Multi-context Systems (MCSs) are a formalism for systems consisting of knowledge bases (possibly heterogeneous and non-monotonic) that are interlinked via bridge rules, where the global system semantics emerges from the local semantics of the knowledge bases (also called contexts) in an equilibrium. While MCSs and related formalisms are inherently targeted for distributed set- tings, no truly distributed algorithms for their evaluation were available. We address this short- coming and present a suite of such algorithms which includes a basic algorithm DMCS, an ad- vanced version DMCSOPT that exploits topology-based optimizations, and a streaming algorithm DMCS-STREAMING that computes equilibria in packages of bounded size. The algorithms be- have quite differently in several respects, as experienced in thorough experimental evaluation of a system prototype. From the experimental results, we derive a guideline for choosing the appropriate algorithm and running mode in particular situations, determined by the parameter settings.

This paper presents a new deterministic approximation technique in Bayesian networks. This method, "Expectation Propagation", unifies two previous techniques: assumed-density filtering, an extension of the Kalman filter, and loopy belief propagation, an extension of belief propagation in Bayesian networks. All three algorithms try to recover an approximate distribution which is close in KL divergence to the true distribution. Loopy belief propagation, because it propagates exact belief states, is useful for a limited class of belief networks, such as those which are purely discrete. Expectation Propagation approximates the belief states by only retaining certain expectations, such as mean and variance, and iterates until these expectations are consistent throughout the network. This makes it applicable to hybrid networks with discrete and continuous nodes. Expectation Propagation also extends belief propagation in the opposite direction - it can propagate richer belief states that incorporate correlations between nodes. Experiments with Gaussian mixture models show Expectation Propagation to be convincingly better than methods with similar computational cost: Laplace's method, variational Bayes, and Monte Carlo. Expectation Propagation also provides an efficient algorithm for training Bayes point machine classifiers.

We present a distributed algorithm for computing equilibria of heterogeneous nonmonotonic multi-context systems (MCS). The algorithm can be parametrized to compute only partial equilibria, which can be used for reasoning tasks like query answering or satisfiability checking that need only partial information and not whole belief states. Furthermore, caching is employed to cut redundant solver calls. As a showcase, we instantiate the MCS framework with answer set program contexts. To characterize equilibria of such MCS, we develop notions of loop formulas that enable reductions to the classical satisfiability problem (SAT). Notably, loop formulas for bridge rules between contexts and for the local contexts can be combined to a uniform encoding of an MCS into a (distributed) SAT instance. As a consequence, we can use SAT solvers for belief set building. We demonstrate this approach by an experimental prototype implementation, which uses an off-the-shelf SAT solver.