In this paper we propose the PCP-like theorem for sub-linear time inapproximability. Abboud et al. have devised the distributed PCP framework for proving sub-quadratic time inapproximability. Here we try to go further in this direction. Staring from SETH, we first find a problem denoted as Ext-$k$-SAT, which can not be computed in linear time, then devise an efficient MA-like protocol for this problem. To use this protocol to prove the sub-linear time inapproximability of other problems, we devise a new kind of reduction denoted as Ext-reduction, and it is different from existing reduction techniques. We also define two new hardness class, the problems in which can be computed in linear-time, but can not be efficiently approximated in sub-linear time. Some problems are shown to be in the newly defined hardness class.

翻译：在本文中,我们建议对亚线性时间使用类似五氯苯酚的理论。 Abboud 等人设计了分布式五氯苯酚框架,以证明亚赤道时间的不协调性。在这里,我们试图朝这个方向更进一步。从SETH看,我们首先发现一个无法以线性时间计算出来的被标为Ext-$k$-SAT的问题,然后为这一问题设计一个高效的MA类协议。为证明亚线性时间对其他问题的不协调性,我们设计了一种新型的减少五氯苯酚框架,称为Ext减少,它与现有的减少技术不同。我们还定义了两种新的硬性类别,在线性时间可以计算出的问题,但在亚线性时间不能有效地接近。有些问题被显示在新定义的硬性类中。