Class Description

MA_POLYLOG: MA With Polylog Verifier

Identical to MA except for that Arthur (the verifier) has random access to the proof string given by Merlin, and is limited to running times of order ${\displaystyle O({\textrm {poly}}(\log n))}$.

This class was used by [SM03] to show that if EXP has circuits of polynomial size, then EXP = MA.

Linked From

EXP: Exponential Time

Equals the union of DTIME(2^p(n)) over all polynomials p.

Also equals P with E oracle.

If L = P then PSPACE = EXP.

If EXP is in P/poly then EXP = MA [BFL91].

Problems complete for EXP under many-one reductions have measure 0 in EXP [May94], [JL95].

There exist oracles relative to which

• EXP = NP = ZPP [Hel84a], [Hel84b], [Hel86],
• EXP = NEXP but still P does not equal NP [Dek76],
• EXP does not equal PSPACE [Dek76].

[BT04] show the following rather striking result: let A be many-one complete for EXP, and let S be any set in P of subexponential density. Then A-S is Turing-complete for EXP.

[SM03] show that if EXP has circuits of polynomial size, then P can be simulated in MA_POLYLOG such that no deterministic polynomial-time adversary can generate a list of inputs for a P problem that includes one which fails to be simulated. As a result, EXP ⊆ MA if EXP has circuits of polynomial size.

[SU05] show that EXP ${\displaystyle \not \subseteq }$ NP/poly implies EXP ${\displaystyle \not \subseteq }$ P^||NP/poly.

In descriptive complexity EXPTIME can be defined as SO(${\displaystyle 2^{n^{O(1)}}}$) which is also SO(LFP)

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