Distributedly Solving Boolean Equations over Networks

In this paper, we propose distributed algorithms that solve a system of Boolean equations over a network, where each node in the network possesses only one Boolean equation from the system. The Boolean equation assigned at any particular node is a private equation known to this node only, and the nodes aim to compute the exact set of solutions to the system without exchanging their local equations. We show that each private Boolean equation can be locally lifted to a linear algebraic equation under a basis of Boolean vectors, leading to a network linear equation that is distributedly solvable. A number of exact or approximate solutions to the induced linear equation are then computed at each node from different initial values. The solutions to the original Boolean equations are eventually computed locally via a Boolean vector search algorithm. We prove that given solvable Boolean equations, when the initial values of the nodes for the distributed linear equation solving step are i.i.d selected according to a uniform distribution in a high-dimensional cube, our algorithms return the exact solution set of the Boolean equations at each node with high probability.