Sangita Kansal

Boolean Petri Nets

Petri net is a graphical tool invented by Carl Adam Petri [13]. These are used for describing, designing and studying discrete event-driven dynamical systems that are characterized as being concurrent, asynchronous, distributed, parallel, random and/or nondeterministic. As a graphical tool, Petri net can be used for planning and designing a system with given objectives, more practically effective than flowcharts and block diagrams. As a mathematical tool, it enables one to set up state equations, algebraic equations and other mathematical models which govern the behavior of discrete dynamical systems. Still, there is a drawback inherent in representing discrete event-systems. They suffer from the state explosion problem as what will happen when a system is highly populated, i.e., initial state consists of a large number of places that are nonempty. This phenomenon may lead to an exponential growth of its reachability graph. This makes us to study the safe systems. The aim of this chapter is to present some basic results on 1-safe Petri nets that generate the elements of a Boolean hypercube as marking vectors. Complete Boolean hypercube is the most popular interconnection network with many attractive and well known properties such as regularity, symmetry, strong connectivity, embeddability, recursive construction, etc. For brevity, we shall call a 1-safe Petri net that generates all the binary n-vectors as marking vectors a Boolean Petri net. Boolean Petri nets are not only of theoretical interest but also are of practical importance, required in practice to construct control systems [1]. In this chapter, we will consider the problems of characterizing the class of Boolean Petri nets as also the class of crisp Boolean Petri nets, viz., the Boolean Petri nets that generate all the binary n-vectors exactly once. We show the existence of a disconnected Boolean Petri net whose reachability tree is homomorphic to the n-dimensional complete lattice Ln. Finally, we observe that characterizing a Boolean Petri net is rather intricate.

Vertex equitable labeling of signed graphs

Abstract A signed graph (or, in short, sigraph) S = ( S u , σ ) consists of an underlying graph S u : = G = ( V , E ) and a function σ : E ( S u ) → { + , − } , called the signature of S. Let S be a signed graph with p vertices and q edges and let A = { 0 , 1 , 2 , … , ⌈ q 2 ⌉ } . A vertex labeling f : V ( S ) → A which is onto, is said to be a vertex equitable labeling of S if it induces a bijective edge labeling f ⁎ : E ( S ) → { 1 , 2 , … , m , − 1 , − 2 , … , − n } defined by f ⁎ ( u v ) = σ ( u v ) ( f ( u ) + f ( v ) ) such that | v f ( a ) − v f ( b ) | ≤ 1 , ∀ a , b ∈ A , where v f ( a ) is the number of vertices with f(v) = a and m, n are number of positive and negative edges respectively in S. A signed graph S is said to be vertex equitable if it admits a vertex equitable labeling. In this paper, we initiate a vertex equitable labeling of signed graphs and study vertex equitable behavior of signed paths, signed stars and signed complete bipartite graphs K 2 , n .

Construction of a Crisp Boolean Petri Net from a 1-safe Petri Net

The concept of a Petri net, a tool for the study of certain discrete dynamical systems, was invented in 1939 by Carl Adam Petri. In the attempt to characterize Boolean Petri nets, we discovered a subclass of Boolean Petri net called the crisp Boolean Petri net, viz., the one that generates every binary vector as its marking vectors exactly once. In this paper, the construction of a crisp Boolean Petri net from a 1-safe Petri net has been shown.

A Disconnected 1-Safe Petri Net Whose Reachability Tree Is Homomorphic to a Complete Boolean Lattice

Petri nets generating all the 2^n binary n-vectors as their marking vectors are not only of theoretical interest but also are of practical importance. In this note, we demonstrate the existence of a disconnected 1-safe Petri net whose reachability tree is homomorphic to the n-dimensional complete lattice L_n. This makes the problem of characterizing the 1-safe Petri nets that generate all the binary n-vectors as marking vectors exactly once appear more intricate.

C-consistent and C-cycle compatible dot-line signed graphs

Abstract A signed graph S = ( S u , σ ) has an underlying graph S u : = G = ( V , E ) and a function σ : E ( S u ) → { + , − } . A marking of S is a function μ : V ( S ) → { + , − } . In canonical marking, denoted μ σ , we assign +(‘−’) sign to a vertex if its negative degree is even(odd). The dot-line (or •-line) signed graph of S, denoted L • ( S ) , is obtained by representing edges of S as vertices, two of these vertices are adjacent if the corresponding edges are adjacent in S and edge e e ′ in L • ( S ) is negative whenever negative degree of a common vertex of edges e and e ′ in S is odd. S is called C -consistent if every cycle in S has an even number of negative vertices under canonical marking. S is called C -cycle compatible if for every cycle Z in S, the product of signs of its vertices equals the product of signs of its edges with respect to canonical marking. In this paper, we establish structural characterizations of signed graph S so that L • ( S ) is C -consistent and C -cycle compatible.

Durated paths in Petrinet

In this paper, a durated path in a Petrinet by thinking of functions from the set of all paths of a Petrinet into the semiring K = N ∪ {∞}, i.e., set of all the natural numbers including zero together with additional element ∞ as duration has been defined and it has been proved that the set of all durations forms a K-module.

Basic Results on Crisp Boolean Petri Nets

The concept of Petri net as a discrete event-driven dynamical system was invented by Carl Adam Petri in his doctoral thesis ‘Communication with Automata’ in 1962. Petri nets are one of the best defined approach to modeling of discrete and concurrent systems. The dynamics of Petri nets represent the long-term behavior of the modeled system. Petri nets combine mathematical concepts with a pictorial representation of the dynamical behavior of the modeled systems. A Petri net is a bipartite directed graph consisting of two type of nodes, namely, place nodes and transition nodes. Directed arcs connect places to transitions and transitions to places to represent flow relation. In this paper, we present some basic results on 1-safe Petri net that generates every binary n-vector exactly once as marking vectors in its reachability tree, known as crisp Boolean Petri net. These results can be used for characterizing crisp Boolean Petri nets.

Results on Lict Signed Graphs Lc(S)

Abstract A signed graph (or, in short, sigraph) S=(Su, α) consists of an underlying graph Su := G = (V, E) and a function a : E(Su) → {+, −}, called the signature of S. The line signed graph of a signed graph S, denoted L(S), is the signed graph having vertex set E(S) in which two of these vertices are adjacent if the corresponding edges are adjacent in S, every such edge ee’ in L(S) is negative whenever both the adjacent edges e and e’ in S are negative. The lict signed graph of a signed graph S, denoted Lc(S), is the signed graph having vertex set E(S) ∪ C(S) in which two of these vertices are adjacent if the corresponding members of S are adjacent or incident, every such edge uv in Lc(S) is negative whenever u, v ∈ E−(S) or u ∈ E−(S) and v is a cut-vertex of S with negative degree odd, here C(S) and E−(S) are cut-vertex set and negative edge set of S respectively. In this paper, we characterize signed graphs on Kp, p ≥ 2, on cycle Cn and on Km,n which are lict signed graphs or line signed graphs, characterize signed graphs S so that Lc(S) and L(S) are balanced. We also establish the characterization of signed graphs S for which S ∼ Lc(S), S ∼ L(S), h(S) ∼ Lc(S) and h(S) ∼ L(S), here h(S) is negation of S and ∼ stands for switching equivalence.

Fuzzy sets, linear logic and petrinets

Abstract In this paper, we study some interconnections amongst the three disciplines mentioned in the title of the paper. To begin with we quote some definitions and observations from Pratt [4]. There are lots of connectives in the closed unit interval [0, 1] , we present some of them. We replace the set of natrual numbers ℕ by a semi ring K = [0, 1] in the definition of a petrinet and discuss the firing rule of this petrinet. We also define a dual net and see that a transition t fires at a marking μ to produce precisely the dual of what it produces when it dual fires at μ . The last section of this paper is devoted exclusively to a “petrinet corresponding to a lineale” (which is a model for linear logic) we will discuss two special cases of a lineale and with the help of these cases, we offer a definition of a ‘complementary net’, a kind of dual, based on the complementation. This is another link to linear logic.

Embedding an Arbitrary 1-safe Petri Net into a Boolean Petri Net

Petri nets are powerful mathematical formalism for designing and studying behaviors of a wide range of discrete dynamical event driven systems. The aim of this paper is to show that an arbitrary1safe Petri net can be embedded as an induced subnet of a Boolean Petri net, viz., the one that generates every binary n-vector as its marking vector.