A Typed Tensor Language for Shared-State Federated Computation
Shared-state federated computations combine client-local tensor computation, mergeable aggregation into shared state, and shared-only post-processing. We introduce a typed tensor language for this class of computations. Its two tensor sorts separate client-partitioned data from globally available values, and typing tracks the partitioned axis. A virtual global tensor serves as a semantic reference for centralized evaluation. We show that typed one-round programs factor through shared tensors whose shapes depend on the program but are independent of client and sample counts. The converse applies to typed-realizable factorizations: each encoder component is represented by an allowed aggregation or contraction with its valid merge, and the decoder is shared-only. The construction extends round by round to programs whose persistent state is shared. For a loss supplied with a client-local per-sample gradient expression, summation represents the empirical gradient. This gives typed programs for server-side first-order updates and, with shared linear algebra, curvature-block updates. The language covers federated analytics and FedSGD. General multi-local-step FedAvg and persistent private client state are outside its scope.