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Algorithm to Count the Number of Signed Paths in an Electrical Network via

Boolean Formulas

Guillermo De Ita, Helena G omez, Belem Merino

Benemerita Universidad Aut onoma de Puebla, M exico

Email: [email protected], [email protected], [email protected]

AbstractThis article presents a practical method to countthe different signed paths to maintain charge on each ofthe lines of energy of an electrical network. We assume acharge on each network node. We model this problem via the#2SAT problem. #2SAT consists on counting models of booleanformulas, where the input Boolean formula is in 2 conjunctiveform. To develop this method we rely on the connectivity graphof an electrical network from which we get its boolean formula

and apply recurrence equations starting from the terminalnodes up to the root node of the network. The recurrenceequations allows to carry the count to indicate the differentways to keep charge on all line of the electrical network.

Keywords-#SAT Problem; Counting models; Electrical Net-work; Graph tree;

I. INTRODUCTION

The problem of propositional satisfiability (SAT) is aclassic NP-complete problem, but there is a wide range of

research on how to identify limited instances for which the

SAT problem can be solved efficiently. The problem of

counting models of a boolean formula (#SAT) is relevantfor approximating reasoning processes tasks.

For example, for estimating the degree of reliability in

a communication network,computing degree of belief in

propositional theories, for the generation of explanations

to propositional queries, in Bayesian inference, in a

truth maintenance systems and for repairing inconsistent

databases [3], [4], [5], [9], [10]. Those previous problems

come from several AI applications such as planning, expert

systems, reasoning, etc.

#SAT is as difficult as the SAT problem, but even when

SAT can be solved in polynomial time, is not known anefficient computational method for #SAT. For example,the 2-SAT problem (SAT limited to consider ( 2)-CFs itcan be solved in linear time, however, the corresponding

counting problem #2-SAT is a #P-complete Problem.

The maximum polynomial class recognized for #2SAT

is the class ( 2, 2)-CF (conjunction of binary or unaryclauses where each variable appears twice at most) [5], [9].

Here, we extend such class for considering the topological

structure of the undirected graph induced by the restrictions

(clauses) of the formula.

We start considering signed graphs (graphs whose

edges are designated positive or negative). That class

of graphs have been very useful for modeling different

class of problems in the social sciences [6], [7], [8].

For example the signed graphs are used for modelingthe interaction among a group of persons and the type of

relationship between certain pair of individuals of the group.

We consider here an extension of the signed graphs

where all edge in the graph has associated a pair of signs.

In this way, any binary clause can be represented by such

signed edges. And we model an electric network by those

extended signed graphs.

We are interested in computing the different assignments

associated to the variables (nodes) in the network which

produces that all edge in the network will be satisfied by at

least one of its two possible signs.

The aim is to develop a method to count the different

signed paths in which an electric network held energy on

each of its electric lines, by counting the number of models

in the Boolean formula associated to the network.

I I . METHODOLOGY

Below we summarize a basic set of concepts through

which we try to improve a more complete understanding of

terms that will be used in this work.

Topology: Systematic study of the components of an

electrical network and its reciprocal links [2].

Element: Individual constituent of a network with two or

more terminals or bounds that are used for interconnection

with other elements [2].

Nodes and branches: The junction points of the

elements in a network are called nodes. Several elements


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can join together and considered as a single unit or a

whole. When several elements are joined together so

that the flow is the same in all of them are said to be

connected in series. The junctions between components

are called internal nodes. The pair of external nodes is the

nodes themselves or simply nodes. The set of elements in

series between the nodes themselves are called branches [2].

Graph: It is the geometric, topological, representation

of a network, forming a backbone of the physical layout of

the network [2]. A graph G is represented by a pair (V, E)where V is the set of nodes and E the set of edges.

A signed graph (also called sigraph) is an orderedpair = (G, ) where G = (V, E) is a graph called theunderlying graph of and : E {+, } is a functioncalled a signature or signing. E+() denotes the set ofedges from E that are mapped by to +, and E()denotes the set of edges from E that are mapped by to -.

The elements of E+() are called positive edges andthose of E() are called negative edges of . A signedgraph is all-positive (respectively, all-negative)if all of

its edges are positive (negative); further, it is said to be

homogeneous if it is either all-positive or all-negative, and

heterogeneous otherwise.

For example the signed graphs are used for modeling

the interaction among a group of persons and the type of

relationship between certain pair of individuals of the group.

Special focus is on the social inequalities. What is it about

such characteristics as sex, race, occupation, education,

and so on that leads to inequalities in social interaction? [8].

Given a set of de n Boolean variables,

X = {x1, x2,...,xn}. Its called a literal to any xvariable or the negation of it x. We use v(l) to indicate thevariable involved by the literal l.

The disjunction of different literals is called a clause.

For k N, a k clause is a clause consisting of exactlyk literals. A variable x X appears in a clause c if x or xis an element of c. Let v(c) = {x X : x appears in c}.

A conjunctive form (CF) is a conjunction of clauses.A k CF is a CF containing only k clausulas and,( k) CF denotes a CF containing clauses with atmost k literals.

Let be a 2-CF, then an assignment s for is a functions : v() {0, 1}. An assignment can also be consideredas a set of non-complementary pairs of literals. If l s,being s an assignment, then s makes l true and makes

l false. A clausula c is satisfied if and only if c s = ,

and if for all l c, l s then s falsifies c.

A CF is satisfiable by an assignment s if eachclausula in is satisfiable by s and is contradictedif it is not satisfied. s is a model of if s is a satisfiedassignment of .

Let SAT() be the set of models than has over v(). is a contradiction or unsatisfiable if SAT() = . Letv()() = |SAT()|, be the cardinality of SAT().Given a CF, the SAT problem consists in determiningif has a model. The #SAT consists of counting thenumber of models of F defined over v(). We will alsodenote v()() by #SAT() [1].

Given a signed network G, we obtain the associated

Boolean formula , this formula can be expressed as atwo Conjunctive Form (2 CF) in the following way.Let G = ((V, E), ) be the signed network, then where

v() = V and for all positive edge {x, y} in the networkthe clause (v(x), v(y)) is formed in , while for a negativeedge {x, y} the clause (v(x), v(y)) is formed in . Thismeans that the vertices of G are the variables of ,and for each clause (x, y) in there is a signed edge(v(x), v(y)) E.

III . BUILDING RECURRENCE EQUATIONS FOR

COUNTING SIGNED PATHS IN AN ELECTRICAL NETWORK

The signed graphs are very useful to represent some kind

of relationships among individuals. In this case, each person

is a node of the graph and there is an arc between {x, y}if x is in some relation to y. Many of the relationships of

interest have natural opposites, for example: likes/dislikes,

associates with/avoids, and so on [8]. For this, two different

signs {+, } are associated with each edge of the graph.

For example, if we consider that x likes y, then a positive

edge is denoted between x and y, but we do not know

what about y with x. y might dislike x or maybe like him.

In order to be more precise in the kind of relationships

between two individuals is better to consider two signs in

each edge, one sign associated with each point end of the

edge.

Furthermore, in order to consider general Boolean

formulas in 2-CF, we consider an extension of the

concept of signed graphs. Instead to consider just one

sign (+ or -) associated with each edge of a signed graph

G = ((V, E), ), we consider here that all edge in E hasassociated a pair of signs. Then the signed function has

now the type : {(+, +), (+, ), (, +), (, )} whichgives a pair of signs to each edge of G.


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In this way, all binary clause {x, y} can be representedby a signed edge in the graph in a natural way without

importance of the signs associated to the variables in the

clause. For example the clause {x, y} will be represented

as the signed edge x +

y and in this case, - is calledthe adjacent sign of x and + the adjacent sign of y in the

edge x + y.

Add more, we can consider those extended signed

graphs as electric networks. = (G, ) whereG = (V, E) is the underlying graph of and : E {(+, +), (+, ), (, +), (, )} is the signaturefunction. That means that all edge in the network is

energized in its endpoints.

We are interested here, in count the different assignments

associated to the nodes of the network in such a way that

all edge in the network will be satisfied by at least one of

its two possible signs of its endpoints.

For this, we start analyzing the way to build satisfy

assignment in the network considering first the most simple

topologies associated with the underlying graph of the

electrical networks.

A. If the Electrical Network is a Linear Path

Let us consider that the electrical network, G = (V, E)is a linear path. Let us write down its associated

formula , without a loss of generality (ordering theclauses and its literals, if it were necessary), as: =

{c1,...,cm} = {x11 , x

12 }, {x

22 , x

23 }, . . . , {x

mm1, x

mm },

where |(ci) (ci+1)| = 1, i [[m 1]], andi, i {0, 1}, i = 1,...,m.

In order to compute the number of signed paths in such

electrical network, we start with a pair of values (i, i)associated with each node i of the network. We call the

charge of node i to the pair (i, i). The value (i)indicates the number of times that node i takes the positive

value and (i) indicate the number of times that node itakes the negative value.

Let fi be a family of clauses of built as follows:

f0 = ,fi = {cj}ji, i [[m]]. Note that fi fi+1,i [[m 1]]. Let SAT(fi) = {s : s satisfies fi}, Ai ={s SAT(fi) : xi s}, Bi = {s SAT(fi) : xi s}.Let i = |Ai|; i = |Bi| and i = |SAT(fi)| = i + i.From the total number of models in i, i [[m]], there arei of which xi takes the logical value true and i models

where xi takes the logical value false.

For example, c1 = (x11 , x

12 ), f1 = {c1}, and

(1, 1) = ( 1, 1) since x1 can take one logical value

true and one logical value false and with whichever

of those values satisfies the subformula f0 while

SAT(f1) = {x11 x

12 , x

111 x

12 , x

11 x

112 }, and then

(2, 2) = (2, 1) if 1 were 1 or rather (2, 2) = (1, 2) if1 were 0.

In general, we compute the values for (i, i) associatedto each node xi, i = 2, . . ,m, according to the signs (i, i) ofthe literals in the clause ci, by the next recurrence equation:

(i, i) =

(i1 ,i1 + i1) if(i, i) = (,)(i1 + i1,i1 ) if(i, i) = (, +)(i1 ,i1 + i1) if(i, i) = (+,)(i1 + i1,i1 ) if(i, i) = (+, +)

(1)

We denote with the application of one of the fourrules of the recurrence (1), so, the expression (2, 3) (5, 2)denotes the application of one of the rules (in this case,

the rule 4), over the pair (i1, i1) = (2, 3) in order toobtain (i, i) = (i1 + i1, i1) = (5, 3).

In recurrence ( 1) the (i, i) represents the signs asso-ciated with the edge joining a child node joining with the

father node. These recurrence equations allow to carry the

count to indicate the different ways to keep the electrical

network powered. In fact, the sum i + i obtained fromthe root node of the network, indicate the total number of

Boolean formula models associated with the network and it

is as well as the number of different ways to keep the charge

on all electric line in the electrical network.

x2x1 x3 x4 x5

a)

b)

+ + + + ++- -

(1,1) (2,3)(2,1) (5,3) (8,5) =13 models

x2x1 x3 x4 x5

+ + - + +-- +

(1,1) (1,3)(2,1) (4,1) (5,1) =6 models

Figure 1. a) Counting models over a positive electrical path networkb) Counting models over a general electrical path network

Example 1: Let = {(x1, x2), (x2, x3), (x3, x4),(x4, x5)} be a path, the series (i, i), i [[5]], is computedaccording to the signs of each clause, as it is illustrated

if the figure (1a). A similar path with 5 nodes but with

different signs in the edges is shown in figure (1b).


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Example 2: Let = {(x1, x2), (x2, x3), (x3, x4),(x4, x5), (x5, x6)} be a 2-CF which conforms alinear path, the series (i, i), i [[6]], is computedas: (1, 1) = (1, 1) (2, 2) = (2, 1) since(1, 1) = ( 1, 1), and the rule 4 has to be applied. Ingeneral, applying the corresponding rule of the recurrence

(1) according to the signs expressed by (i, i), i = 3,..., 6,we have (2, 1) (3, 3) = ( 1, 3) (4, 4) =(3, 4) (5, 5) = (3, 7) (6, 6) = (10, 7), and then,#SAT() = () = 6 = 6 + 6 = 10 + 7 = 17.

If is a path , we apply (1) in order to compute(). The procedure has a linear time complexity over thenumber of variables of , since (1) is applied while we aretraversing the chain, from the initial node y0 to the final

node ym.

B. If the Electrical Network is a Tree

The charge of the nodes (i, i) for i = 1...n, arecalculated as the nodes are visited applying a post-order

traversals. The first pair of values (i, i) for the terminalnodes of the network (tree leaves) are initialized with the

value (1, 1). So, to traverse the tree in post-order, we startby the left tree first, followed by the right tree, and then

finally the root node. At each visit of a child node i 1 toa father node i, the new values (i, i) are computed forthe father node from the following formula [1]:

Algorithm Count Models ( A )

Input: A the tree defined by the depth-search overconnectivity graph of the electrical network G

Output: The number of signed paths of an electrical

network.

Procedure: Traversing A in depth-fist, and when a

node v A is left (all of its edges have been processed)assign:

1. (v, v) = (1, 1), If v is a leaf node in A

2. Ifv is a father node with a list of child nodes associated,

i.e., u1, u1,...uk, are the child nodes of v, then as we have

already visited all the child nodes, then each pair (uj , uj )j = 1,...,k has been defined based on (1), (vi , vi) isobtained by apply (2) over (i1, i1) = (uj , uj ). Thisstep is iterated until computes all the values (vj , vj ),

j = 1, , k. And finally, let v =j=1

k vj y v =j=1

k vj

3. If v is the root node of A the return (v + v)

Figure 2. The tree graph for the formula of the example 1

This procedure returns the number of signed paths for an

electrical network in time O(n + m) which is the necessarytime for traversing G in depth-first.

Example 3: Let = {(x1, x2), (x2, x3), (x2, x4),(x2, x5), (x4, x6), (x6, x7), (x6, x8)} be a 2 CF, obtainedfrom connectivity graph of an electrical network and con-

sider us that the depth-search starts in the node x1. The

tree generated by the depth-search as well as the number

of signed paths in each level of the tree is showed in

Figure 2. The procedure Count Models returns for x1 =

168, x1 = 84 and the total number of signed paths is#SAT() = 168 + 84 = 252.

IV. CONCLUSION

We show that the problem of counting the number of

models of a Boolean formula can be applied to count the

number of signed paths in an electrical network and that

the complexity of the algorithm is linear time on the length

of the network nodes.

We also have presented different linear procedures to

compute the number of energized paths in an electrical

network, based on the formula in 2-CF, obtained from thegraph of the electric network. And we will continue working

on different applications and repercussions than might have

this research in the future.

REFERENCES

[1] De Ita G., Bello P. y Contreras M. New Polyno-mial Classes for #2SAT Established Via Graph-TopologicalStructure, Engeneering Letters (2007). Available fromwww.engineeringletters.com/issues v15/issue 2/EL 15 2 11.pdf


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[2] Lopina M., Rodrguez C. Ecuaciones de redes Topologa, Facultadde Ingenieria - Universidad de Buenos Aires (1994). Available fromhttp://materias.fi.uba.ar/6509/Ecuaciones%20de%20Redes%20I%20al%2006-05-07.pdf

[3] Angelsmark O., Jonsson P., Improved Algorithms for Counting So-lutions in Constraint Satisfaction Problems, In ICCP: Int. Conf. onConstraint Programming, 2003.

[4] Darwiche, A., On the Tractability of Counting Theory Models andits Application to Belief Revision and Truth Maintenance, Jour. of

Applied Non-classical Logics, pp. 11-34, N11(1-2),2001.

[5] De Ita G., Polynomial Classes of Boolean Formulas for Comput-ing the Degree of Belief, Advances in Artificial Intelligence LNAI3315,pp.430-440,2004.

[6] Germina K.A., Hameed S.K., On signed paths, signed cycles and theirenergies, Applied Mathematical Sciences, Vol. 4:(70), 2010, pp. 3455-3466.

[7] Kota P. S., Subramanya M.S., Note on path signed graphs, Notes onNumber Theory and Discrete Mathematics 15:(4), 2009, pp. 1-6

[8] Fred S. Roberts, Graph Theory and Its Applications to Problems ofSociety, Edit. SIAM Collection, 1978.

[9] Roth, D., On the hardness of approximate reasoning, ArtificialIntelligence 82,pp 273-302,1996.

[10] Vadhan, S. P., The complexity of Counting in Sparse, Regular, andPlanar Graphs, SIAM Journal on Computing,pp. 398-427, V31, N2,2001.

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