Who Asked Us? How the Theory of Computing Answers Questions about Analysis

Lutz, Jack H.; Lutz, Neil

doi:10.1007/978-3-030-41672-0_4

Cited by 7 publications

(2 citation statements)

References 50 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example this theory defines, for every subset X of C, a quasipolynomial-time (i.e., n polylog n -time) dimension dim qp (X) in such a way that dim(X | EXP) = dim qp (X ∩ EXP) is a coherent notion of the dimension of X within the complexity class EXP = TIME(2 polynomial ). The second method [26], algorithmic dimension (also called constructive dimension or effective dimension) has to date been more widely investigated, partly because of its interactions with algorithmic randomness (i.e., Martin-Löf ran-domness [35]) and partly because of its applications to classical fractal geometry [29,31]. Algorithmic dimension plays a motivating role in this paper, but resource-bounded dimension is our main topic.…”

Section: Introductionmentioning

confidence: 99%

Dimension and the Structure of Complexity Classes

Lutz

Mayordomo

2022

Theory Comput Syst

Self Cite

View full text Add to dashboard Cite

We prove three results on the dimension structure of complexity classes.1. The Point-to-Set Principle, which has recently been used to prove several new theorems in fractal geometry, has resource-bounded instances. These instances characterize the resource-bounded dimension of a set X of languages in terms of the relativized resource-bounded dimensions of the individual elements of X, provided that the former resource bound is large enough to parametrize the latter. Thus for example, the dimension of a class X of languages in EXP is characterized in terms of the relativized p-dimensions of the individual elements of X. 2. Every language that is ≤ P m -reducible to a p-selective set has p-dimension 0, and this fact holds relative to arbitrary oracles. Combined with a resourcebounded instance of the Point-to-Set Principle, this implies that if NP has positive dimension in EXP, then no quasipolynomial time selective language is ≤ P m -hard for NP.

show abstract

Section: Introductionmentioning

confidence: 99%

Dimension and the Structure of Complexity Classes

Lutz

Mayordomo

2022

Theory Comput Syst

Self Cite

View full text Add to dashboard Cite

show abstract

“…For example this theory defines, for every subset X of C, a quasipolynomial-time (i.e., n polylog n -time) dimension dim qp (X) in such a way that dim(X | EXP) = dim qp (X ∩ EXP) is a coherent notion of the dimension of X within the complexity class EXP = TIME (2 polynomial ). The second method [27], algorithmic dimension (also called constructive dimension or effective dimension) has to date been more widely investigated, partly because of its interactions with algorithmic randomness (i.e., Martin-Löf randomness [35]) and partly because of its applications to classical fractal geometry [30,31]. Algorithmic dimension plays a motivating role in this paper, but resource-bounded dimension is our main topic.…”

Section: Introductionmentioning

confidence: 99%

Dimension in complexity classes

Lutz

Proceedings 15th Annual IEEE Conference on Computational Complexity

119

View full text Add to dashboard Cite

We prove three results on the dimension structure of complexity classes.1. The Point-to-Set Principle, which has recently been used to prove several new theorems in fractal geometry, has resource-bounded instances. These instances characterize the resource-bounded dimension of a set X of languages in terms of the relativized resourcebounded dimensions of the individual elements of X, provided that the former resource bound is large enough to parameterize the latter. Thus for example, the dimension of a class X of languages in EXP is characterized in terms of the relativized p-dimensions of the individual elements of X.2. Every language that is ≤ P m -reducible to a p-selective set has p-dimension 0, and this fact holds relative to arbitrary oracles. Combined with a resource-bounded instance of the Point-to-Set Principle, this implies that if NP has positive dimension in EXP, then no quasipolynomial time selective language is ≤ P m -hard for NP. 3. If the set of all disjoint pairs of NP languages has dimension 1 in the set of all disjoint pairs of EXP languages, then NP has positive dimension in EXP.

show abstract

Extending the reach of the point-to-set principle

Lutz

Mayordomo

2023

Information and Computation

View full text Add to dashboard Cite

Who Asked Us? How the Theory of Computing Answers Questions about Analysis

Cited by 7 publications

References 50 publications

Dimension and the Structure of Complexity Classes

Dimension and the Structure of Complexity Classes

Dimension in complexity classes

Extending the reach of the point-to-set principle

Contact Info

Product

Resources

About