We study the capabilities of probabilistic finite-state machines that act as verifiers for certificates of language membership for input strings, in the regime where the verifiers are restricted to toss some fixed nonzero number of coins regardless of the input size. Say and Yakary{\i}lmaz showed that the class of languages that could be verified by these machines within an error bound strictly less than 1/2 is precisely NL, but their construction yields verifiers with error bounds that are very close to 1/2 for most languages in that class. We characterize a subset of NL for which verification with arbitrarily low error is possible by these extremely weak machines. It turns out that, for any $\varepsilon>0$, one can construct a constant-coin, constant-space verifier operating within error $\varepsilon$ for every language that is recognizable by a linear-time multi-head finite automaton (2nfa($k$)). We discuss why it is difficult to generalize this method to all of NL, and give a reasonably tight way to relate the power of linear-time 2nfa($k$)'s to simultaneous time-space complexity classes defined in terms of Turing machines.

中文翻译：

---

具有任意小误差的恒定空间、恒定随机性验证器

我们研究了概率有限状态机作为输入字符串语言成员资格证书验证器的能力，在该机制中，验证器被限制为无论输入大小如何都可以抛出一些固定的非零数量的硬币。Say 和 Yakary{\i}lmaz 表明，这些机器可以在严格小于 1/2 的误差范围内验证的语言类别恰好是 NL，但他们的构造产生的验证器的误差范围非常接近 1/ 2 对于该课程中的大多数语言。我们描述了 NL 的一个子集，这些极弱的机器可以对其进行任意低误差的验证。事实证明，对于任何 $\varepsilon>0$，都可以构造一个常量硬币，对于线性时间多头有限自动机 (2nfa($k$)) 可识别的每种语言，常量空间验证器都在错误 $\varepsilon$ 内运行。我们讨论了为什么很难将这种方法推广到所有 NL，并给出了一种相当紧密的方法来将线性时间 2nfa($k$) 的能力与根据图灵机定义的同时时空复杂度类相关联.