scholarly journals Resolvable Networks—A Graphical Tool for Representing and Solving SAT

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2597
Author(s):  
Gábor Kusper ◽  
Csaba Biró ◽  
Benedek Nagy
Keyword(s):  
Directed Graphs ◽  
Important Application ◽  
Boolean Logic ◽  
Wireless Sensor ◽  
Incoming Edge ◽  
Sat Problem ◽  
Graphical Tool ◽  
Strongly Connected ◽  
Information Bank ◽  
Source And Sink

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.

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

2020 ◽  
Author(s):  
Gábor Kusper ◽  
Csaba Biró
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Deep Insight ◽  
Communication Graph ◽  
Sat Problem ◽  
Weak Model ◽  
Know How ◽  
Black And White ◽  
Communication Graphs ◽  
Strongly Connected

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 Balatonbogl\'{a}r 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 Blask-and-White SAT problems. We show that the Balatonbogl\'{a}r model is between the strong and the weak model, and it generates $3$-SAT problems, so the Balatonbogl\'{a}r 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 Balatonbogl\'{a}r 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 Balatonbogl\'{a}r 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: SAT problem and directed graphs.

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

Algorithms ◽  
2020 ◽  
Vol 13 (12) ◽  
pp. 321
Author(s):  
Gábor Kusper ◽  
Csaba Biró
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Deep Insight ◽  
Communication Graph ◽  
Sat Problem ◽  
Weak Model ◽  
Know How ◽  
Black And White ◽  
Communication Graphs ◽  
Strongly Connected

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.

scholarly journals A kommunikációs gráfok és a fekete-fehér SAT probléma közti összefüggések vizsgálata

2021 ◽  
Author(s):  
Franciska Rajna
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Normal Form ◽  
Directed Graphs ◽  
Conjunctive Normal Form ◽  
Wireless Sensor ◽  
Sat Problem ◽  
Black And White ◽  
Communication Graphs ◽  
The Relationship

Ebben a cikkben a kommunikációs gráfok és a fekete-fehér SAT probléma közötti összefüggéseket vizsgálom. A kommunikációs gráfok olyan speciális hurokélmentes irányított gráfok, amelyeknek csúcsai logikai változók, az élei pedig a kommunikációt reprezentálják. Ilyen típusú gráfokkal lehet többek között vezeték nélküli szenzorhálózatokat is modellezni. A cikkben bemutatom a fekete-fehér SAT problémát. A fekete-fehér SAT problémák olyan logikai formulák, amelyek majdnem kielégíthetetlenek, csak két megoldásuk van, az úgynevezett fehér hozzárendelés, ahol minden változó igaz, és a fekete hozzárendelés, amelyben minden változó hamis. A fekete-fehér SAT problémák ekvivalensek az olyan konjunktív normálformában lévő logikai formulákkal, amelyekben minden klózban pozitív és negatív literálok vegyesen szerepelnek (például ilyen 3SAT klózok a -++, --+), de sem a fehér klóz, amelyben minden literál pozitív, sem a fekete klóz, amelyben minden literál negatív, nem vezethető le. Továbbá ismertetem, és hatékonyság szempontjából elemzem a kommunikációs gráfok különböző logikai modelljeit (Erős modell, Balatonboglár modell, Egyszerűsített BB modell, Gyenge modell). ----- Investigation of the relationship between communication graphs and the black and white sat ----- In this article, I examine the relationships between communication graphs and the black-andwhite SAT problem. Communication graphs are special loop-free directed graphs whose vertices are logical variables and whose edges represent communication. These types of graphs can be used to model wireless sensor networks (WSNs), among other things. I present the black-and-white SAT problem. Black-and-white SAT problems are logical formulas that are almost unsatisfiable, they have only two solutions, the so-called white assignment, where all variables are true, and the black assignment, in which all variables are false. Black-and-white SAT problems are equivalent to logical formulas in a conjunctive normal form in which positive and negative literals are mixed in each clause (e.g., such 3-SAT clauses are - ++, - +), but not the white clause in which all literals are positive, nor the black clause in which all literals are negative cannot be deduced. I also describe and analyze the different logical models of communication graphs (Strong model, Balatonboglár model, Simplified BB model, Weak model) in terms of efficiency.

Cost Minimization of Sensor Placement and Routing in Wireless Sensor Networks

2020 ◽  
pp. 1286-1301
Author(s):  
Tata Jagannadha Swamy ◽  
Garimella Rama Murthy
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Cost Minimization ◽  
Real Life ◽  
Sensor Nodes ◽  
Wireless Sensor ◽  
Wireless Sensor Nodes ◽  
Technological Advances ◽  
Strongly Connected ◽  
Placement And Routing

Wireless Sensor Nodes (WSNs) are small in size and have limited energy resources. Recent technological advances have facilitated widespread use of wireless sensor networks in many real world applications. In real life situations WSN has to cover an area or monitor a number of nodes on a plane. Sensor node's coverage range is proportional to their cost, as high cost sensor nodes have higher coverage ranges. The main goal of this paper is to minimize the node placement cost with the help of uniform and non-uniform 2D grid planes. Authors propose a new algorithm for data transformation between strongly connected sensor nodes, based on graph theory.

Strongly Connected Spanning Subgraph for Almost Symmetric Networks

2017 ◽  
Vol 27 (03) ◽  
pp. 207-219
Author(s):  
A. Karim Abu-Affash ◽  
Paz Carmi ◽  
Anat Parush Tzur
Keyword(s):  
Directed Graph ◽  
Euclidean Distance ◽  
Directed Graphs ◽  
Minimum Weight ◽  
Shortest Paths ◽  
Strong Connectivity ◽  
Spanning Subgraph ◽  
Strongly Connected ◽  
Set Of Points ◽  
Symmetric Networks

In the strongly connected spanning subgraph ([Formula: see text]) problem, the goal is to find a minimum weight spanning subgraph of a strongly connected directed graph that maintains the strong connectivity. In this paper, we consider the [Formula: see text] problem for two families of geometric directed graphs; [Formula: see text]-spanners and symmetric disk graphs. Given a constant [Formula: see text], a directed graph [Formula: see text] is a [Formula: see text]-spanner of a set of points [Formula: see text] if, for every two points [Formula: see text] and [Formula: see text] in [Formula: see text], there exists a directed path from [Formula: see text] to [Formula: see text] in [Formula: see text] of length at most [Formula: see text], where [Formula: see text] is the Euclidean distance between [Formula: see text] and [Formula: see text]. Given a set [Formula: see text] of points in the plane such that each point [Formula: see text] has a radius [Formula: see text], the symmetric disk graph of [Formula: see text] is a directed graph [Formula: see text], such that [Formula: see text]. Thus, if there exists a directed edge [Formula: see text], then [Formula: see text] exists as well. We present [Formula: see text] and [Formula: see text] approximation algorithms for the [Formula: see text] problem for [Formula: see text]-spanners and for symmetric disk graphs, respectively. Actually, our approach achieves a [Formula: see text]-approximation algorithm for all directed graphs satisfying the property that, for every two nodes [Formula: see text] and [Formula: see text], the ratio between the shortest paths, from [Formula: see text] to [Formula: see text] and from [Formula: see text] to [Formula: see text] in the graph, is at most [Formula: see text].

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