Oracles are subtle but not malicious
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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 “MA EXP ”?

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Oracles are subtle but not malicious

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?


What are you waiting for?? circuit lower bounds?

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If you answered “yes” to any of these questions, take a look at ECCC TR05-040.

  • Oracle where PP has linear-size circuitsOur 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 eitherExtra Bonus: Not even quantum circuits with quantum advice


  • Oracle where has linear-size circuits circuit lower bounds?

  • 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…


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