Separability and one-way functions

Fortnow, Lance; Rogers, John D.

doi:10.1007/s00037-002-0173-4

Cited by 13 publications

(17 citation statements)

References 23 publications

(25 reference statements)

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“…We will use UP-generics as developed by Fortnow and Rogers [FR94]. To create a UP-generic start with an oracle like TQBF that makes P = PSPACE and add a generic set restricted to have at most one string at lengths that are towers of 2 and no strings at any other lengths.…”

Section: Limitations Of P Np[1] = P Np[2]mentioning

confidence: 99%

Two queries

Buhrman¹,

Fortnow²

Proceedings. Thirteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference

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Section: Limitations Of P Np[1] = P Np[2]mentioning

confidence: 99%

Two queries

Buhrman¹,

Fortnow²

Proceedings. Thirteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference

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“…Then P = PSPACE relative to B, and since Theorem 6.18 also holds relative to B, rerelativizing with a Cohen generic G gives us all the collapses above relative to B ⊕ G with no assumptions. In this and other papers [FR94,FR99], when trying to show that an oracle exists for a certain property, we often assume that P = PSPACE unrelativized before we define the oracle. If our oracle construction is relativizable (and it always is), then this assumption costs us nothing, since it can be discharged by rerelativization.…”

Section: Collapses For Cohen Genericsmentioning

confidence: 99%

“…In Section 7.2, we created a specialized form of generic, an MA-generic, to solve a specific problem. Other specialized generic sets, including UP-generic and (NP ∩ co-NP)-generic sets, have been applied by Fortnow and Rogers [FR94] to get simultaneous collapses and separations of various subclasses of NP. Buhrman and Fortnow also used a UP-generic set to find an oracle where NP = co-NP but P NP[1] = P NP[2] = PSPACE [BF99].…”

Section: Further Work and Open Problemsmentioning

confidence: 99%

An oracle builder’s toolkit

Fenner

Fortnow

Kurtz

et al. 2003

Information and Computation

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We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions;2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SP-generics;3. we show that under additional assumptions these collapses also occur relative to Cohen generics;4. we show that relative to SP-generics, ULIN ∩ co-ULIN ⊆ DTIME(n k ) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ co-NP) ⊆ P relative to these generics;5. we show that there is an oracle relative to which NP/1∩co-NP/1 ⊆ (NP∩co-NP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ⊇ MA.

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“…Indeed, if Proposition Q is true and PH is infinite under an oracle then P = NP and P = NP ∩ coNP under that oracle. The oracles having the latter property were constructed in [1,4,7,12]. It is not hard to see that PH is infinite if and only if Proposition Q is false for all oracles constructed in those papers.…”

Section: Introductionmentioning

confidence: 99%

Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?

et al. 2008

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The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomial-time does this imply the polynomial-time hierarchy collapses? By computing a multivalued function in deterministic polynomial-time we mean on every input producing one of the possible values of the function on that input.We give a relativized negative answer to this question by exhibiting an oracle under which TFNP functions are easy to compute but the polynomial-time hierarchy is infinite. We also show that relative to this same oracle, P = UP and TFNP NP functions are not computable in polynomial-time with an NP oracle.

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Separability and one-way functions

Cited by 13 publications

References 23 publications

Two queries

Two queries

An oracle builder’s toolkit

Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?

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