Reachability graph

In the context of reachability analysis, this graph is called symbolic reachability graph of the automaton A and can be searched using shortest path search techniques as widely used in computer science. Hence, the task of finding the cost-optimal schedule is to find the shortest (or cheapest) path in a (priced) symbolic reachability graph. 

The search starts from the initial state qo. The set W is a data structure which stores symbolic states which are not yet explored. The functionfinal decides whether the given symbolic state is a target state in which the production is completed. The symbolic reachability graph is step-wise constructed by evaluating the successor relation Succ(q), which computes the successor symbolic states of q. The best solution is returned in cost. Existing tools implement numerous extensions of the standard algorithm to improve the efficiency ... 

Fig. 10.6 The initial part of the reachability graph of A. The node names give information about the locations of the automata R-, / 2, R3 / is the idle and b is the busy location. The arcs are annotated with the time transitions preceding the discrete transitions. The bold arcs represent the optimal trace.

Table 10.1 The optimal path of the reachability graph. The length is 15 nodes and the makespan is 10. The clock valuations of the individual clocks C are not shown, only the absolute time is presented in the second column.

The systems behaviour in case of one of the possible CCF can be modeled according to the 1 oo2-system model. Note that the corresponding Markov model had 83 states 3 states that the individual components may be in and two CCF-states. In comparison to this, the Petri net has 3-4 = 12 states for the individual components and two states for the CCF. Certainly, the reachability graph that may automatically be generated has 83 states as well. 

B contains all nodes reachable from node m B is graph isomorphic to B... 

Scheme P is a graph hononorphic image of the scheme P in Example IV-3, which is tree-like. Scheme P is formed from P by duplicating nodes when anomalous connections occur. Thus the direct connection from the node labelled T(x ) which follows D, to B is anonalous in P. In P, this connection is removed and a direct connection to a copy of B is substituted. The copy of B leads into a copy of the subscheme reachable from B until A occurs. We do not need to duplicate A since any connection to A must be acceptable hence the diagram loops lack to A. Similarly, the anomalous direct connection from the node T(x ) which follows C to E is anonalous and is replaced by a connection to a copy of E and the node following E, TCxg) the connections from this instance of I(x2) are now legal and so no duplicates are needed.

The maximal loops in the graph can be found from the reachability matrix by finding those sets of vertices that satisfy the following conditions (1) r j = rfi = 1, where i and j take on all possible combinations of the vertex numbers in the set (2) no other vertices, not included in the set, satisfy condition (1). The first condition requires that each vertex in the set is reachable by some path from every other vertex in the set. The second condition requires that there is no path from a vertex in the set to a vertex outside the set... 

Let me define the chemical Actual , and Adjacent Possible . Consider a set of N molecular species in a volume and the associated reaction graph. Call this set the Actual . Now let the molecules react. It may be that new molecular species, reachable in a single reaction step from the N, arise. Call these novel molecules the Adjacent Possible . The union of the Actual with this first Adjacent Possible creates a new Actual with a new Adjacent Possible. At issue is the flow of material from any initial Actual. 

Multiway Decision Graphs, Reachability Analysis, Recurrent domains, p-terms... 

This information can be obtained automatically from the CPN model through the construction of an occurrence graph. Such graph contains all reachable states of the model from a given initial state. It can then be examined for a research of potentially parallel and concurrent paths from one state to the other. This information can be helpful for determining alternative operation procedures as well as for the detection of non determinism in the choice of possible paths. The occurrence graph for the presented model is shown on Figure 5. 

In bond graphs of hybrid system models, some storage elements in integral causality may be connected to a detector only via a causal path through a switch model Sw m or even through several switch models. Clearly, the necessary reachability condition is only satisfied when the switch involved in the causal path is closed. That is, a model is not fully state-observable in those system modes in which a switch being part of the only causal path from a storage element to a detector is off, i.e. m = 0 for the switched MTF of that switch model. 

Inspection of the bond graph in Fig. 2.18 shows that there is a direct causal path from the effort source to the I-element and a causal path from the effort source through the I-element to the capacitor. That is, both storage elements are reachable from the effort source independently of the switch states. The sufficient condition is also satisfied. Both storage elements take derivative causality in the bond graph with preferred derivative causality in Fig. 3.1. Hence, the model of the circuit in Fig. 2.17 with two independent switches is structurally completely state controllable with the one effort source for all four system modes. 

Fig. 1. The figures shows a particular RAF set on (a) the complete graph substrates/ products-reactions-catalysis and on (b) the simplified catalyst-product graph representation. In (a) solid lines represent materials production/consumption, whereas dotted lines represent catalysis AA, AB and B species are needed for the production of species AAB and AAAB and are provided by the environment. Scheme in (b) represents the same process, that in this case constitutes a SCC (Strongly Connected Component- a structure where each node is directly or indirectly reachable starting from any other node of the same structure) this partial representation does not allow the recognition that some materials have to be present to allow the system s growth...

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