Oracles Are Subtle But Not Malicious

Scott Aaronson

(no affiliation)

Are you frustrated by the scarcity of nonrelativizing circuit lower bounds?

Dissatisfied by results that apply only to made-up classes like “MAEXP”?

Do you hunger for an oracle relative to which PP has linear-size circuits?

Or … do you feel like it’s easy to construct a relativized world where anything is true—like you never get more out of oracle results than you put in to them?

If you answered “yes” to any of these questions, take a look at

ECCC TR05-040. Oracle where PP has linear-size circuits

Our competitors can only promise this for MA and PNP

Oracle where PNP = P = PEXP

New proof that in the unrelativized world, PP does NOT have linear-size circuits

Bonus! PP doesn’t have linear-size quantum circuits either

Extra Bonus: Not even quantum circuits with quantum advice

What are you waiting for??

DOWNLOAD NOW!

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Oracle where has linear-size circuits

If P=NP, then given any Boolean function f with polynomial-size circuits, you can learn such a circuit in

The implications are endless: computational learning theory (Bshouty et al.’s algorithm), circuit minimization, Karp-Lipton collapses…

NPBPP||

fNPP||

NP

MA

AM

NPP||

NPZPP||

NPBPP||NPP

NPZPP

NPBPP

pS2

pathBPP

PP

PPBP

BLACK-B

OX B

ARRIER

Successful nonrelativizing

incursion

RELATIVIZ

ATION B

ARRIER

NP

MA

AM

NPP||

NPZPP||

NPBPP||NPP

NPZPP

NPBPP

pS2

pathBPP

PP

PPBP

BLACK-B

OX B

ARRIER

Successful nonrelativizing

incursion

RELATIVIZ

ATION B

ARRIER