16. Functional Programming

This module will discuss some gates-that-work-on-gates and other assorted operators that are commonly recognized as functional programming tools.

Given a gate, you can manipulate it to accept a different number of values than its sample formally requires, or otherwise modify its behavior. These techniques mirror some of the common tasks used in other functional programming languages like Haskell, Clojure, and OCaml.

Functional programming, as a paradigm, tends to prefer rather mathematical expressions with explicit modification of function behavior. It works as a formal system of symbolic expressions manipulated according to given rules and properties. FP was derived from the lambda calculus, a cousin of combinator calculi like Nock. (See also APL.)

Changing Arity

If a gate accepts only two values in its sample, for instance, you can chain together multiple calls automatically using the ;: miccol rune.

> (add 3 (add 4 5))
12

> :(add 3 4 5)
12

> (mul 3 (mul 4 5))
60

> :(mul 3 4 5)
60

This is called changing the arity of the gate. (Does this work on +mul:rs?)

Binding the Sample

"Currying" describes taking a function of multiple arguments and reducing it to a set of functions that each take only one argument. "Binding", an allied process, is used to set the value of some of those arguments permanently.

If you have a gate which accepts multiple values in the sample, you can fix one of these. To fix the head of the sample (the first argument), use +cury; to bind the tail, use +curr.

Consider calculating a x² + b x + c, a situation we earlier resolved using a door. We can resolve the situation differently using currying:

> =full |=([x=@ud a=@ud b=@ud c=@ud] (add (add (mul (mul x x) a) (mul x b)) c))

> (full 5 4 3 2)
117

> =one (curr full [4 3 2])  

> (one 5)  
117

One can also +cork a gate, or arrange it such that it applies to the result of the next gate. This pairs well with ;: miccol. (There is also +corl, which composes backwards rather than forwards.) This example decrements a value then converts it to @ux by corking two gates:

> ((cork dec @ux) 20)  
0x13

Exercise: Bind Gate Arguments

Create a gate +inc which increments a value in one step, analogous to +dec.

Exercise: Chain Gate Values

Write an expression which yields the parent galaxy of a planet's sponsoring star by composing two gates.

Working Across +lists

The +turn function takes a list and a gate, and returns a list of the products of applying each item of the input list to the gate. For example, to add 1 to each item in a list of atoms:

> (turn `(list @)`~[11 22 33 44] |=(a=@ +(a)))
~[12 23 34 45]

Or to double each item in a list of atoms:

> (turn `(list @)`~[11 22 33 44] |=(a=@ (mul 2 a)))
~[22 44 66 88]

+turn is Hoon's version of Haskell's +map.

We can rewrite the Caesar cipher program using turn:

|=  [a=@ b=tape]
^-  tape
?:  (gth a 25)
  $(a (sub a 26))
%+  turn  b
|=  c=@tD
?:  &((gte c 'A') (lte c 'Z'))
  =.  c  (add c a)
  ?.  (gth c 'Z')  c
  (sub c 26)
?:  &((gte c 'a') (lte c 'z'))
  =.  c  (add c a)
  ?.  (gth c 'z')  c
  (sub c 26)
c

+roll and +reel are used to left-fold and right-fold a list, respectively. To fold a list is similar to +turn, except that instead of yielding a +list with the values having had each applied, +roll and +reel produce an accumulated value.

> (roll `(list @)`[1 2 3 4 5 ~] add)
15

> (reel `(list @)`[1 2 3 4 5 ~] mul)
120

Exercise: Calculate a Factorial

Use +reel to produce a gate which calculates the factorial of a number.

Classic Operations

Functional programmers frequently rely on three design patterns to produce operations on collections of data:

1. Map. The Map operation describes applying a function to each item of a set or iterable object, resulting in the same final number of items transformed. In Hoon terms, we would say slamming a gate on each member of a +list or +set. The standard library arms that accomplish this include +turn for a list, +run:in for a +set, and +run:by for a +map.

2. Reduce. The Reduce operation applies a function as a sequence of pairwise operations to each item, resulting in one summary value. The standard library arms that accomplish this are +roll and +reel for a list, +rep:in for a +set, and +rep:by for a +map.

3. Filter. The Filter operation applies a true/false function to each member of a collection, resulting in some number of items equal to or fewer than the size of the original set. In Hoon, the library arms that carry this out include +skim, +skid, +murn for a list, and +rib:by for a +map.