In graph theoryreachability refers to the ability to get from one vertex to another within a graph. In an undirected graph, reachability between all pairs of vertices can be determined by identifying the connected components of the graph. The connected components of an undirected graph can be identified in linear time.

The remainder of this article focuses on the more difficult problem of determining pairwise reachability in a directed graph which, incidentally, need not be symmetric. Algorithms for determining reachability fall into two classes: those that require preprocessing and those that do not.

## Toolbox Description

If you have only one or a few queries to make, it may be more efficient to forgo the use of more complex data structures and compute the reachability of the desired pair directly. This can be accomplished in linear time using algorithms such as breadth first search or iterative deepening depth-first search.

If you will be making many queries, then a more sophisticated method may be used; the exact choice of method depends on the nature of the graph being analysed. Three different algorithms and data structures for three different, increasingly specialized situations are outlined below. The Floydâ€”Warshall algorithm [5] can be used to compute the transitive closure of any directed graph, which gives rise to the reachability relation as in the definition, above.

This algorithm is not solely interested in reachability as it also computes the shortest path distance between all pairs of vertices. For graphs containing negative cycles, shortest paths may be undefined, but reachability between pairs can still be noted. For planar digraphsa much faster method is available, as described by Mikkel Thorup in This algorithm can also supply approximate shortest path distances, as well as route information. An outline of the reachability related sections follows.

The proof that such separators can always be found is related to the Planar Separator Theorem of Lipton and Tarjan, and these separators can be located in linear time. At each level of the recursion, only linear work is needed to identify the separators as well as the connections possible between vertices.

## Formal for Over-Constraint and Reachability Analysis

An even faster method for pre-processing, due to T. Kameda in[7] can be used if the graph is planaracyclicand also exhibits the following additional properties: all 0- indegree and all 0- outdegree vertices appear on the same face often assumed to be the outer faceand it is possible to partition the boundary of that face into two parts such that all 0-indegree vertices appear on one part, and all 0-outdegree vertices appear on the other i. Preprocessing performs the following steps. For each vertex we store the list of adjacencies out-edges in order of the planarity of the graph for example, clockwise with respect to the graph's embedding.

During this traversal, the adjacency list of each vertex is visited from left-to-right as needed. The depth-first traversal is then repeated, but this time the adjacency list of each vertex is visited from right-to-left.

The breadth-first search technique works just as well on such queries, but constructing an efficient oracle is more challenging.These examples are from the Cambridge English Corpus and from sources on the web.

Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.

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Word Lists. Choose your language. Tell us about this example sentence:. The word in the example sentence does not match the entry word. The sentence contains offensive content. Cancel Submit. Your feedback will be reviewed. Add a definition. The objective here is to study reachabilityminimization and minimal realization in these bicategories.

From the Cambridge English Corpus. In each case we study reachability and minimization, and prove a minimal realization theorem.

The objective here is to study functorial aspects of reachabilityminimisation and minimal realisation.

Moreover, as is usual in automata theory, we require reachability for our compositional minimisation. The proof of the reachability is similar for all of the remaining cases, so we will omit arguing it in the rest of the discussion. The reachability property is clearly satisfied, since the output store is the same as the input store. The second part of the theorem expresses what we will call the reachability property.

Our verification method makes use of symbolic representations of infinite set of system states and of symbolic backward reachability. For the sake of brevity, we will illustrate the connection between provability and reachability in the extended setting through the following example.Latest news.

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**Reachability Analysis of Pushdown Systems - Ahmed Bouajjani**

KukovecSean McLaughlinJ. ReedNeha RungtaJ. SizemoreM. StalzerP. SrinivasanP.In this paper we study the reachability problem of sub- and superconservative discrete state chemical reaction networks d-CRNs. It is known that a subconservative network has bounded reachable state space, while that of a superconservative one is unbounded. The reachability problem of superconservative reaction networks is traced back to the reachability of subconservative ones.

We consider network structures composed of reactions having at most one input and one output species beyond the possible catalyzers. We give a proof that, assuming all the reactions are charged in the initial and target states, the reachability problems of sub- and superconservative reaction networks are equivalent to the existence of nonnegative integer solution of the corresponding d-CRN state equations. Therefore, the number of feasible trajectories satisfying the reachability relation can be counted in polynomial time in the number of species and in the distance of initial and target states, assuming fixed number of reactions in the system.

Employing deterministic ordinary differential equation systems to characterize the dynamical behavior of complex networks of chemically interacting components species is a widely used approach in mathematical and computational systems biology [ 1 â€” 3 ]. Such a continuous state modeling approach assumes high molecular count of species and their homogeneous well-mixed distribution in the surrounding media [ 4 ].

However, in several bio chemically interesting systems, such as some enzymatic and genetic networks, the molecular count of different species is relatively low e. Hence it is necessary to introduce a discrete state model capable of keeping track of the individual molecular counts in order to properly characterize the qualitative dynamical behavior of bio chemical networks of species with low number of molecules [ 910 ].

There exist several mathematical models describing the state evolution of discrete state chemical reactions networks, such as Markov chain models [ 810 ] and stochastic Petri nets [ 11 ]. In the context of chemical reaction networks of several interacting components, in order to completely characterize the system it is needed to simultaneously study the dynamical behavior and the underlying network structure as well.

Moreover, it is also important to examine how the dynamical behavior and the network structure are related to each other, and how we can predict the dynamical behavior e.

For continuous state reaction networks obeying the law of mass action, it is recognized that the network structure i. In the case of discrete state reaction networks the so-called reachability is a strictly related problem to the dynamical behavior; namely, is it possible to reach a prescribed target state from a given initial one through a finite sequence of transition reactions?

It is known that the reachability relation between any pair of nonnegative initial and target states is determined by the network structure itself. Through the reachability analysis several problems of great importance can be analyzed; one of them having high interest is the existence of so-called extinction events: the existence of trajectories resulting in the irreversible extinction of some species from the system.

It has been shown that under some conditions on the network structure a discrete state chemical reaction network exhibits an extinction event from any point of its state space [ 91617 ].

The properties of recurrence the ability of returning to any initial state and irreducibility the ability of reaching any state from any other one are also examined in the context of discrete state reaction networks [ 1819 ]. The mathematical model of discrete state chemical reaction networks is equivalent to an important model of theoretical computer science, namely, the so-called vector addition systems with states VASS or equivalently Petri nets [ 2021 ].

Hence the discrete chemical reaction network reachability problem is equivalent to the extensively studied problem of vector addition system VAS reachability. Unfortunately, contrary to the proven polynomial time complexity of reachability of rate independent continuous state chemical reaction networks [ 21 ], in the case of discrete state reaction networks it is not known whether there exists an algorithm of primitive-recursive time complexity deciding this problem [ 27 ].

The aim of this paper is to study of the reachability problem of sub- and superconservative d-CRNs.

We make use of the relation between the sub- and superconservative properties. In Propositions 15 and 17we give necessary and sufficient conditions on the network structure and the initial and target states under which the reachability is equivalent to the nonnegative integer solution of the d-CRN state equation. Then these results in Corollaries 16 and 18 are extended to a subclass of superconservative d-CRNs.

### Reachability Analysis for AWS-based Networks

The paper is organized as follows. Section 3 discusses the classes of sub- and superconservative d-CRNs and their duality as well. In Section 4 the reachability problem of sub- and superconservative d-CRNs is examined. Firstly the case of low state space-dimensional d-CRNs is discussed, followed by the extension to the general case when the dimension of the state space is arbitrarily high.

In Section 5 our findings are illustrated in a representative example. In Table 1 we summarize the notations and concepts of discrete chemical reaction networks which will be extensively used later.

For each complexthe stoichiometric coefficients of the species can be represented as a vector of the following form: For eacha reaction vector can be associated with the track of the net molecular count changes of the species upon firing the reaction: so that and are the corresponding source and product complexes of. We will also assume that for all the examined d-CRNs a fixed order of the reaction vectors is given; i.

A d-CRN can also be represented by a directed graph such that the vertices and edges correspond to the complexes and the reactions, respectively, i. For each edge a weight corresponding to the reaction rate constant also called intensity or propensity corresponding to the respective reaction can also be associated. Beyond the above representations it is also possible to describe a d-CRN in an algebraic way by means of its unique stoichiometric matrix.

Definition 1.Reachability analysis is a solution to the reachability problem in the particular context of distributed systems. It is used to determine which global states can be reached by a distributed system which consists of a certain number of local entities that communicated by the exchange of messages. Reachability analysis was introduced in a paper of for the analysis and verification of communication protocols.

This paper was inspired by a paper by Bartlett et al. This protocol belongs to the Link layer and, under certain assumptions, provides as service the correct data delivery without loss nor duplication, despite the occasional presence of message corruption or loss.

For reachability analysis, the local entities are modeled by their states and transitions. An entity changes state when it sends a message, consumes a received message, or performs an interaction at its local service interface. In the simplest case, the medium between two entities is modeled by two FIFO queues in opposite directions, which contain the messages in transit that are sent, but not yet consumed.

Reachability analysis considers the possible behavior of the distributed system by analyzing all possible sequences of state transitions of the entities, and the corresponding global states reached [4]. The result of reachability analysis is a global state transition graph also called reachability graph which shows all global states of the distributed system that are reachable from the initial global state, and all possible sequences of send, consume and service interactions performed by the local entities.

However, in many cases this transition graph is unbounded and can not be explored completely. The transition graph can be used for checking general design flaws of the protocol see belowbut also for verifying that the sequences of service interactions by the entities correspond to the requirements given by the global service specification of the system [1].

Boundedness: The global state transition graph is bounded if the number of messages that may be in transit is bounded and the number states of all entities is bounded. The question whether the number of messages remains bounded in the case of finite state entities is in general not decidable [5]. One usually truncates the exploration of the transition graph when the number of messages in transit reaches a given threshold.

As an example, we consider the system of two protocol entities that exchange the messages mambmc and md with one another, as shown in the first diagram. The protocol is defined by the behavior of the two entities, which is given in the second diagram in the form of two state machines.

Here the symbol "! The initial states are the states "1". The third diagram shows the result of the reachability analysis for this protocol in the form of a global state machine. Each global state has four components: the state of protocol entity A leftthe state of the entity B right and the messages in transit in the middle upper part: from A to B; lower part: from B to A.

Each transition of this global state machine corresponds to one transition of protocol entity A or entity B. The initial state is [1, - -1] no messages in transit. One sees that this example has a bounded global state space - the maximum number of messages that may be in transit at the same time is two. This protocol has a global deadlock, which is the state [2, - -3]. If one removes the transition of A in state 2 for consuming message mbthere will be an unspecified reception in the global states [2, ma mb ,3] and [2, - mb ,3].

The design of a protocol has to be adapted to the properties of the underlying communication medium, to the possibility that the communication partner fails, and to the mechanism used by an entity to select the next message for consumption. The communication medium for protocols at the Link level is normally not reliable and allows for erroneous reception and message loss modeled as a state transition of the medium.Skip to Main Content. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity.

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Email Address. Sign In. Reachability Analysis of Networked Finite State Machine With Communication Losses: A Switched Perspective Abstract: Networked finite state machine takes into account communication losses in industrial communication interfaces due to the limited bandwidth. The reachability analysis of networked finite state machine is a fundamental and important research topic in blocking detection, safety analysis, communication system design and so on.

This paper is concerned with the impact of arbitrary communication losses on the reachability of networked finite state machine from a switched perspective. First, to model the dynamics under arbitrary communication losses in communication interfaces from the controller to the actuatorby resorting to the semi-tensor product STP of matrices, a switched algebraic model of networked finite state machine with arbitrary communication losses is proposed, and the reachability analysis can be investigated by using the constructed model under arbitrary switching signal.

Subsequently, based on the algebraic expression and its transition matrix, necessary and sufficient conditions for the reachability are derived for networked finite state machine. Finally, some typical numerical examples are exploited to demonstrate the effectiveness of the proposed approach. Note that current results provide valuable clues to design reliable and convergent Internet-of-Things networks. Article :. Date of Publication: 16 March DOI: Need Help?CORA integrates various vector and matrix set representations and operations on them as well as reachability algorithms of various dynamic system classes.

The software is designed such that set representations can be exchanged without having to modify the code for reachability analysis. CORA is designed using the object oriented paradigm, such that users can safely use methods without concerning themselves with detailed information hidden inside the object.

The following points summarize the main capabilities of the CORA toolbox:. CORA computes reachable sets for linear systems, nonlinear systems as well as for systems with constraints. Continuous as well as discrete time models are supported. Uncertainty in the system inputs as well as uncertainty in the model parameters can be explicitly considered. In addition, CORA also provides capabilities for the simulation of dynamical models. The toolbox is also capable to calculate the reachable sets for hybrid systems.

All implemented dynamic system classes can be used to describe the different continuous flows for the discrete system states. Further, multiple different methods for the calculation of the intersections with guard sets are implemented in CORA. CORA has a modular design, making it possible to use the capabilities of the various set representations for other purposes besides reachability analysis. The toolbox implements vector set representation, e. The following points summarize the main capabilities of the CORA toolbox: Reachability Analysis for Continuous Systems CORA computes reachable sets for linear systems, nonlinear systems as well as for systems with constraints.

Reachability Analysis for Hybrid Systems The toolbox is also capable to calculate the reachable sets for hybrid systems. Geometric Sets CORA has a modular design, making it possible to use the capabilities of the various set representations for other purposes besides reachability analysis. Althoff, D. Grebenyuk and N. In Proc. Althoff and D. Implementation of interval arithmetic in CORA An introduction to CORA

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