State Based Control of Timed Discrete Event Systems using Binary ...

Chapter 5. BDD based implementation 41In order to do these computations we employ binary decision diagrams (BDDs) as therepresentation of predicates.The algorithm for finding the reachable states is as follows:ReachableStates := InitialState;DO{tmp := ReachableStates;new := NextStates(ReachableStates, T );ReachableStates := ReachableStates ∨ new;}While tmp ≠ ReachableStates;Actually, we have started from the initial state and find all the states we can reachfrom it.As stated in Chapter 3, ordering of the variables in a BDD is one of the most importantoptimizations.The variables we have in finding the transition predicate are:act, t σ1 , ..., t σn , act ′ , t ′ σ 1, ..., t ′ σ n. Since the number of elements in the domain of each variableis greater than 2, each variable itself consists of a number of BDD variables. Themore closely coupled two variables are, the “closer” the two components should be placedin the BDD. In our algorithm, the variables representing t σ and t ′ σ should be adjacentbecause in most of the transitions t σ and t ′ σ differ only in one or zero unit of time. If weuse a different ordering of variables, the number of BDD nodes will be increased andthe algorithm may become much too slow.One of the main problems in working with BDDs is the problem of “intermediate nodenumber explosion” [30][18], i.e. although the number of BDD nodes in the final resultmay not be large, it could be tens of times larger during the synthesis process, or moreaccurately, during the generation of reachable states. In order to reduce this problem in