Studying large-scale collaborative systems engineering projects across teams with differing intellectual property clearances, or healthcare solutions where sensitive patient data needs to be partially shared, or similar multi-user information systems over databases, all boils down to a common mathematical framework. Updateable views (lenses) and more generally bidirectional transformations are abstractions to study the challenge of exchanging information between participants with different read access privileges. The view provided to each participant must be different due to access control or other limitations, yet also consistent in a certain sense, to enable collaboration towards common goals. A collaboration system must apply bidirectional synchronization to ensure that after a participant modifies their view, the views of other participants are updated so that they are consistent again. While bidirectional transformations (synchronizations) have been extensively studied, there are new challenges that are unique to the multidirectional case. If complex consistency constraints have to be maintained, synchronizations that work fine in isolation may not compose well. We demonstrate and characterize a failure mode of the emergent behaviour, where a consistency restoration mechanism undoes the work of other participants. On the other end of the spectrum, we study the case where synchronizations work especially well together: we characterize very well-behaved multidirectional transformations, a non-trivial generalization from the bidirectional case. For the former challenge, we introduce a novel concept of controllability, while for the latter one, we propose a novel formal notion of faithful decomposition. Additionally, the paper proposes several novel properties of multidirectional transformations.

研究具有不同知识产权许可的团队之间的大规模协作系统工程项目，或需要部分共享敏感患者数据的医疗保健解决方案，或基于数据库的类似多用户信息系统，所有这些都可以归结为一个通用的数学框架。可更新的视图（镜头）和更一般的双向转换是一种抽象方法，用于研究在具有不同读取访问权限的参与者之间交换信息的挑战。由于访问控制或其他限制，提供给每个参与者的视图必须有所不同，但在某种意义上也必须保持一致，以实现朝着共同目标的协作。协作系统必须应用双向同步，以确保参与者修改视图后，其他参与者的观点也会更新，以便再次保持一致。尽管已经广泛研究了双向转换（同步），但是对于多方向情况，存在着新的挑战。如果必须维护复杂的一致性约束，那么孤立运行良好的同步可能无法很好地实现。我们演示并描述了紧急行为的失败模式，其中一致性恢复机制使其他参与者的工作无法进行。在频谱的另一端，我们研究了同步特别好地协同工作的情况：我们描述了行为良好的多向转换，这是双向情况的一个重要总结。对于前一个挑战，我们引入了一种新颖的可控性概念，而对于后一个挑战，我们提出了一种忠实分解的新形式概念。此外，本文提出了多向变换的几种新颖性质。