## Archive for the ‘**Mathematica**’ Category

## Scala Expressiveness

At its heart, Mathematica has deep support for functional programming. As a computer algebra system, the functional programming support in Mathematica is unmatched in similar products, specifically, when it comes to pattern matching and rule based programming.

This post is a walk through the differences and similarities between the approaches that can be taken to solve a problem using both Scala and Mathematica, with a focus on the expressiveness and fluency of the solution code.

### Quick Sort

To get into the context, the following is an implementation of the Quick Sort algorithm in Mathematica.

The code demonstrates pattern matching, pattern guards and rule based programming. Cases is the partioning function, while a rule is used to capture the resulting partitions tuple. The *qsort* function itself is a pattern based function similar to Haskell and Alice style of function definitions.

The Scala implementation uses the partition method to split the list around the pivot *p*

def qsort[T <% Ordered[T]](list: List[T]): List[T] = list match { case Nil => Nil case p :: xs => val (lesser, greater) = xs partition (_ <= p) qsort(lesser) ++ List(p) ++ qsort(greater) }

Another Scala version could have been implemented using list comprehension, which is much similar to the Mathematica one:

def qsort2[T <% Ordered[T]](list: List[T]): List[T] = list match { case Nil => Nil case p :: xs => qsort2(for(x <- xs if x <= p) yield x) ++ List(p) ++ qsort2(for(x <- xs if x > p) yield x) }

### Power Set

A Power set is one of *Neat Examples* that demonstrates the power of Mathematica’s *Fold* higher-order function.

The Scala implementation as almost as expressive, in fact, you might find it more readable and easier to understand.

def powerSet[A](s: Set[A]) = s.foldLeft(Set(Set.empty[A])) { (set, element) => set union (set map (_ + element)) }

Here is a example of this implementation in action:

val ps = powerSet(Set('α', 'β', 'γ')) println(ps.asString)

Which will result into

[[],[α],[α,γ],[γ],[α,β,γ],[β],[β,γ],[α,β]]

### Mathematica Cases

The Cases function in Mathematica can be used to perform filtering based on pattern matching, for example:

is used to select those expressions in the form of base and exponent, only when the exponent is equal to the constant 2 regardless of the base. Because Mathematica is a symbolic language, the mathematical expressions are first class constructs, and Mathematica is designed to perform operations on such expressions. This is obviously not what we are trying to compare, so we need to define the ADT Power:

case class Power(val base: Symbol, val exp: Int)

In this case, the Scala equivalent is pretty straight forward,

val l = Power('a, 2) :: Power('b, 3) :: Power('c, 4) :: Nil val cases0 = l flatMap { case p@Power(_, 2) => Some(p) case _ => None }

Taking this a little bit further, the following code tries to match any expression provided that the exponent is even, notice the */;* before the predicate *EvenQ*

Still, pretty simple in Scala:

l flatMap { case p@Power(_, n) if n % 2 == 0 => Some(p) case _ => None }

One interesting feature of the Cases function, is that it allows direct transformation of matched elements, for example:

Here, the pattern matches terms with odd exponent, and transforms them into even ones by adding 1 to the exponent. Again, the Scala equivalent is not that hard to implement or less expressive:

l flatMap { case p@Power(x, n) if n % 2 != 0 => Some(Power(x, n + 1)) case _ => None }

### NestWhileList

NestWhileList is one of the powerful and commonly used higher-order function in Mathematica.

Here is a simple example of the function:

Starting with an intial value 32, NestWhileList nests the application of the actual argument pure function, reporting the result of each application as a list, and terminating when the argument predicate returns *true*.

This is basically what a *Stream* in Scala is designed to do, but without the termination, so, a simple implementation of NestWhileList in Scala can look like the following:

def nestWhileList[A](f: A => A, initial: A, p: A => Boolean) = { val result = Stream.iterate(initial)(f).takeWhile(p).toList result :+ f(result.last) }

An equivalent call to the Mathematica call might look like this:

nestWhileList((_: Int) / 2, 32, (_: Int) != 1)

Now consider the definition of the this function:

The function checks if a list is palindrome. An algorithm known as Algorithm 196 that generates palindromes can be defined by:

Which can be expressed in Scala as:

val alg196 = (n: Int) => n + n.toString.reverse.toInt @tailrec def isPalindrome[A <% Ordered[A]](a: List[A]): Boolean = a match { case Nil | _ :: Nil => true case x :: xs => x == xs.last && isPalindrome(xs.init) }

To generate a list of palindromes in Mathematica, one could write:

Which can similarly be done in Scala as:

nestWhileList(alg196, 77, (_: Int) < 10000000). filter (n => isPalindrome(n.toString.toList))

More examples on the usage of NextWhileList follows.

### Pascal Triangle

In Mathematica, the Pascal Triangle can be defined by:

But let’s define it in the naive way, assuming we have no idea what a binomial is, notice what the combination of the *tuples* and *NestWhileList* functions will result into:

Which forms a model for a nice manipulation

The implementation in Scala is almost identical, concerning both our *tuples* and *pascalTriangle* functions:

def pascalTriangle(n: Int) = { def tuple(l: List[Int]): List[Int] = l match { case Nil | _ :: Nil => Nil case x :: y :: ys => (x + y) :: tuple(y :: ys) } nestWhileList[List[Int]](l => (1 :: tuple(l)) :+ 1, List(1), _.length < n) }

### 3N+1 Problem

There are several ways to play with the Collatz conjecture in Mathematica, the following is the definition in terms of a piecewise function:

Again, the Scala implementation is as expressive:

def collatz(n: Int) = n match { case n if n % 2 == 0 => n / 2 case n => (3 * n) + 1 }

val cs = nestWhileList(collatz(_: Int), 200, (_: Int) != 1) println(cs.asString)

[200,100,50,25,76,38,19,58,29,88,44,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1]

There is a nice thread here if you would like to see different solutions to this problem in several programming languages.

### Bubble Sort

Orginal code of this bubble sort implementation is posted here

The problem with this Mathematica implementation is that Scala cannot do pattern matching involving arbitrary length prefix, to do this we need to define our version of suffix match method:

*The suffixMatch method*

@tailrec def suffixMatch[A, B](l: List[A], prefix: List[A] = Nil) (f: PartialFunction[List[A], B]): Option[(List[A], B)] = l match { case Nil => None case xs if f isDefinedAt xs => Some((prefix, f(xs))) case x :: xs => suffixMatch(xs, prefix :+ x)(f) }

The method tries to match the suffix of a list by recursively invoking an argument matching function, dropping the prefix on each iteration. If the function succeeds, the *suffixMatch* method returns both the dropped prefix and the matched suffix.

We can then define a nice version of this function, to be used more naturaly as the match keyword:

implicit def matchOps[A](l: List[A]) = new { def smatch[B](f: PartialFunction[List[A], B], prefix: List[A] = Nil) = suffixMatch(l, prefix)(f) }

*Bubble Sort in Scala*

A typical translation of the Mathematica version can then be expressed in Scala this way:

@tailrec def bubbleSort(l: List[Int]): List[Int] = { l smatch { case x :: y :: ys if x > y => y :: x :: ys } match { case None => l case Some((p, s)) => bubbleSort(p ++ s) } }

### Run Length Encoding

An amazing, famous run length encoding implementation in Mathematica is:

Using our new *smatch* method, the implementation can be ported to Scala, almost as is:

@tailrec def runLengthEncoding[A](l: List[(A, Int)]): List[(A, Int)] = { l smatch { case (x1, n) :: (x2, m) :: ys if x1 == x2 => (x1, (m + n)) :: ys } match { case None => l case Some((p, s)) => runLengthEncoding(p ++ s) } }

### Conclusion

Scala is an amazingly expressive and, more importantly, beautiful language. The design and implementation of the core idea of expanding the library not the language, makes Scala a extremely powerful and expressive tool for modeling solutions to simple and complex problems alike.

## Mathematica Conference, Egypt 2010

Here are the slides of my recent talk at Mathematica Conference, Egypt 2010

The talk is an introduction to Functional Programming in Mathematica

Please leave a comment if you’re interested in the source notebook

## Mathematica Algebraic Data Types

I have been doing lots of Mathematica lately, but because of my statically structured inflexible mind, I needed ADTs and static typing even in Mathematica. To check if this is possible, let’s test a very simple algorithm implementation. The problem is to list all paths of a rooted tree starting at the root and ending with the leafs. This can be solved with a basic depth first traversal implementation, so I started by modeling the solution in Haskell.

Our three basic ADTs are:

- Vertex, to model the graph nodes
- Forest to model leafs and branches
- Graph to model the graph itself

Here is the Haskell implementation, straight forward as always:

data Vertex a = Vertex a data Forest a = Leaf (Vertex a) | Branch (Vertex a) [Forest a] data Graph a = Graph [ ( Vertex a, [Vertex a] ) ]

On the Mathematica side, a Vertex can be modeled as:

VertexT will hold as the Vertex Haskell equivalent, but will never get defined in Mathematica terms, this is basically a hack to define an ADT with the name VertexT, but it turns out to be very useful. For the definition of the Graph type in Mathematica, VertexT can be used in the constructor definition as:

So that defines a list of tuples, with the first element being a vertex and the second is a list of vertices.

Both Leaf and Branch can be implemented the same way:

The Leaf:

And the Branch:

The Haskell branch function is:

branch :: Vertex a -> [Forest a] -> Forest a branch v [] = Leaf v branch v fs = Branch v fs

Yes, Haskell shines so far, but let’s go on with our implementation. We need a function to identify the successor of a vertex, as in Haskell:

scc :: Eq a => Graph a -> Vertex a -> [Vertex a] scc (Graph rep) v = case ( map(snd) . filter ((v==) . fst) $ rep ) of [] -> [] (x:_) -> x

While in Mathematica:

Not too bad, our hack is starting to pay off, it’s time to walk the tree:

dft :: Eq a => Graph a -> Vertex a -> Forest a dft g v = branch v . map (dft g) $ scc g v

And in Mathematica, it is strikingly similar:

We will need a concatMap version for Mathematica later on, let’s implement one:

We can know build the tree chains or paths:

chains :: Forest a -> [[Vertex a]] chains (Leaf v) = [[v]] chains (Branch v fs) = concatMap (map (v:) . chains $) fs

With a similar Mathematica implementation:

Testing the Haskell code with a simple example:

*Main> let g = Graph [ (Vertex 'r', [Vertex 'a', Vertex 'b', Vertex 'c']) ] *Main> let b = dft g (Vertex 'r') *Main> chains b [[Vertex 'r',Vertex 'a'],[Vertex 'r',Vertex 'b'],[Vertex 'r',Vertex 'c']]

Before we can work with the Mathematica code, we need a few functions to convert our graph model to Mathematica’s own model:

And a nice rendering function:

So, let’s create an instance of our graph model:

And plot it:

Let’s see if the whole thing works:

How about a nice plot of the chains:

Now we have at least basic ADT support in Mathematica. 🙂