Theoretical Computer Science ( IF 0.9 ) Pub Date : 2021-06-03 , DOI: 10.1016/j.tcs.2021.05.030 Oded Goldreich , Tom Gur
Universal locally testable codes (s), recently introduced in our companion paper (CJTCS, 2018), are codes that admit local tests for membership in numerous subcodes, allowing for testing properties of the encoded message. Unfortunately, s suffer strong limitations, which motivate us to initiate, in this work, the study of the “NP analogue” of these codes, wherein the testing procedures are also given free access to a short proof, akin the proofs of proximity of Gur and Rothblum (Computational Complexity 2018). We call such codes “universal locally verifiable codes” (s).
A for a family of functions is a code such that, for every , membership in the subcode can be verified locally using explicit access to a short (sublinear length) proof. A can be viewed as providing an encoding of inputs under which a large family of properties of the encoded inputs can be locally testable using a short proof.
We show s of block length for the family of all functions expressible by t-ary constraint satisfaction problems (t-CSP) over n constraints and k variables, with proof length and query complexity , where and . In addition, we prove a lower bound of for every polynomial length , having proof complexity p and query complexity q, for such CSP functions.
We give an application of s for interactive proofs of proximity (), introduced by Rothblum, Vadhan, and Wigderson (STOC 2013), which are interactive proof systems wherein the verifier queries only a sublinear number of input bits to the end of asserting that, with high probability, the input is close to an accepting input. Specifically, we show a 3-round for the set of assignments that satisfy fixed CSP instances, with sublinear communication and query complexity, which we derive from our for CSP functions.
中文翻译:
CSP 的通用本地可验证代码和 3 轮交互式邻近证明
通用本地可测试代码(s),最近在我们的配套论文 ( CJTCS , 2018) 中引入,是允许对众多子代码中的成员资格进行本地测试的代码,允许测试编码消息的属性。很遗憾,s 受到很大的限制,这促使我们在这项工作中开始研究这些代码的“NP 类似物”,其中测试程序也可以免费访问一个简短的证明,类似于 Gur 和 Rothblum 接近的证明(计算复杂性2018)。我们称此类代码为“通用本地可验证代码”(s)。
一种 对于函数族 是这样的代码,对于每个 , 子代码中的成员资格 可以使用对短(次线性长度)证明的显式访问在本地进行验证。一种可以被视为提供输入的编码,在这种编码下,可以使用简短的证明在本地测试编码输入的一大类属性。
我们展示 块长度 s 为家庭的所有功能可表达由吨进制约束满足问题(吨-csp)超过Ñ约束和ķ变量,以及证明长度和复杂性的查询, 在哪里 和 . 此外,我们证明了下界 对于每个多项式长度 ,对于此类 CSP 函数,具有证明复杂度p和查询复杂度q。
我们给出一个应用 s 用于邻近性的交互式证明(),由 Rothblum、Vadhan 和 Wigderson ( STOC 2013)引入,它们是交互式证明系统,其中验证器仅查询次线性数量的输入位,直到断言很可能输入接近接受输入. 具体来说,我们展示了一个 3 轮 对于满足固定 CSP 实例的一组分配,具有亚线性通信和查询复杂性,我们从我们的 用于 CSP 功能。