It may surprise you, but you might consider algebraic topology. See Maurice Herlihy's marvellous little book, Distributed Computing through Combinatorial Topology.

The idea is that the state of a distributed computation can be represented as a simplicial complex (think "graph" only in higher dimensions), and that steps in a protocol correspond to continuous transformations applied to the complex.

The result is that significant distributed computing problems get reduced to well-known results from combinatorial topology, and they "just fall out".

Logic, predicate calculus, and temporal logic for a start.

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