理性交互证明的结构复杂性
Structural Complexity of Rational Interactive Proofs
AI总结：
本文研究理性交互证明的结构复杂性，证明DRMA[k]的预言机等价性，并揭示对数通信单轮理性证明类等于PP，且黑盒采样Parity-P需线性通信。
摘要：
本文是提交至2023年欧洲可计算性会议（CiE 2023）论文的完整版本，其中包含了该会议版本中省略的所有证明。2012年，P. D. Azar和S. Micali提出了一种新的交互证明模型，称为“理性交互证明”。在该模型中，证明者既非诚实也非恶意，而是理性的，即其目标是最大化期望奖励。本文探讨了这一领域与经典复杂性结果之间的联系。在本文的第一部分，我们重新审视了计数层级与常数轮理性证明层级之间的关联。我们证明了具有对DRMA[k]的预言机访问权限的多项式时间机器恰好可判定DRMA[k]中的语言，这一巧合对于计数层级的各层而言是未知的。在第二部分，我们研究了单轮理性证明的通信复杂性。我们证明了由对数通信量的单轮理性证明所定义的类与PP类一致。我们还表明，将Parity-P中的问题视为随机变量的黑盒采样器的单轮理性协议，至少需要线性的通信比特数。
英文摘要：
This is the full version of a paper submitted to the Computability in Europe (CiE 2023) conference, with all proofs omitted there. In 2012 P. D. Azar and S. Micali introduced a new model of interactive proofs, called "Rational Interactive Proofs". In this model the prover is neither honest nor malicious, but rational in terms of maximizing his expected reward. In this article we explore the connection of this area with classic complexity results. In the first part of this article we revise the ties between the counting hierarchy and the hierarchy of constant-round rational proofs. We prove that a polynomial-time machine with oracle access to DRMA[k] decides exactly languages in DRMA[k], a coincidence unknown for levels of the counting hierarchy. In the second part we study communication complexity of single-round rational proofs. We show that the class defined by logarithmic-communication single-round rational proofs coincides with PP. We also show that single-round rational protocols that treat problems in Parity-P as black-box samplers of a random variable require at least a linear number of bits of communication.
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