scholarly journals Polynomial-Time Random Oracles and Separating Complexity Classes

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.

ON HIGHER ARTHUR-MERLIN CLASSES

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.

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

1995 ◽  
Vol 60 (1) ◽  
pp. 103-121 ◽  
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).

scholarly journals Impossibility of blind quantum sampling for classical client

2019 ◽  
Vol 19 (9&10) ◽  
pp. 793-806
Author(s):  
Tomoyuki Morimae ◽  
Harumichi Nishimura ◽  
Yuki Takeuch ◽  
Seiichiro Tani
Keyword(s):  
Quantum Computing ◽  
Polynomial Time ◽  
Open Problem ◽  
Input Output ◽  
Information Theoretic ◽  
Single Qubit ◽  
The Third ◽  
The One ◽  
Qubit States ◽  
Polynomial Time Hierarchy

Blind quantum computing enables a client, who can only generate or measure single-qubit states, to delegate quantum computing to a remote quantum server in such a way that the input, output, and program are hidden from the server. It is an open problem whether a completely classical client can delegate quantum computing blindly (in the information theoretic sense). In this paper, we show that if a completely classical client can blindly delegate sampling of subuniversal models, such as the DQC1 model and the IQP model, then the polynomial-time hierarchy collapses to the third level. Our delegation protocol is the one where the client first sends a polynomial-length bit string to the server and then the server returns a single bit to the client. Generalizing the no-go result to more general setups is an open problem.

scholarly journals Circuit Depth Relative to a Random Oracle

1991 ◽  
Vol 20 (378) ◽  
Author(s):  
Peter Bro Miltersen
Keyword(s):  
Random Oracle ◽  
Complexity Classes ◽  
Random Oracles ◽  
Circuit Depth

<p>The study of separation of complexity classes with respect to random oracles was initiated by Bennett and Gill and continued by many other authors.</p><p> </p><p>Wilson defined relativized circuit depth and constructed various oracles A for which</p><p> </p><p> </p><ul> <li> P^A ¬ NC^A </li><li> NC^A_k ¬ NC^A_k+varepsilon, </li><li> AC^A_k ¬ AC^A_k+varepsilon, </li><li> AC^A_k ¬ subset= AC^A_k+1-varepsilon, </li></ul> and <p>NC^A_k not subset= AC^A_ k-varepsilon,</p><p>for all positive rational <em>k</em> and varepsilon, thus separating those classes for which no trivial argument shows inclusion. In this note we show that as a consequence of a single lemma, these separations (or improvements of them) hold with respect to a random oracle A.</p>

Constructivizing Membership Proofs in Complexity Classes

1997 ◽  
Vol 08 (04) ◽  
pp. 433-442 ◽  
Author(s):  
V. Arvind
Keyword(s):  
Turing Machine ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Complexity Class ◽  
The Self ◽  
Complexity Classes ◽  
Machine Code ◽  
Computational Problem ◽  
Program Checking

A computational problem is said to have the Ptime self-witnessing property if we can design a Turing machine code M such that if the problem is polynomial-time computable, then M actually encodes a polynomial-time algorithm for it. This notion captures constructivizing proofs of membership in P. In this paper we define and study analogous notions of self-witnessing corresponding to other complexity classes like DLOG, PSPACE, and NC. In particular, we show that logspace self-reducible sets are DLOG self-witnessing and wdq-self-reducible sets are PSPACE self-witnessing. As a consequence of this we derive that for any complexity class [Formula: see text], if [Formula: see text] then [Formula: see text] is constructively equal to DLOG. Likewise, we show that is PSPACE = EXP then PSPACE is constructively equal to EXP. We also show connections between the self-witnessing property and self-helping and program checking

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