Tacit programming

Tacit programming can be illustrated with the following Python code. A sequence of operations such as the following:

def example(x):
    return baz(bar(foo(x)))

... can be written in point-free style as the composition of a sequence of functions, without parameters:[3]

from functools import partial, reduce
def compose(*fns):
    return partial(reduce, lambda v, fn: fn(v), fns)

example = compose(foo, bar, baz)

For a more complex example, the Haskell code p = ((.) f) . g can be translated as:

p = partial(compose, partial(compose, f), g)

A simple example (in Haskell) is a program which computes the sum of a list of numbers. We can define the sum function recursively using a pointed style (cf. value-level programming) as:

sum (x:xs) = x + sum xs
sum [] = 0

However, using a fold we can replace this with:

sum xs = foldr (+) 0 xs

And then the argument is not needed, so this simplifies to

sum = foldr (+) 0

which is point-free.

Another example uses function composition:

p x y z = f (g x y) z

The following Haskell-like pseudo-code exposes how to reduce a function definition to its point-free equivalent:

p = \x -> \y -> \z -> f (g x y) z
  = \x -> \y -> f (g x y)
  = \x -> \y -> (f . (g x)) y
  = \x -> f . (g x)
  (* Here the infix compose operator "." is used as a curried function. *)
  = \x -> ((.) f) (g x)
  = \x -> (((.) f) . g) x

p = ((.) f) . g

Finally, to see a complex example imagine a map filter program which takes a list, applies a function to it, and then filters the elements based on a criterion

mf criteria operator list = filter criteria (map operator list)

It can be expressed point-free[4] as

mf = (. map) . (.) . filter

Note that, as stated previously, the points in 'point-free' refer to the arguments, not to the use of dots; a common misconception.[5]

A few programs have been written to automatically convert a Haskell expression to a point-free form.

In J, the same sort of point-free code occurs in a function made to compute the average of a list (array) of numbers:

avg=: +/ % #

+/ sums the items of the array by mapping (/) summation (+) to the array. % divides the sum by the number of elements (#) in the array.

Euler's formula ${\displaystyle e^{ix}=\cos x+i\sin x,}$ expressed tacitly:

cos =: 2 o. ]
sin =: 1 o. ]
Euler =: ^@j. = cos j. sin

(j. is a primitive function whose monadic definition is 0j1 times x and whose dyadic definition is x+0j1×y.) The same tacit computations expressed in Dyalog APL:

avg ← +⌿ ÷ ≢

cos ← 2 ○ ⊢
sin ← 1 ○ ⊢
EulerCalc←   cos + 0j1 × sin   ⍝ 0j1 is what's usually written as i 
EulerDirect← *0J1×⊢            ⍝ Same as ¯12○⊢ 
⍝ Do the 2 methods produce the same result? 
EulerCheck← EulerDirect=EulerCalc
EulerCheck ¯1 1 2 3     
1 1 1 1   
⍝ Yes, so far so good!

In stack-oriented programming languages (and concatenative ones, most of which are stack based), point-free methods are commonly used. For example, a procedure to compute the Fibonacci numbers might look like the following in PostScript:

/fib
{
   dup dup 1 eq exch 0 eq or not
   {
      dup 1 sub fib
      exch 2 sub fib
      add
   } if
} def

In Unix scripting the functions are computer programs which receive data from standard input and send the results to standard output. For example,

sort | uniq -c | sort -rn

is a tacit or point-free composition which returns the counts of its arguments and the arguments, in the order of decreasing counts. The 'sort' and 'uniq' are the functions, the '-c' and '-rn' control the functions, but the arguments are not mentioned. The pipe '|' is the composition operator.

Due to the way pipelines work, it is only normally possible to pass one "argument" at a time in the form of a pair of standard input/output stream. Although extra file descriptors can be opened from named pipes, this no longer constitutes a point-free style.

jq is a JSON-oriented programming language in which the '|' symbol is used to connect filters to form a pipeline in a familiar way. For example:

   [1,2] | add

evaluates to 3. (Yes, the JSON array is a jq filter that evaluates to an array.)

Although similar to Unix pipelines, jq pipelines allow the incoming data to be sent to more than one recipient on the RHS of the '|' as though in parallel. For example, the program `add/length` will compute the average of the numbers in an array, so that:

   [1,2] | add/length

evaluates to 1.5

Similarly:

   [1,2] | [length, add, add/length]

evaluates to [2,3,1.5]

A dot ('.') can be used to define an attachment point on the RHS, e.g.:

   1 | [., .]

evaluates to [1,1]

and similarly:

   2 | pow(.; .)

evaluates to 4 since pow(x;y) is x to the power y.

A tacit jq program for generating the Fibonacci sequence would be:

   [0,1] | recurse( [last, add] ) | first

Here, [0,1] is the initial pair to be taken as the first two items in the Fibonacci sequence. (The pair [1,1] could likewise be used for the variant definition.)

The alphabetic tokens are built-in filters: `first` and `last` emit the first and last elements of their input arrays respectively; and `recurse(f)` applies a filter, f, to its input recursively.

jq also allows new filters to be defined in a tacit style, e.g.:

   def fib: [0,1] | recurse( [last, add] ) | first;

In the section on Python in this article, the following Python definition is considered:

def example(x):
    return baz(bar(foo(x)))

In point-free style, this could be written in Python as:

example = compose(foo, bar, baz)

In jq, the equivalent point-free definition would be:

   def example: foo | bar | baz;