Tacit programming

Tacit programming, also called point-free style, is a programming paradigm in which function definitions do not identify the arguments (or "points") on which they operate. Instead the definitions merely compose other functions, among which are combinators that manipulate the arguments. Tacit programming is of theoretical interest, because the strict use of composition results in programs that are well adapted for equational reasoning.[1] It is also the natural style of certain programming languages, including APL and its derivatives,[2] and concatenative languages such as Forth. The lack of argument naming gives point-free style a reputation of being unnecessarily obscure, hence the epithet "pointless style."[1]

UNIX scripting uses the paradigm with pipes.

For example, a sequence of operations in an applicative language like the following:

def example(x):
  y = foo(x)
  z = bar(y)
  w = baz(z)
  return w

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

def flow(fns):
    def reducer(v, fn):
        return fn(v)

    return functools.partial(functools.reduce, reducer, fns)

example = flow([baz, bar, foo])

The key idea in tacit programming is to assist in operating at the appropriate level of abstraction. That is, to translate the natural transformation given by currying

into computer functions, where the left represents the uncurried form of a function and the right the curried. CA denotes the functionals from A to C (see also exponential object), while A × B denotes the Cartesian product of A and B.

Examples

Functional programming

A simple example (in Haskell) is a program which takes a sum of a list. A programmer might define a sum recursively using a pointed (cf. value-level programming) method as:

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

However, by noting this as a fold the programmer could replace this with:

sum xs = foldr (+) 0 xs

And then the argument is not needed, so this can be replaced with

sum = foldr (+) 0

which is point-free.

Another example uses the dot operator:

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)
  = \x -> ((.) f) (g x)
  = ((.) f) . g

so 

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]

APL family

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 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 ○ ⊢
j   ← {⍺←0 ⋄ ⍺+0j1×⍵}  ⍝ this part is not tacit
Euler ← *∘j = cos j sin

Stack-based

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:

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

UNIX pipeline

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 '|' is the composition operator.

See also

References

  1. 1 2 Manuel Alcino Cunha (2005) Point-free Program Calculation
  2. ↑ W. Neville Holmes, ed. (2006) Computers and People
  3. ↑ "Name code not values". Concatenative.org. Retrieved 13 September 2013.
  4. ↑ pipermail
  5. ↑ "Pointfree - HaskellWiki". wiki.haskell.org. Retrieved 2016-06-05.
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