A Formal Model of Asynchronous Broadcast Communication

We present a mathematical model, called Asynchronous Broadcast Networks (ABN), of distributed computation based on topology-dependent and asyn-chronous communication. Our model combines three main features: a graph representation of a network configuration decoupled from the specification of individual process behavior, a topology-dependent semantics of synchronization, the use of local mailboxes to deliver messages to individual nodes. The resulting communication layer is similar to that of languages like AWN [9]. As in other protocol models like [16,17] and AHN [5], our main abstraction comes from considering protocols defined via a communicating finite-state automaton replicated on each node of the network. Formally, we consider a finite set ¦ of messages, and different disciplines for handling the mailbox (message buffer), e.g., unordered mailboxes that we represent as bags over ¦, and ordered mailboxes that we represent as words over ¦. The initial configuration is any graph in which all the nodes are in the initial control state and all local buffers are empty. Even if the set of control states is finite, there are infinitely many possible initial configurations. We next formalize the above intuition. A mailbox structure is a tuple Å Å Ð Ð ℄, where Å is a denumerable set of elements denoting possible mailbox contents on some fixed finite alphabet ¦, and, for ¾ ¦ and Ñ ¾ Å: ´ ѵ denotes the mailbox obtained by adding to Ñ, дѵ is true if can be removed from Ñ; дѵ denotes the mailbox obtained by removing from Ñ when possible, undefined otherwise. Finally, ℄ ¾ Å denotes the empty mailbox. We call an element of Ñ Ú×××Ð when дѵ ØÖÙÙ. The semantics and corresponding properties change with the type of mailbox considered. A protocol is defined by a process È ÉÉ ¦ Ê Õ ¼ , where É is a finite set of control states, ¦ is a finite message alphabet, Ø ¾ ¦, Ê É ¢ Ø ¢ É is the transition relation, Õ ¼ ¾ É is an initial control state. The label represents the ability of performing an internal action, while [] represents the ability of broadcasting [receiving] a message ¾ ¦. Configurations are undirected É¢Å-graphs. A É¢Å-graph ­ is a tuple Î Ä, where Î is a finite set of nodes, Î ¢Î is a finite set of edges (self-loops are forbidden to model half-duplex communication), and Ä Î É ¢ Å is a labeling function. We …