scholarly journals Lower and upper bounds of shortest paths in reachability graphs

2004 ◽  
Vol 2004 (57) ◽  
pp. 3023-3036 ◽  
Author(s):  
P. K. Mishra
Keyword(s):  
Petri Nets ◽  
Shortest Path ◽  
Petri Net ◽  
Free Choice ◽  
Shortest Paths ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Marked Graphs ◽  
Marked Graph

We prove the following property for safe marked graphs, safe conflict-free Petri nets, and live and safe extended free-choice Petri nets. We prove the following three results. If the Petri net is a marked graph, then the length of the shortest path is at most(|T|−1)⋅|T|/2. If the Petri net is conflict free, then the length of the shortest path is at most(|T|+1)⋅|T|/2. If the petrinet is live and extended free choice, then the length of the shortest path is at most|T|⋅|T+1|⋅|T+2|/6, whereTis the set of transitions of the net.

Characterizing Liveness Monotonicity for Weighted Petri Nets in Terms of Siphon-Based Properties

2003 ◽  
Vol 14 (04) ◽  
pp. 641-658 ◽  
Author(s):  
Li Jiao ◽  
To-Yat Cheung
Keyword(s):  
Petri Nets ◽  
Petri Net ◽  
Free Choice ◽  
Sufficient Condition

A Petri net (N, M0) is monotonically live (m-live) if it remains live when the values of its initial marking M0 are increased. N is structurally m-live if there exists an initial marking M0 such that (N, M0) is m-live. Three new siphon-based characterizations for these properties are obtained: (1) For a weighted net N, the ST-property (i.e., every siphon contains a trap) is a necessary but not sufficient condition for N to be structurally m-live. (2) For a weighted net N, a necessary but not sufficient condition for (N, M0) to be m-live is that every siphon of N contains an M0-controlled trap (i.e., for every reachable marking M, the trap contains a place whose token value is not smaller than the least weight of its outgoing arcs). (3) A homogeneous asymmetric choice net (N, M0) is m-live if and only if every minimal siphon of N contains an M0-controlled trap. Characterization (3) is a generalization of Commoner's Theorem from ordinary liveness for ordinary free choice nets to m-liveness for homogeneous asymmetric choice nets.

On a Decidable Class of Partially Controlled Petri Nets With Liveness Enforcing Supervisory Policies

2013 ◽  
Vol 43 (5) ◽  
pp. 1256-1261 ◽  
Keyword(s):  
Petri Nets ◽  
Petri Net ◽  
Free Choice ◽  
Closed Set ◽  
Construction Procedure ◽  
Transition Set

We identify a class of partially controlled Petri net (PN) structures, which is denoted by G, that strictly includes the class of partially controlled free-choice (FC) PN structures. We show that there is a supervisory policy that enforces liveness in an arbitrary instance N(m0), where N ∈ G, if and only if there is a similar policy for an FCPN that results when the construction procedure enunciated in this paper is executed with N and its controllable transition set as input. Since the existence of a supervisory policy in an arbitrary partially controlled FCPN is decidable, it follows that the existence of similar policies for any N(m0), where N ∈ G, is also decidable. Furthermore, when it exists, the minimally restrictive supervisory policy that enforces in a member of G is characterized by a right-closed set of markings.

scholarly journals On the geodetic hull number for complementary prisms II

2020 ◽  
Author(s):  
Diane Castonguay ◽  
Erika Morais Martins Coelho ◽  
Hebert Coelho ◽  
Julliano Nascimento
Keyword(s):  
Convex Hull ◽  
Shortest Path ◽  
Disjoint Union ◽  
Perfect Matching ◽  
Convex Set ◽  
Upper Bounds ◽  
The Other ◽  
Lower And Upper Bounds ◽  
Split Graph ◽  
Complementary Prism

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is \textit{convex} if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The \textit{convex hull} $H(S)$ of $S$ is the smallest convex set containing $S$. If $H(S) = V(G)$, then $S$ is a \textit{hull set}. The cardinality $h(G)$ of a minimum hull set of $G$ is the \textit{hull number} of $G$. The \textit{complementary prism} $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. A graph $G$ is \textit{autoconnected} if both $G$ and $\overline{G}$ are connected. Motivated by previous work, we study the hull number for complementary prisms of autoconnected graphs. When $G$ is a split graph, we present lower and upper bounds showing that the hull number is unlimited. In the other case, when $G$ is a non-split graph, it is limited by $3$.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

scholarly journals Relationship between Age and Value of Information for a Noisy Ornstein–Uhlenbeck Process

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 940
Author(s):  
Zijing Wang ◽  
Mihai-Alin Badiu ◽  
Justin P. Coon
Keyword(s):  
Value Of Information ◽  
Markov Models ◽  
Upper Bounds ◽  
Distribution Functions ◽  
Lower And Upper Bounds ◽  
Uhlenbeck Process ◽  
Source Data ◽  
Non Linear ◽  
Queue Model ◽  
Selection Of

The age of information (AoI) has been widely used to quantify the information freshness in real-time status update systems. As the AoI is independent of the inherent property of the source data and the context, we introduce a mutual information-based value of information (VoI) framework for hidden Markov models. In this paper, we investigate the VoI and its relationship to the AoI for a noisy Ornstein–Uhlenbeck (OU) process. We explore the effects of correlation and noise on their relationship, and find logarithmic, exponential and linear dependencies between the two in three different regimes. This gives the formal justification for the selection of non-linear AoI functions previously reported in other works. Moreover, we study the statistical properties of the VoI in the example of a queue model, deriving its distribution functions and moments. The lower and upper bounds of the average VoI are also analysed, which can be used for the design and optimisation of freshness-aware networks. Numerical results are presented and further show that, compared with the traditional linear age and some basic non-linear age functions, the proposed VoI framework is more general and suitable for various contexts.

The Lower and Upper Bounds of Turán Number for Odd Wheels

2021 ◽  
Vol 37 (3) ◽  
pp. 919-932
Author(s):  
Byeong Moon Kim ◽  
Byung Chul Song ◽  
Woonjae Hwang
Keyword(s):  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Turán Number
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