Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question

Baker, Theodore P.; Gill, John; Solovay, Robert M.

doi:10.1137/0204037

Cited by 642 publications

(348 citation statements)

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“…Our constructions, based on the theorems of Baker, Grill, and Solovay [2], are similar to the "slow diagonalizations" of Schöning and Book [13] and Balcâzar [3], and yield sets in different probabilistic classes that are immune with respect to other classes. As a conséquence, strong séparations of the classes under considéra-tion are obtained.…”

Section: Introductionmentioning

confidence: 99%

Immunity and simplicity in relativizations of probabilistic complexity classes

Balcázar¹,

Russo²

1988

RAIRO-Theor. Inf. Appl.

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Section: Introductionmentioning

confidence: 99%

Immunity and simplicity in relativizations of probabilistic complexity classes

Balcázar¹,

Russo²

1988

RAIRO-Theor. Inf. Appl.

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“…Np A with probability one relative to a random oracle A [BGS75]. Thus, since Corollary 3.20 relativizes:…”

Section: Proofmentioning

confidence: 94%

On the structure and complexity of infinite sets with minimal perfect hash functions

Goldsmith

Hemachandra

Kunen³

1991

Lecture Notes in Computer Science

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This paper studies the class of infinite sets that have minimal perfect hash functions one-to-one onto maps between the sets and E·-computable in polynomial time. We show that all standard NP-complete sets have polynomial-time computable minimal per fect hash functions, and give a structural condition sufficient to ensure that all infinite NP sets have polynomial-time computable minimal perfect hash functions: If E = Ef, then all infinite NP sets have polynomial-time computable minimal perfect hash functions. On the other hand, we present evidence that some infinite NP sets, and indeed some infinite P sets, do not have polynomial-time computable minimal perfect hash functions: if an infinite NP set A has polynomial-time computable perfect minimal hash functions, then A has an infinite sparse NP subset, yet we construct a relativized world in which some infinite NP sets lack infinite sparse NP subsets. This world is built upon a result that is of interest in its own right; we determine optimally-with respect to any relativizable proof technique--the complexity of the easiest infinite sparse subsets that infinite P sets are guaranteed to have.

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“…There are proofs that certain techniques will not suffice. These include techniques from computability theory [9], current methods with circuits [58], and a hybrid of the two [4].…”

Section: Seriously Can You Give a More Enlightening Answer To Havementioning

confidence: 99%

Classifying Problems into Complexity Classes

Gasarch

2014

Advances in Computers

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A fundamental problem in computer science is, stated informally: Given a problem, how hard is it?. We measure hardness by looking at the following question: Given a set A whats is the fastest algorithm to determine if "x ∈ A?" We measure the speed of an algorithm by how long it takes to run on inputs of length n, as a function of n. For example, sorting a list of length n can be done in roughly n log n steps.Obtaining a fast algorithm is only half of the problem. Can you prove that there is no better algorithm? This is notoriously difficult; however, we can classify problems into complexity classes where those in the same class are of roughly the same complexity.In this chapter we define many complexity classes and describe natural problems that are in them. Our classes go all the way from regular languages to various shades of undecidable. We then summarize all that is known about these classes.

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Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question

Cited by 642 publications

References 2 publications

Immunity and simplicity in relativizations of probabilistic complexity classes

Immunity and simplicity in relativizations of probabilistic complexity classes

On the structure and complexity of infinite sets with minimal perfect hash functions

Classifying Problems into Complexity Classes

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