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AC 0  ACC 0  PP, also TC 0  PP NC 1  PSPACE, … , NL  PSPACE

LANDSCAPE OF COMPUTATIONAL COMPLEXITY Spring 2008 State University of New York at Buffalo Department of Computer Science & Engineering Mustafa M. Faramawi, MBA Dr. Kenneth W. Regan.

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AC 0  ACC 0  PP, also TC 0  PP NC 1  PSPACE, … , NL  PSPACE

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  1. LANDSCAPE OF COMPUTATIONAL COMPLEXITY Spring 2008 State University of New York at Buffalo Department of Computer Science & Engineering Mustafa M. Faramawi, MBA Dr. Kenneth W. Regan A complete language for EXPSPACE: PIM, “Polynomial Ideal Membership”—the simplest natural completeness level that is known not to have polynomial-size circuits. Arithmetical Hierarchy (AH) PIM n n TOT = {Me : Me is total, i.e. halts for all inputs} K(2) TOT EXPSPACE Succinct 3SAT Unknown but Commonly Believed: 2 2 RECRE • AC0  ACC0 PP, also TC0 PP • NC1 PSPACE, … , NL  PSPACE • P  EXP, NP  NEXP • PSPACE  EXPSPACE • L ≠ NL …………………….. L ≠ PH • P ≠ NP  co-NP ……… P ≠ PSPACE • NP ≠ p2  p2 ……… NP ≠ EXP NEXP co-NEXP = .REC = .REC K D nxn Chess D = {Turing machines Me: Me does not accept e} = the complement of K. (“Diagonal Language”) EXP RE co-RE Best Known Separations: REC For any fixed k, there is a problem in this intersection that can NOT be solved by circuits of size O(nk) QBF = .REC = .REC PSPACE BQP: Bounded-Error Quantum Polynomial Time. Believed larger than P since it has FACTORING, but not believed to contain NP. Polynomial Hierarchy (PH) p2 p2 WS5 p2 p2 L The levels of AH and PH are analogous, except that we believe NP  co-NP ≠ P and p2 p2 ≠ PNP, which stand in contrast to RE  co-RE = REC and 2 2 = RECRE BPP: Bounded-Error Probabilistic Polynomial Time. Many believe BPP = P. = poly.P = poly.P TAUT SAT NC1 NLIN NP co-NP TC0 QBF NTIME [n2] PARITY PP Probabilistic Polynomial Time PNP FACT NP PSPACE ACC0 MAJ-SAT CVP P AC0 P Analogy between Arithmetical and Polynomial Hierarchies C Low-Level Classes NP co-NP GAP D REG NL = poly.P AC0 ≠ ACC0 = poly.P UGAP p2 p2 L etc P WS5 DTIME [n3] co-NP NP DLIN REG Deterministic and Nondeterministic Time Hierarchies Within NP BQP FACTORING is not believed to be complete for BQP or for NP  co-NP. A DTIME [n2] REG ≠ L B Complexity “Main Sequence” FACT WS5, the word problem for the symmetric group S5, is a regular language that is complete for NC1 under AC0 many-one reductions. P BPP E Realm of Feasibility?

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