foundation Grants 9610461 and 9988483.In 1919, Hausdorff =-=[7]-=- augmented classical Lesbesgue measure theory with a theory of dimension. This theory assigns to every subset X of a given metric space a real number dimH(X), which is now called the Hausdorff dimensi... |

23 |
A survey of the theory of random sequences
- Schnorr
- 1977
(Show Context)
Citation Context ...(δ)[0... n − 1]) ≤ 2 l ⌋ d(λ) 15whence R(δ) ∈ S ∞ [d]. ✷ We conclude this section by mentioning some relevant earlier work relating martingales and supermartingales to computable dimension. Schnorr =-=[14, 15]-=- defined a martingale d to have exponential order on a sequence (equivalently, language) S if lim sup n→∞ log d(S[0..n − 1]) n > 0 (4.1) and proved that no computable martingale can have exponential o... |

16 |
The complexity and effectiveness of prediction problems
- Ryabko
- 1994
(Show Context)
Citation Context ...equivalent to the existence of an s < 1 for which the s-gale d (s) (w) = 2 (s−1)|w| d(w) succeeds on S. Thus Schnorr’s result says that dimcomp({S}) = 1 for every Church-stochastic sequence S. Ryabko =-=[13]-=- and Staiger [17] defined the exponent of increase λd(S) of a martingale d on a sequence S to be the left-hand side of (4.1). (We are using Staiger’s notation here.) Both papers paid particular attent... |

14 | Complete sets and structure in subrecursive classes
- Buhrman, Torenvliet
- 1998
(Show Context)
Citation Context ...re in 1991 [9], the investigation of the internal, measure-theoretic structure of complexity classes has produced a rapidly growing body of new insights and results. As indicated by the survey papers =-=[1, 3, 10, 12]-=-, this line of inquiry has shed light on a wide variety of topics in computational complexity. The ongoing fruitfulness of this research is not surprising because resource-bounded measure is a complex... |

9 |
On the synthesis of contact networks
- Lupanov
- 1958
(Show Context)
Citation Context ...ircuits of fewer than 2 n n gates. Proof: Let m = n + logs. Then there are 2 2 m = 2s2 n dierent sets C f0; 1g m . If we let " = s2 , so thats= (1 2"), then for all suciently large n, Lupa=-=nov [8] has shown that e-=-ach of these sets is decided by a circuit of at most 2 m m (1 + ") gates. Now for suciently large n, 2 m m =s2 n n + logs= 2 n n (1 2") n n + logs 2 n n (1 "); so 2 m m (1 + ")s2... |

6 |
Dimension und ausseres
- Hausdor
- 1919
(Show Context)
Citation Context ...e both these limitations were already present in classical Lesbesgue measure theory. This research was supported in part by National Science Foundation Grants 9610461 and 9988483. In 1919, Hausdor [7=-=-=-] augmented classical Lesbesgue measure theory with a theory of dimension. This theory assigns to every subset X of a given metric space a real number dimH (X), which is now called the Hausdor dimensi... |

5 |
The fractional dimension of a set de by decimal properties
- Eggleston
- 1949
(Show Context)
Citation Context ...ws that dim(FREQ( ) j R()) H() : The case = all of Theorem 5.3 says simply that the classical Hausdor dimensions of FREQ() and FREQ( ) are both H(). This was proven in 1949 by Eggleston[5, 2]. The proof here is a new proof, using gales, of this classical result. However, it is complexity-theoretic results of the following kind that are of interest in this paper. Corollary 5.4. 1. For all ... |

2 |
A stronger Kolmogorov zero-one law for resource-bounded measure
- Dai
- 2001
(Show Context)
Citation Context ...sure can provide in computational complexity. One of these limitation arises from the resource-bounded Kolmogorov zero-one law, which was proven by Lutz [11] and has recently been strengthened by Dai =-=[4-=-]. For any class C in which resourcebounded measure is dened, and for any set X of languages that is | like most sets of interest in computational complexity | closed undersnite variations, the zero-o... |