Byzantine Generals in the Permissionless Setting

We write down a formal framework for the analysis of both permissioned and permissionless systems. Within this framework, we then consider the Byzantine Generals Problem as a formalisation of the problem of reaching consensus, and address a programme of research that asks, "Under what adversarial conditions, and for what types of permissionless protocol, is consensus possible?" We prove a number of results for this programme, our main result being that deterministic consensus is not possible for decentralised permissionless protocols. To close, we give a list of seven open questions.