Resolvable Networks—A Graphical Tool for Representing and Solving SAT

Gábor Kusper,Benedek Nagy,Csaba Biró

doi:10.3390/math9202597

Gábor Kusper, Benedek Nagy + Show 1 more

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https://doi.org/10.3390/math9202597

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Abstract

In this paper, we introduce the notion of resolvable networks. A resolvable network is a digraph of subnetworks, where subnetworks may overlap, and the inner structure of subnetworks are not interesting from the viewpoint of the network. There are two special subnetworks, Source and Sink, with the following properties: there is no incoming edge to Source, and there is no outgoing edge from Sink. Any resolvable network can be represented by a satisfiability problem in Boolean logic (shortly, SAT problem), and any SAT problem can be represented by a resolvable network. Because of that, the resolution operation is valid also for resolvable networks. We can use resolution to find out or refine the inner structure of subnetworks. We give also a pessimistic and an optimistic interpretation of subnetworks. In the pessimistic case, we assume that inside a subnetwork, all communication possibilities are represented as part of the resolvable network. In the optimistic case, we assume that each subnetwork is strongly connected. We show that any SAT problem can be visualized using the pessimistic interpretation. We show that transitivity is very limited in the pessimistic interpretation, and in this case, transitivity corresponds to resolution of clauses. In the optimistic interpretation of subnetworks, we have transitivity without any further condition, but not all SAT problems can be represented in this case; however, any such network can be represented as a SAT problem. The newly introduced graphical concept allows to use terminology and tools from directed graphs in the field of SAT and also to give graphical representations of various concepts of satisfiability problems. A resolvable network is also a suitable data structure to study, for example, wireless sensor networks. The visualization power of resolvable networks is demonstrated on some pigeon hole SAT problems. Another important application field could be modeling the communication network of an information bank. Here, a subnetwork represents a dataset of a user which is secured by a proxy. Any communication should be done through the proxy, and this constraint can be checked using our model.

Highlights

Complex networks can be found everywhere, and they belong to an interdisciplinary field of science [1,2,3]
We introduce the novel concept of resolvable networks, which are structures that have the following three properties: a resolvable network (1) consists of subnetworks, which are a set of nodes, and since subnetworks are sets, their inner structure is unconstrained; (2) non-overlapping subnetworks may communicate by passing messages over directed edges; and (3) subnetworks may overlap
SAT problems and resolvable networks are equivalent, i.e., any SAT problem can be represented by a resolvable network and vice versa

Summary

Introduction

Complex networks can be found everywhere, and they belong to an interdisciplinary field of science [1,2,3]. Their modeling and analysis is of great importance. From a mathematical point of view, the networks appear in the theory of graphs. Topology can represent and characterize the properties of the entire network structure. A topology represents a real network and usually it is converted to either a directed or an undirected graph. The topological models are usually calculated based on some probabilities for large networks [4,5,6,7]

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