AI 中文总结
本文在随机预言机模型中证明，闭包维度为零的可数语言集合可实现零错误信息论生成，但多项式查询的均匀生成器会面临指数级期望错误下界，即信息论易处理性与受限访问计算难度可共存。
AI 中文摘要
在Kleinberg和Mullainathan[KM24]的模型中，极限生成保证对每一个可数的无限语言集合都能最终生成，而闭包维度则表征更强的信息论保证[RLT25]，两者均未对每个输出的计算进行限制。错误受限生成中的累积错误目标将有限失败前缀量化[KPR26]，且每个输出的查询预算揭示了其计算来源。目前已知针对奇偶函数、合取式以及具有多项式个最大项的单调函数存在多项式时间算法[JKO26]。本文探究信息论上的易处理性是否能与受限访问下的计算难度共存。相对于随机预言机H，我们通过构造一个闭包维度为零的可数无限语言集合C*给出肯定回答。在相同的H下几乎必然地，无界生成器对每个目标及每个完整的不同枚举都产生零错误。设λ为目标种子长度，每个固定的均匀生成器G若具有关于λ和输出索引i的多项式个预言机查询，则存在常数c_G>0，使得对于每个足够大的λ，某个目标在其前2(⌈2^{c_Gλ}⌉+1)个规范输出中会产生超过2^{c_Gλ}的期望错误。无限偶然一致性支持穷举搜索，稀疏查询隐藏了新的目标值。因此，在随机预言机模型中，零错误信息论生成与多项式查询访问下依赖于生成器的最坏情况期望错误的指数下界共存。
英文摘要
Generation in the limit guarantees eventual generation for every countable collection of infinite languages in the model of Kleinberg and Mullainathan [KM24], while closure dimension characterizes stronger information-theoretic guarantees [RLT25]. Neither restricts per-output computation. The cumulative-mistake objective in mistake-bounded generation makes finite failure prefixes quantitative [KPR26], and a per-output query budget exposes their computational source. Polynomial-time algorithms are known for parities, conjunctions, and monotone functions with polynomially many maxterms [JKO26]. We ask whether information-theoretic ease can coexist with bounded-access computational hardness. Relative to a random oracle $H$, we answer yes by constructing a countable collection $C^\star$ of infinite languages with closure dimension zero. Almost surely on the same $H$, an unbounded generator makes zero mistakes on every target and every complete distinct enumeration. Yet, writing $λ$ for the target-seed length, every fixed uniform generator $G$ with polynomially many oracle queries in $λ$ and the output index $i$ has a constant $c_G>0$ such that, for every sufficiently large $λ$, some target incurs more than $2^{c_Gλ}$ expected mistakes within its first $2(\lceil 2^{c_Gλ}\rceil+1)$ canonical outputs. Infinite accidental agreement enables exhaustive search; sparse queries hide fresh target values. Thus, in the random-oracle model, zero-mistake information-theoretic generation coexists with a generator-dependent exponential lower bound on worst-case expected mistakes under polynomial-query access.
