CIF: Small: Fundamental limits in ambiguous communication

噪声，无论是由环境引起的，还是由于固有的随机性引起的，是用于在双方或多方之间传输数据的信息通道的一个不可避免的特征。这种信道的所谓零差错容量是当允许多次使用该信道时能够以零差错概率通过该信道传输信息的最佳速率。虽然在实践中噪声的影响随着时间的推移而变化，但到目前为止可用的容量理论主要集中在信道连续使用独立的简化图解上。该项目旨在建立理论基础，以便能够在依赖于状态的机制中估计零差错容量和密切相关的参数，其中信道的每个单独使用取决于其先前的使用和周围噪声的当前状态。它正在使用开发的框架来研究一种特殊的信息渠道，这种渠道被视为博弈的策略--称为两个证明者博弈--在这种博弈中，两个参与者通过问答过程，共同拥有某一知识。到目前为止，这样的比赛只被认为是在无记忆的情况下进行的，即比赛的连续几轮是彼此独立进行的。本课题研究的是两个验证者博弈的胜率，其中玩家和验证者都有记忆，游戏规则--被检验的知识--可能会随着时间的推移而演变；在信息通道的设置中，本课题定义和研究了有记忆通道的零错容量，它作为一种特例将无记忆通道的零错容量作为一次参数。这是通过引入依赖于状态的信道的混淆图来实现的，而该参数的估计和界是通过发展有限图的众所周知的Lovász数的可测量版本来获得的。在两个证明者游戏的背景下，该项目正在开发有记忆的玩家之间的无信号关联，包括有用的子类，如量子关联的类别，作为有限问答集游戏的玩家之间无信号关联的深远推广。这种方法允许在一般框架中考虑价值分离问题和识别玩家的渐近行为，允许应用泛函分析和算子理论工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Noise, either arising from the environment or due to inherent randomness, is an unavoidable feature of information channels, which are used to transmit data between two or more parties. So-called zero-error capacities of such channels are the optimal rates at which information can be transmitted via the channel with a zero probability of error when multiple uses of the channel are allowed. While in practice the effects of noise change with time, the capacity theories available until now focus mostly on the simplified picture where the successive uses of the channel are independent. This project aims at establishing the theoretical foundations that would allow the estimation of zero-error capacities and closely related parameters in the state-dependent regime, where each individual use of the channel depends on its previous uses and the current state of the surrounding noise. It is using the developed framework to study a special case of information channels, which are viewed as strategies for games -- called two-prover games -- in which two players are tested by a verifier, through a question-answer process, for the joint possession of a certain piece of knowledge. Such games have until now been considered only in the memoryless situation where the successive rounds of the game are played independently from each other. The project is studying the winning rates of two-prover games, where the players and the verifier possess memory and the rules of the game -- the tested knowledge -- may evolve with time.In the setting of information channels, the project is defining and studying the zero-error capacity of a channel with memory, which covers as a special case the usual zero-error capacity of a memoryless channel as a one-shot parameter. This is being achieved through the introduction of the confusability graph of a state-dependent channel, while estimations and bounds for this parameter are obtained by developing a measurable version of the well-known Lovász number of a finite graph. In the setting of two prover games, the project is developing no-signaling correlations between players with memory, including useful subclasses such as the class of quantum correlations, as a far-reaching generalization of no-signaling correlations between players of games with finite question-answer sets. This approach allows the consideration of value-separation questions and the identification of the players' asymptotic behavior in a general framework, allowing the application of functional analytic and operator theoretic tools.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.