AI 中文总结
该研究探讨依赖目标的局部验证中信息与证明长度的权衡，推导相关信息下界，分析自适应分支的非自适应模拟，将所得界应用于dPCP接口并得出测试配置文件菜单的必要条件。
AI 中文摘要
我们研究了带有依赖目标的局部测试的固定布局局部验证。设M是{0,1}^K上的随机变量，S记录每个坐标处选择的测试。对于每个s∈supp(S)，设F_s是对应的目标纤维，并令D_fib=max_s VCdim(F_s)。我们证明H(M|S)≤log₂(∑_{j=0}^{D_fib} C(K,j))。一个击碎d个坐标的纤维会产生一个弱松弛的局部可译码，其消息长度为d，在原始证明字母表上的块长度为d+P。对于均匀K位目标、固定证明字母表Q和σ，Goldberg-Gur-Saraogi下界意味着，若I(M;S)≤γK（γ<1为固定值），则P=Ω(K^(1+1/a)/(log K)^(2+2/a))，其中a=⌈Q/σ⌉。若P≤K(log K)^c，则I(M;S)≥K-O(K^(a/(a+1))(log K)^(3+ac/(a+1)))=K-o(K)。任何确定S的离散验证器状态T都满足相同的信息下界。有界随机性自适应分支可通过暴露其决策树以非自适应方式模拟。一个使用至多r个随机位和q个自适应证明查询的分支，会产生一个具有完美完备性且查询数至多为1+2^(r+1)∑_{j<q}A^j的解码器。在I(M;S)≤γK下，近线性证明长度要求该数量为Ω(log K/log log K)；二元单查询情形给出P=2^(Ω(K))。将我们的界应用于Gur-Minzer-Weissenberg-Zheng的全局列表健全dPCP接口，表明固定的非依赖目标的L个测试配置文件菜单必须满足log₂L≥K-o(K)。依赖证明的列表和额外的依赖目标的选择数据必须包含在测量状态中。
英文摘要
We study fixed-layout local verification with target-dependent local tests. Let $M$ be a random variable on $\{0,1\}^K$, and let $S$ record the test selected at each coordinate. For each $s\in\operatorname{supp}(S)$, let $F_s$ be the corresponding target fiber and set $D_{\mathrm{fib}}=\max_s\operatorname{VCdim}(F_s)$. We prove $H(M\mid S)\le \log_2\!\left(\sum_{j=0}^{D_{\mathrm{fib}}}\binom Kj\right)$. A fiber that shatters $d$ coordinates yields a weak relaxed locally decodable code with message length $d$ and block length $d+P$ over the original proof alphabet. For a uniform $K$-bit target and fixed proof alphabet, $Q$, and $σ$, the Goldberg--Gur--Saraogi lower bound implies that $I(M;S)\leγK$, for fixed $γ<1$, forces $P=Ω\!\left(K^{1+1/a}/(\log K)^{2+2/a}\right)$, where $a=\lceil Q/σ\rceil$. If $P\le K(\log K)^c$, then $I(M;S)\ge K-O\!\left(K^{a/(a+1)}(\log K)^{3+ac/(a+1)}\right)=K-o(K)$. Any discrete verifier state $T$ determining $S$ satisfies the same information lower bound. Bounded-randomness adaptive branches can be simulated nonadaptively by exposing their decision trees. A branch using at most $r$ random bits and $q$ adaptive proof queries yields a decoder with perfect completeness and at most $1+2^{r+1}\sum_{j<q}A^j$ queries. Under $I(M;S)\leγK$, near-linear proof length requires this quantity to be $Ω(\log K/\log\log K)$; the binary one-query case gives $P=2^{Ω(K)}$. Applied to a global list-sound dPCP interface of Gur--Minzer--Weissenberg--Zheng, our bound shows that a fixed target-independent menu of $L$ test profiles must satisfy $\log_2L\ge K-o(K)$. Proof-dependent lists and additional target-dependent selection data must be included in the measured state.
Comments15 pages. Submitted to computational complexity
