Our goal is to create a propositional logic formula to model a directed graph and use a SAT solver to analyse it. This model is similar to the well-know one of Aspvall et al., but they create a directed graph from a 2-SAT problem, we generate a 2-SAT problem from a directed graph. In their paper if the 2-SAT problem is unsatisfiable, then the generated directed graph is strongly connected, in our case, if the directed graph is strongly connected, then the generated 2-SAT problem is a black-and-white 2-SAT problem, which has two solutions: where each variable is true (the white assignment), and where each variable is false (the black one). If we see a directed graph as a communication model of a network, then we can ask in our model whether a node can send a message to another one through the network. More specifically we can ask whether all nodes can send messages to all other ones, i.e., the graph is strongly connected or not.

No abstract

In a previous paper we defined the black and white SAT problem which has exactly two solutions, where each variable is either true or false. We showed that black and white 2-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonboglár model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a black and white SAT problem. We prove a powerful theorem, the so called transitions theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as black and white SAT problems. We show that the Balatonboglár model is between the strong and the weak model, and it generates 3-SAT problems, so the Balatonboglár model represents strongly connected communication graphs as black and white 3-SAT problems. Our motivation to study these models is the following: The strong model generates a 2-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonboglár model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonboglár model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: the SAT problem and directed graphs.

We describe the use of temporal logic formulas as runtime assertions in a parallel debugging environment. The user asserts in a message passing program the expected system behavior by one or several such formulas. The debugger allows by "macro-stepping" to interactively elaborate the execution tree (i.e., the set of possible execution paths) which arises from the use of non-deterministic communication operations. In each macro-step, a temporal logic checker verifies that the once asserted temporal formulas are not violated by the current program state. Our approach thus introduces powerful runtime assertions into parallel and distributed debugging by incorporating ideas from the model checking of temporal formulas.

No abstract