A Theory of Distributed Systems

The theory θ presented here is the smallest theory in the temporal logic TLB [10] that all distributed systems, according to our definition of a distributed system, must satisfy. θ is an instance of the classical modal logic S4.2. The central theorems of θ are stated here without proof. Proofs will appear in [10]. Logics like TLA [14] and TLRCS [18] are used for specifying computer programs and reasoning about their behaviors. Their usefulness for large distributed systems has yet to be proven. Systems with multi-thousand node networks exhibit inherently asynchronous concurrency. Logics like TLA and TLRCS only provide a sequential (interleaved) concurrent execution model. Hence TLA and the future-tense part of TLRCS should be instances of the modal logic S4.3.1 [11, p. 179], but we are unaware if this has been proven. Not having a fully asynchronous concurrent execution model makes proofs about distributed systems within these logics suspect. Like quantum systems, distributed systems have pairs of observables that are not simultaneously measurable which leads to inherent uncertainty in their behaviors. θ has an asynchronous concurrent execution model and accounts for this inherent uncertainty. Unlike S4.2, TLB has a second primitive modal operator called the “everywhere sometime” operator that mixes space and time. The reduction formulas of θ allow us to reduce distributed correctness proofs to finitely many program proofs running on single computers. For distributed systems with the peer process architecture, θ allow us to reduce distributed correctness proofs to a single program proof running on a single computer.