AbstractWe introduce and study interactive proofs in the framework of real number computations as introduced by Blum, Shub, and Smale. Ivanov and de Rougemont started this line of research showing that an analogue of Shamir’s result holds in the real additive Blum–Shub–Smale model of computation when only Boolean messages can be exchanged. Here, we introduce interactive proofs in the full BSS model in which also multiplications can be performed and reals can be exchanged. The ultimate goal is to give a Shamir-like characterization of the real counterpart $${{\rm IP}_\mathbb{R}}$$IPR of classical IP. Whereas classically Shamir’s result implies IP = PSPACE = PAT = PAR, in our framework a major difficulty arises: In contrast to Turing complexity theory, the real number classes $${{\rm PAR}_\mathbb{R}}$$PARR and $${{\rm PAT}_\mathbb{R}}$$PATR differ and space resources considered separately are not meaningful. It is not obvious how to figure out whether at all $${{\rm IP}_\mathbb{R}}$$IPR is characterized by one of the above classes—and if so by which. We obtain two main results, an upper and a lower bound for the new class $${{\rm IP}_\mathbb{R}.}$$IPR. As upper bound we establish $${{{\rm IP}_\mathbb{R}} \subseteq {\rm MA\exists}_\mathbb{R}}$$IPR⊆MA∃R, where $${{\rm MA} \exists_\mathbb{R}}$$MA∃R is a real complexity class introduced by Cucker and Briquel satisfying $${{\rm PAR}_\mathbb{R} \subsetneq {\rm MA}\exists_{\mathbb{R}} \subseteq {\rm PAT}_\mathbb{R}}$$PARR⊊MA∃R⊆PATR and conjectured to be different from $${{\rm PAT}_\mathbb{R}}$$PATR. We then complement this result and prove a non-trivial lower bound for $${{\rm IP}_\mathbb{R}}$$IPR. More precisely, we design interactive real protocols verifying function values for a large class of functions introduced by Koiran and Perifel and denoted by UniformVPSPACE$${^{0}.}$$0. As a consequence, we show $${{\rm PAR}_\mathbb{R} \subseteq {\rm IP}_\mathbb{R}}$$PARR⊆IPR, which in particular implies co-$${{\rm NP}_\mathbb{R} \subseteq {\rm IP}_\mathbb{R}}$$NPR⊆IPR, and $${{\rm P}_\mathbb{R}^{Res} \subseteq {\rm IP}_\mathbb{R}}$$PRRes⊆IPR, where Res denotes certain multivariate Resultant polynomials. Our proof techniques are guided by the question in how far Shamir’s classical proof can be used as well in the real number setting. Towards this aim results by Koiran and Perifel on UniformVPSPACE$${^{0}}$$0 are extremely helpful.

中文翻译：

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用于实数计算的交互式证明和类似 Shamir 的结果

摘要 我们在 Blum、Shub 和 Smale 介绍的实数计算框架中介绍和研究交互式证明。Ivanov 和 de Rougemont 开始了这一系列研究，表明当只能交换布尔消息时，Shamir 结果的类似物适用于真正的可加 Blum-Shub-Smale 计算模型。在这里，我们在完整的 BSS 模型中引入了交互式证明，其中还可以执行乘法并可以交换实数。最终目标是对经典 IP 的真实对应物 $${{\rm IP}_\mathbb{R}}$$IPR 进行类似 Shamir 的表征。虽然经典的 Shamir 的结果意味着 IP = PSPACE = PAT = PAR，但在我们的框架中出现了一个主要困难：与图灵复杂性理论相反，实数类 $${{\rm PAR}_\mathbb{R}}$$PARR 和 $${{\rm PAT}_\mathbb{R}}$$PATR 不同，单独考虑的空间资源没有意义. 如何确定 $${{\rm IP}_\mathbb{R}}$$IPR 是否具有上述类别之一的特征并不明显——如果是，则由哪个类别表征。我们获得了两个主要结果，新类 $${{\rm IP}_\mathbb{R}.}$$IPR 的上限和下限。作为上限，我们建立 $${{{\rm IP}_\mathbb{R}} \subseteq {\rm MA\exists}_\mathbb{R}}$$IPR⊆MA∃R，其中 $${{ \rm MA} \exists_\mathbb{R}}$$MA∃R 是由 Cucker 和 Briquel 引入的满足 $${{\rm PAR}_\mathbb{R} \subsetneq {\rm MA}\存在_{\mathbb{R}} \subseteq {\rm PAT}_\mathbb{R}}$$PARR⊊MA∃R⊆PATR 并且推测与$${{\rm PAT}_\mathbb{R 不同}}$$PATR。然后我们补充这个结果并证明 $${{\rm IP}_\mathbb{R}}$$IPR 的非平凡下界。更准确地说，我们设计了交互式真实协议，验证 Koiran 和 Perifel 引入的一大类函数的函数值，并用 UniformVPSPACE$${^{0}.}$$0 表示。因此，我们显示 $${{\rm PAR}_\mathbb{R} \subseteq {\rm IP}_\mathbb{R}}$$PARR⊆IPR，这特别意味着 co-$${{ \rm NP}_\mathbb{R} \subseteq {\rm IP}_\mathbb{R}}$$NPR⊆IPR，和 $${{\rm P}_\mathbb{R}^{Res} \ subseteq {\rm IP}_\mathbb{R}}$$PRRes⊆IPR，其中 Res 表示某些多元结果多项式。我们的证明技术以 Shamir 的经典证明在实数设置中可以使用到什么程度这一问题为指导。Koiran 和 Perifel 在 UniformVPSPACE$${^{0}}$$0 上针对这一目标的结果非常有帮助。