Advice classes in computational complexity have frequently been used to model real-world scenarios encountered in cryptography, quantum computing and machine learning, where some computational task may be broken down into a preprocessing and deployment phase, each associated with a different complexity. However, in these scenarios, the advice given by the preprocessing phase must still be generated by some (albeit more powerful) bounded machine, which is not the case in conventional advice classes. To better model these cases we develop `bounded advice classes', where a more powerful Turing machine generates advice for another, less powerful, Turing machine. We then focus on the question of when various classes generate useful advice, to answer this we connect bounded advice to unary languages. This connection allows us to state various conditional and unconditional results on the utility of advice generated by $\mathsf{EXP}$, $\mathsf{NP}$, $\mathsf{BQP}$, $\mathsf{PSPACE}$, and more. We study the relations between bounded advice classes, quantum bounded advice classes, and randomised bounded advice. We also examine how each of these concepts interact with recently introduced classes, like $\mathsf{BPP/samp}$. Our results also improve the state of the art in existing research on the complexity of advice functions.

翻译：计算复杂性中的建议类常被用于建模密码学、量子计算和机器学习中遇到的实际场景，在这些场景中，计算任务可分解为预处理和部署两个阶段，每个阶段对应不同的复杂性。然而，在这些场景中，预处理阶段提供的建议仍需由某个（尽管更强大的）有界机器生成，而传统建议类并非如此。为了更好地建模这些情况，我们发展了“有界建议类”，其中更强大的图灵机为另一台能力较弱的图灵机生成建议。随后，我们聚焦于各类何时能生成有用建议的问题；为回答此问题，我们将有界建议与一元语言联系起来。这一关联使我们能够就$\mathsf{EXP}$、$\mathsf{NP}$、$\mathsf{BQP}$、$\mathsf{PSPACE}$等类生成建议的效用，陈述多种条件性与无条件性结果。我们研究了有界建议类、量子有界建议类与随机化有界建议之间的关系。同时，我们也检验了这些概念如何与近期引入的类别（如$\mathsf{BPP/samp}$）相互作用。我们的结果还改进了现有关于建议函数复杂性研究的技术水平。