Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

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.

Resolvable Networks—A Graphical Tool for Representing and Solving SAT

“…At communication graphs, the word node is also used as a synonym of vertex. Now, we briefly recall a crucial property of the weak model of communication graphs from [10]. If we have a circuit (a 1 , a 2 , .…”

“…The semantics of resolvable networks are motivated by the concept of message sending, which is a very basic operation of a network. In these semantics, vertices are called nodes, and nodes can be used as Boolean variables, similarly as in [10] and as we have already mentioned.…”

“…If inside B the nodes cannot send messages to each other, then the answer is probably no. If inside B we find a strongly connected communication graph, see [10], then the answer is yes. See also the pessimistic and the optimistic semantics of subnetworks.…”

“…We believe that if we could visualize the SAT problem, then we would gain more insight on it that could help also to find more efficient solutions for them, at least in some subcases. This paper is inspired by one of the recent papers of the above project, which introduces the so-called weak model of communication graphs [10].…”

“…The weak model has some nice properties; for example, if the communication graph is strongly connected, then its weak model is a black and white SAT problem [10]. On the other hand, the weak model is computationally very expensive, because we have to find all circuits in the input communication graph to be able to generate its weak model.…”