Recursion is a common pattern for solving certain problems in most programming languages, and Hoon is no exception. One of the classically recursive problems is that of factorial. The factorial of n is the product of all positive integers less than or equal to n. Thus the factorial of 5, denoted as 5!, would be:

5 * 4 * 3 * 2 * 1 = 120

Factorial In Hoon

Let's implement a factorial calculator in Hoon. To do so, we'll write a recursive gate (Hoon's equivalent of a function) to perform the relevant computation. Save the code below as factorial.hoon in the /gen directory of the %base desk on your ship.

|=  n=@ud
?:  =(n 1)
  1
(mul n $(n (dec n)))

Our gate takes one sample (argument) n that must nest inside @ud, the unsigned-integer type.

Next we check to see if n is 1. If so, the result is just 1, since 1 * 1 = 1.

If, however, n is not 1, then we branch to the final line of the code, (mul n $(n (dec n))), where the recursion logic lives. Here, we multiply n by the recursion of n minus 1. $ initiates recursion: it calls the gate that we're already in, but replaces its sample.

In our example, we multiply n with the product of this entire gate all over again with $(n (dec n)), except when that new gate has its sample decremented by one. This works recursively because each new gate will, of course, itself contain the code to call a further-decremented gate. Gates will continue to call new, further-decremented gates until n is 1, and that 1 will be the final number to be multiplied by.

Let's run the program in the Dojo:

> +factorial 5
120

It may help to visualize the operation of our example gate. The pseudo-Hoon below illustrates what happens when we use it to find the factorial of 5:

(factorial 5)
(mul 5 (factorial 4))
(mul 5 (mul 4 (factorial 3)))
(mul 5 (mul 4 (mul 3 (factorial 2))))
(mul 5 (mul 4 (mul 3 (mul 2 (factorial 1)))))
(mul 5 (mul 4 (mul 3 (mul 2 1))))
(mul 5 (mul 4 (mul 3 2)))
(mul 5 (mul 4 6))
(mul 5 24)
120

It's easy to see how we're "floating" gate calls until we reach the final iteration of such calls that only produces a value. The mul n component of the gate leaves something like mul 5, waiting for the final series of terms to be operated upon. The $(n (dec n)) component is what expands out the expression, as illustrated by (factorial 4). Once the expression cannot be expanded out further, the operations work backwards, successively feeding values into the mul functions behind them.

Tail-Call Optimization

Our last example isn't a very efficient use of computing resources. The pyramid-shaped illustration approximates what's happening on the call stack, a memory structure that tracks the instructions of the program. In our example code, every time a parent gate calls another gate, the gate being called is "pushed" to the top of the stack in the form of a frame. This process continues until a value is produced instead of a function, completing the stack.

                  Push order      Pop order
(fifth frame)         ^               |
(fourth frame)        |               |
(third frame)         |               |
(second frame)        |               |
(first frame)         |               V

Once this stack of frames is completed, frames "pop" off the stack starting at the top. When a frame is popped, it executes the contained gate and passes produced data to the frame below it. This process continues until the stack is empty, giving us the gate's output.

When a program's final expression uses the stack in this way, it's considered to be not tail-recursive. This usually happens when the last line of executable code calls more than one gate, our example code's (mul n $(n (dec n))) being such a case. That's because such an expression needs to hold each iteration of $(n (dec n) in memory so that it can know what to run against the mul function every time.

To reiterate: if you have to manipulate the result of a recursion as the last expression of your gate, as we did in our example, the function is not tail-recursive, and therefore not very efficient with memory. A problem arises when we try to recurse more times than we have space on the stack. This will result in our computation failing and producing a stack overflow. If we tried to find the factorial of 5.000.000, for example, we would almost certainly run out of stack space.

But the Hoon compiler, like most compilers, is smart enough to notice when the last statement of a parent can reuse the same frame instead of needing to add new ones onto the stack. If we write our code properly, we can use a single frame that simply has its values replaced with each recursion.

A Tail-Recursive Gate

With a bit of refactoring, we can write a version of our factorial gate that is tail-recursive and can take advantage of this feature:

|=  n=@ud
=/  t=@ud  1
|-
?:  =(n 1)
    t
$(n (dec n), t (mul t n))

The above code should look familiar. We are still building a gate that takes one argument n. This time, however, we are also putting a face on a @ud and setting its initial value to 1. The |- here is used to create a new gate with one arm $ and immediately call it. Think of |- as the recursion point.

We then evaluate n to see if it is 1. If it is, we return the value of t. In case that n is anything other than 1, we perform our recursion:

$(n (dec n), t (mul t n))

All we are doing here is recursing our new gate and modifying the values of n and t. t is used as an accumulator variable that we use to keep a running total for the factorial computation.

Let's use pseudo-Hoon to illustrate how the stack is working in this example for the factorial of 5.

(factorial 5)
(|- 5 1)
(|- 4 5)
(|- 3 20)
(|- 2 60)
(|- 1 120)
120

We simply multiply t and n to produce the new value of t, and then decrement n before repeating. Since this $ call is the final and solitary thing that is run in the default case and since we are doing all computation before the call, this version is properly tail-recursive. We don't need to do anything to the result of the recursion except recurse it again. That means that each iteration can be replaced instead of held in memory.

A Note on $

$ (pronounced "buc") is, in its use with recursion, a reference to the gate that we are inside of. That's because a gate is just a core with a single arm named $. The subject is searched depth-first, head before tail, with faces skipped, and stopping on the first result. In other words, the first match found in the head will be returned. If you wished to refer to the outer $ in this context, the idiomatic way would be to use ^$. The ^ operator skips the first match of a name.

Exercises

1. Write a recursive gate that produces the first n Fibonacci numbers

2. Write a recursive gate that produces a list of moves to solve the Tower of Hanoi problem. Disks are stacked on a pole by decreasing order of size. Move all of the disks from one pole to another with a third pole as a spare, moving one disc at a time, without putting a larger disk on top of a smaller disk.