Interactive proof systems whose verifiers are constant-space machines have interesting features that do not have counterparts in the better studied case where the verifiers operate under reasonably large space bounds. The language verification power of finite-state verifiers is known to be sensitive to the difference between private and public randomization. These machines also lack the capability of imposing worst-case superlinear bounds on their own runtime, and long interactions with untrustable provers can involve the risk of being fooled to loop forever. We analyze such verifiers under different bounds on the numbers of private and public random bits that they are allowed to use. This separate accounting for the private and public coin budgets as resource functions of the input length provides interesting characterizations of the collections of the associated languages. When the randomness bound is constant, the verifiable class is $\rm NL$ for private-coin machines, but equals just the regular languages when one uses public coins. Increasing the public coin budget while keeping the number of private coins constant augments the power: We show that the set of languages that are verifiable by such machines in expected polynomial time (with an arbitrarily small positive probability of looping) equals the complexity class $\rm P$. This hints that allowing a minuscule probability of looping may add significant power to polynomial-time finite-state automata, since it is still not known whether those machines can verify all of $\rm P$ when required to halt with probability 1, even with no bound on their private coin usage. We also show that logarithmic-space machines which hide a constant number of their coins are limited to verifying the languages in $\rm P$.

翻译：验证器为恒定空间机器的交互式证明系统具有一些有趣特性，这些特性在验证器运行于较大空间限制下这一更受关注的情形中并无对应体现。已知有限状态验证器的语言验证能力对私有随机化与公开随机化之间的差异十分敏感。这些机器也缺乏对其自身运行时间施加最坏情况超线性界限的能力，且与不可信证明者进行长时间交互可能涉及被欺骗而陷入无限循环的风险。我们在允许使用的私有与公开随机比特数受到不同限制的条件下分析此类验证器。将私有与公开硬币预算作为输入长度的资源函数进行单独核算，为相关语言集合提供了有趣的刻画。当随机性界限为常数时，私有硬币机器可验证的类别为 $\rm NL$，而使用公开硬币时则仅为正则语言。在保持私有硬币数量不变的同时增加公开硬币预算会增强能力：我们证明，此类机器在期望多项式时间内（具有任意小的正概率循环）可验证的语言集合等于复杂性类 $\rm P$。这暗示允许极小的循环概率可能为多项式时间有限状态自动机增添显著能力，因为目前尚不清楚这些机器在要求以概率 1 停机时（即使对其私有硬币使用无限制）是否能够验证所有 $\rm P$ 类语言。我们还证明，隐藏常数数量硬币的对数空间机器仅限于验证 $\rm P$ 类中的语言。