Subject: [xsl] The Holy Trinity of Functional Programming ... Is there a way to define recursive data types in XSLT 2.0? From: "Costello, Roger L." <costello@xxxxxxxxx> Date: Wed, 22 Aug 2012 09:41:37 +0000 
Hi Folks, Professor Richard Bird has written extensively on functional programming. In one of his books he has a fascinating discussion on three key aspects of functional programming, which he calls the holy trinity of functional programming. The first key aspect is: Userdefined recursive data types He gives an example (Haskell notation): data Nat = Zero  Succ Nat The elements of this data type include: Zero, Succ Zero, Succ (Succ Zero), Succ (Succ (Succ Zero)), ... Is there a way in XSLT 2.0 to define recursive data types? If yes, how would the Nat data type be defined in XSLT 2.0? /Roger P.S. For those interested, below is my summary of Bird's discussion on the holy trinity of functional programming.  The Holy Trinity of Functional Programming  These three ideas constitute the holy trinity of functional programming: 1. Userdefined recursive data types. 2. Recursively defined functions over recursive data types. 3. Proof by induction: show that some property P(n) holds for each element of a recursive data type. Here is an example of a userdefined recursive data type. It is a declaration for the natural numbers 0, 1, 2, ...: data Nat = Zero  Succ Nat The elements of this data type include: Zero, Succ Zero, Succ (Succ Zero), Succ (Succ (Succ Zero)), ... To understand this, when creating a Nat value we have a choice of either Zero or Succ Nat. Suppose we choose Succ Nat. Well, now we must choose a value for the Nat in Succ Nat. Again, we have a choice of either Zero or Succ Nat. Suppose this time we choose Zero, to obtain Succ Zero. The ordering of the elements in the Nat data type can be specified by defining Nat to be a member of the Ord class: instance Ord Nat where Zero < Zero = False Zero < Succ n = True Succ m < Zero = False Succ m < Succ n = (m < n) Here is how the Nat version of the expression 2 < 3 is evaluated: Succ (Succ Zero) < Succ (Succ (Succ Zero))  Nat version of 2 < 3 = Succ Zero < Succ (Succ Zero)  by the 4th equation defining order = Zero < Succ Zero  by the 4th equation defining order = True  by the 2nd equation defining order Here is a recursively defined function over the data type; it adds two Nat elements: (+) :: Nat > Nat > Nat m + Zero = m m + Succ n = Succ(m + n) Here is how the Nat version of 0 + 1 is evaluated: Zero + Succ Zero  Nat version of 0 + 1 = Succ (Zero + Zero)  by the 2nd equation defining + = Succ Zero  by the 1st equation defining + Here is another recursively defined function over the data type; it subtracts two Nat elements: () :: Nat > Nat > Nat m  Zero = m Succ m  Succ n = m  n If the Nat version of 0  1 is executed, then the result is undefined: Zero  Succ Zero The "undefined value" is denoted by this symbol: __ (also known as "bottom") Important: __ is an element of *every* data type. So we must expand the list of elements in Nat: __, Zero, Succ Zero, Succ (Succ Zero), Succ (Succ (Succ Zero)), ... There are still more elements of Nat. Suppose we define a function that returns a Nat. Let's call the function undefined: undefined :: Nat undefined = undefined It is an infinitely recursive function: when invoked it never stops until the program is interrupted. This function undefined is denoted by the symbol __ Recall how we defined the ordering of values in Nat: instance Ord Nat where Zero < Zero = False Zero < Succ n = True Succ m < Zero = False Succ m < Succ n = (m < n)
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