The reliability function of a channel is the maximum achievable exponential rate of decay of the error probability as a function of the transmission rate. In this work, we derive bounds on the reliability function of discrete memoryless multiple-access channels (MAC) with noiseless feedback. We show that our bounds are tight for a variety of MACs, such as $m$-ary additive and two independent point-to-point channels. The bounds are expressed in terms of a new information measure called ``variable-length directed information". The upper bound is proved by analyzing stochastic processes defined based on the entropy of the message, given the past channel's outputs. Our method relies on tools from the theory of martingales, variable-length information measures, and a new technique called time pruning. We further propose a variable-length achievable scheme consisting of three phases: (i) data transmission, (ii) hybrid data-confirmation, and (iii) full confirmation. We show that two-phase-type schemes are strictly suboptimal in achieving the MAC's reliability function. Moreover, we study the shape of the lower-bound and show that it increases linearly with respect to a specific Euclidean distance measure defined between the transmission rate pair and the capacity boundary. As side results, we derive an upper bound on the capacity of MAC with noiseless feedback and study a new problem involving a hybrid of hypothesis testing and data transmission.

翻译：信道的可靠性函数是指错误概率随传输速率增大而指数衰减的最大可达速率。本文研究了带无噪声反馈的离散无记忆多址接入信道（MAC）的可靠性函数边界。我们证明了这些边界对多种多址接入信道（如 $m$ 进制加性信道和两个独立点对点信道）是紧致的。边界由一种称为“可变长度有向信息”的新信息度量表示。上界通过分析基于给定历史信道输出的消息熵所定义的随机过程得到证明。我们的方法依赖于鞅理论、可变长度信息度量工具以及一种称为“时间剪枝”的新技术。此外，我们提出了一种由三个阶段组成的可变长度可达方案：（i）数据传输，（ii）混合数据确认，以及（iii）完全确认。我们证明了两阶段型方案在实现MAC可靠性函数方面是严格次优的。进一步，我们研究了该下界的形状，并证明它随传输速率对与容量边界之间特定欧几里得距离度量线性增长。作为辅助结果，我们推导了带无噪声反馈的MAC容量上界，并研究了涉及假设检验与数据传输混合的新问题。