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Special nodes are used for true and false at the leaves: we ll write these as T and F Every BDD is constructed with respect to a global variable ordering, so x AND NOT y can be represented as (x => (y => F | T) | F) if x comes before y in this ordering and as (y => F | (x => T | F)) if y comes before x The variable ordering can be critical for performance of the representation BDDs are efficient because they use some of the language representation techniques you saw in 9 In particular, they work by uniquely memoizing all BDD nodes that are identical, which works by representing a BDD as an integer index into a lookup table that stores the real information about the node.

// delete the corresponding value in the sort index _sortIndex.RemoveAt(index); // now fix up all index pointers in the sort index foreach (ListItem item in _sortIndex) if (item.BaseIndex > baseIndex) item.BaseIndex -= 1; OnListChanged(new ListChangedEventArgs( ListChangedType.ItemDeleted, index)); _initiatedLocally = false; } else _list.RemoveAt(index); } Remember that _bindingList is a reference to the original collection object s implementation of the IBindingList interface. So this code merely sets _initiatedLocally to true and then delegates the AddNew() call to the original collection. Similarly, the RemoveAt() call is delegated to the original collection through its IList<T> interface. I ve also optimized the RemoveAt() implementation, so when an item is removed from the middle of the list, the entire sorted index isn t rebuilt. This offers substantial performance improvements when dealing with larger-sized lists.

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Furthermore, negative indexes are used to represent the negation of a particular BDD node without creating a separate entry for the negated node Listing 12-15 shows our implementation of BDDs Fully polished BDD packages are often implemented in C It is easy to access those packages from F# using the techniques described in 19 Here we are content with a clear and simple implementation entirely in F# code Listing 12-15 Implementing Binary Decision Diagrams open SystemCollectionsGeneric let memoize f = let tab = new Dictionary<_,_>() fun x -> if tabContainsKey(x) then tab[x] else let res = f x in tab.

The title attributes for the abbreviations in the document are hidden. Some browsers will display the titles as tool tips when you mouse over the abbreviations. The default browser behavior for abbreviations is as unpredictable as the default browser styling. In the same way that you can override the default styles with CSS, you can override the default behavior with the DOM.

The important thing here is that SortedBindingList doesn t maintain a local copy of the collection s data. Instead, it delegates all calls directly to the original collection itself.

[x] <- res; res type type type type BddIndex = int Bdd = Bdd of BddIndex BddNode = Node of Var * BddIndex * BddIndex BddBuilder(order : Var -> Var -> int) = // The core data structures that preserve uniqueness let uniqueTab = new Dictionary<BddNode,BddIndex>() let nodeTab = new Dictionary<BddIndex,BddNode>() // Keep track of the next index let mutable nextIdx = 2 let trueIdx = 1 let falseIdx = -1.

The original collection performs the requested operation and adds or removes the child object. Of course, that triggers a ListChanged event from the original collection. Recall that in the constructor of SortedBindingList, the original collection s ListChanged event was handled by the SourceChanged() method. I ll cover the SourceChanged() method in a moment, and you ll see how the ListChanged event is suppressed when the add or remove operation is initiated by the view itself. The Insert() method is simpler: public void Insert(int index, T item) { _list.Insert(index, item); } void System.Collections.IList.Insert(int index, object value) { Insert(index, (T)value); } When a new item is inserted into the view, it is really inserted into the original collection. This results in the original collection raising its ListChanged event, and in turn the view then raises its ListChanged event (in the SourceChanged() method in the following code). The end result is that the view raises the ListChanged event exactly once, which is the desired goal.

The source collection s ListChanged event is handed by the SourceChanged() method. This allows SortedBindingList to re-sort the data if it is changed in the original collection, thus keeping the view current. It also means that the event can be reraised by the view so any UI components bound to the sorted view are also aware that the underlying data has changed.

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