Minimax games, spin glasses, and the polynomial-time hierarchy of complexity classes

Peter Varga

doi:10.1103/physreve.57.6487

ON HIGHER ARTHUR-MERLIN CLASSES

International Journal of Foundations of Computer Science ◽

10.1142/s0129054104002273 ◽

2004 ◽

Vol 15 (01) ◽

pp. 3-19

Author(s):

JIN-YI CAI ◽

DENIS CHARLES ◽

A. PAVAN ◽

SAMIK SENGUPTA

Keyword(s):

Polynomial Time ◽

Upper Bounds ◽

Complexity Classes ◽

Polynomial Time Hierarchy

We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R and coR. We investigate the complexity classes they define, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy.

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Delineating classes of computational complexity via second order theories with weak set existence principles. I

Journal of Symbolic Logic ◽

10.2307/2275511 ◽

1995 ◽

Vol 60 (1) ◽

pp. 103-121 ◽

Cited By ~ 1

Author(s):

Aleksandar Ignjatović

Keyword(s):

Computational Complexity ◽

Polynomial Time ◽

Second Order ◽

Complexity Classes ◽

Bounded Arithmetic ◽

Recursive Functions ◽

Speed Up ◽

Bounded Complexity ◽

Polynomial Time Hierarchy ◽

Comprehension Axiom

AbstractIn this paper we characterize the well-known computational complexity classes of the polynomial time hierarchy as classes of provably recursive functions (with graphs of suitable bounded complexity) of some second order theories with weak comprehension axiom schemas but without any induction schemas (Theorem 6). We also find a natural relationship between our theories and the theories of bounded arithmetic (Lemmas 4 and 5). Our proofs use a technique which enables us to “speed up” induction without increasing the bounded complexity of the induction formulas. This technique is also used to obtain an interpretability result for the theories of bounded arithmetic (Theorem 4).

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Polynomial-Time Random Oracles and Separating Complexity Classes

ACM Transactions on Computation Theory ◽

10.1145/3434389 ◽

2021 ◽

Vol 13 (1) ◽

pp. 11-16

Author(s):

John M. Hitchcock ◽

Adewale Sekoni ◽

Hadi Shafei

Keyword(s):

Polynomial Time ◽

Open Problem ◽

Random Oracle ◽

Complexity Class ◽

Complexity Classes ◽

Random Oracles ◽

Class Separation ◽

Polynomial Time Hierarchy

Bennett and Gill [1981] showed that P A ≠ NP A ≠ coNP A for a random oracle A , with probability 1. We investigate whether this result extends to individual polynomial-time random oracles. We consider two notions of random oracles: p-random oracles in the sense of martingales and resource-bounded measure [Lutz 1992; Ambos-Spies et al. 1997], and p-betting-game random oracles using the betting games generalization of resource-bounded measure [Buhrman et al. 2000]. Every p-betting-game random oracle is also p-random; whether the two notions are equivalent is an open problem. (1) We first show that P A ≠ NP A for every oracle A that is p-betting-game random. Ideally, we would extend (1) to p-random oracles. We show that answering this either way would imply an unrelativized complexity class separation: (2) If P A ≠ NP A relative to every p-random oracle A , then BPP ≠ EXP. (3) If P A ≠ NP A relative to some p-random oracle A , then P ≠ PSPACE. Rossman, Servedio, and Tan [2015] showed that the polynomial-time hierarchy is infinite relative to a random oracle, solving a longstanding open problem. We consider whether we can extend (1) to show that PH A is infinite relative to oracles A that are p-betting-game random. Showing that PH A separates at even its first level would also imply an unrelativized complexity class separation: (4) If NP A ≠ coNP A for a p-betting-game measure 1 class of oracles A , then NP ≠ EXP. (5) If PH A is infinite relative to every p-random oracle A , then PH ≠ EXP. We also consider random oracles for time versus space, for example: (6) L A ≠ P A relative to every oracle A that is p-betting-game random.

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