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member g.Build(f) = match f with | And(x,y) -> g.And(g.Build x, g.Build y) | Var(p) -> g.Var(p) | Not(x) -> g.Not(g.Build x) | False -> g.False | Exists(v,p) -> failwith "Exists node" member g.Equiv p1 p2 = (g.Build(p1) = g.Build(p2)) You can now install a pretty-printer and inspect the BDDs for some simple formulae: > let bddBuilder = BddBuilder(compare);; val bddBuilder: BddBuilder > fsi.AddPrinter(fun bdd -> bddBuilder.ToString(bdd));; val it: unit = () > bddBuilder.Build(var "x");; val it : Bdd = (x => T | F) > bddBuilder.Build(var "x" &&& var "x");; val it : Bdd = (x => T | F) > bddBuilder.Build(var "x") = bddBuilder.Build(var "x" &&& var "x");; val it : bool = true > (var "x") = (var "x" &&& var "x");; val it : bool = false > bddBuilder.Build(var "x" &&& var "y");; val it : Bdd = (x => (y => T | F) | F) > bddBuilder.Equiv (var "x") (var "x" &&& var "x");; val it : bool = true Note that the BDD representations of x and x AND x are identical, while the Prop representations are not. The Prop representation is an abstract syntax representation, while the BDD representation is more of a semantic or computational representation. The BDD representation incorporates all the logic necessary to prove propositional formula equivalent; in other words, this logic is built into the representation itself.

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Why not simply create a relational parent and child table instead The data stored in tables containing nested table columns may not be accessible to many third-party tools that don t understand the syntax to retrieve and store data in them Your application may become less open and accessible from third-party tools that understand only the more standard relational SQL DMLs can run substantially slower and scale much less when you use nested tables rather than relational tables We will do a comparison of different alternatives later in section Object Views vs Relational Tables vs Nested Tables to establish this fact Most of the preceding arguments hold true in principle when you use varrays in tables as well.

Figure 10-6. A dirty read from the database The question is: How should the updating process deal with this situation How can the updating process even learn of it And once it does, what should be done These questions must always be asked when inventing an application s architecture. How they are solved depends on the business requirements of the application, and the amount of control the development team has over the database whose data is being consumed. The problem is much different when it s a custom database than it is when the database is vendor supplied and changes cannot be made to it. These are the factors that determine which of the following solutions will be employed.

You can now use BDDs to perform circuit verification. For example, the following verifies that you can swap the x and y inputs to an 8-bit adder:

In addition, varrays are even less flexible when it comes to updating them, since they don t allow selective updates; you have to replace the entire varray even if you want to change only one value in the varray Besides, you can t create indexes on individual varray object columns Thus, you can conclude that using nested tables or varrays as a mechanism to persist your data isn t really a good idea The question becomes, is there a middle ground In other words, can you have the flexibility of storing data in relational tables, and yet access them as objects when needed Object views, covered in the next section, may provide you with one such mechanism..

> bddBuilderEquiv (nBitCarryRippleAdder 8 (vec 8 "x") (vec 8 "y") (vec 8 "sum") (vec 9 "carry")) (nBitCarryRippleAdder 8 (vec 8 "y") (vec 8 "x") (vec 8 "sum") (vec 9 "carry"));; val it : bool = true Thirty-three variables are involved in this circuit A naive exploration of this space would involve searching a truth table of more than eight billion entries The BDD implementation takes moments on any modern computer Efficient symbolic representations pay off! A more substantial verification problem involves checking the equivalence of circuits that have substantial structural differences To explore this, let s take a different implementation of addition called a carry select adder.

Summary

Object views allow you to access and manipulate relational data as if the data were stored in tables containing object type or collection columns. Object views give you the flexibility to store data in relational tables. Thus you can selectively choose to use object features when it makes sense (e.g., to retrieve data directly as objects for your Java applications). For the most part, you can use relational SQL directly on the underlying relational schema, thus avoiding the code complexity and performance problems associated with storing data in tables containing objects. The process of creating object views on relational tables involves the following steps: 1. Define an object type. In this step, we define an object type where each attribute corresponds to a relational table column. This object type will be used in creating the object view.