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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.

One common view of good UI design holds that the user should be free to enter arbitrary text, and it is up to the application to make sense of the entry. Nowhere is this truer than with date values, and the SmartDate type is designed to simplify how a business developer uses dates and exposes them to the UI. Examples of free-form date entry are easy to find. Just look at widely used applications like Microsoft Money or Intuit s Quicken. In these applications, users are free to enter dates in whatever format is easiest for them. Additionally, various shortcuts are supported; for example, the + character means tomorrow, while means yesterday.

10. Append the definition title to the definition list. 11. Append the definition description to the definition list. 12. Append the definition list to the body of the document. Let s write a function to do this.

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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:

Most users find this approach more appealing than being forced to enter a date in a strict format through a masked edit control, or having to always use the mouse to use a graphical calendar control. Of course, being able to additionally support a calendar control is also a great UI design choice. Date handling is also quite challenging because the standard DateTime data type doesn t have any comprehension of an empty or blank date. Many applications have date values that may be empty for a time and are filled in later. Consider a sales order, in which the shipment date is unknown when the order is entered. That date should remain blank or empty until an actual date is known. Without having the concept of an empty date, an application will require the user to enter an invalid placeholder date until the real date is known; and that s just poor application design.

In the early 1990s, I worked at a company where all far-future dates were entered as 12/31/99. Guess how much trouble the company had around Y2K, when all of its never-to-be-delivered orders started coming due!

> 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.

It is true that the Nullable<T> type can be applied to a DateTime value like this: Nullable<DateTime>. This allows a date to be empty in a limited sense. Unfortunately, that isn t enough for many applications, since an actual date value can t be meaningfully compared to a null value. Is the null value greater than or less than a given date With Nullable<T>, the answer is an exception, which is not a very useful answer. Additionally, data binding doesn t deal well with null values, and so exposing a null value from a business object s property often complicates the UI code.

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