Tacit programming

Tacit programming,also calledpoint-free style,is aprogramming paradigmin which function definitions do not identify thearguments(or "points" ) on which they operate. Instead the definitions merelycomposeother functions, among which arecombinatorsthat manipulate the arguments. Tacit programming is of theoretical interest, because the strict use of composition results in programs that are well adapted forequationalreasoning.It is also the natural style of certainprogramming languages,includingAPLand its derivatives,andconcatenative languagessuch asForth.The lack of argument naming gives point-free style a reputation of being unnecessarily obscure, hence the epithet "pointless style".

Unixscriptinguses the paradigm withpipes.

Examples

Python

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

defexample(x):
returnbaz(bar(foo(x)))

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

fromfunctoolsimportpartial,reduce
defcompose(*fns):
returnpartial(reduce,lambdav,fn:fn(v),fns)

example=compose(foo,bar,baz)

For a more complex example, the Haskell codep = ((.) f). gcan be translated as:

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

Functional programming

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

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

However, using afoldwe can replace this with:

sumxs=foldr(+)0xs

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

sum=foldr(+)0

which is point-free.

Another example usesfunction composition:

pxyz=f(gxy)z

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

p=\x->\y->\z->f(gxy)z
=\x->\y->f(gxy)
=\x->\y->(f.(gx))y
=\x->f.(gx)
(*Heretheinfixcomposeoperator"."isusedasacurriedfunction.*)
=\x->((.)f)(gx)
=\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

mfcriteriaoperatorlist=filtercriteria(mapoperatorlist)

It can be expressed point-freeas

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.

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

APL family

InJ,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$e^{ix}=\cos x+i\sin x,$expressed tacitly:

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

(j.is a primitive function whose monadic definition is0j1times x and whose dyadic definition isx+0j1×y.) The same tacit computations expressed inDyalog 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¯1123
1111
⍝ Yes, so far so good!

Stack-based

Instack-oriented programming languages(andconcatenative ones,most of which are stack based), point-free methods are commonly used. For example, a procedure to compute theFibonacci numbersmight look like the following inPostScript:

/fib
{
dupdup1eqexch0eqornot
{
dup1subfib
exch2subfib
add
}if
}def

Pipelines

Unix pipeline

Main article:Pipeline (Unix)

In Unix scripting the functions are computer programs which receive data fromstandard inputand send the results tostandard 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 extrafile descriptorscan be opened fromnamed pipes,this no longer constitutes a point-free style.

jq

jqis 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.

Fibonacci sequence

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 Alpha betic 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;

Composition of unary functions

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

defexample(x):
returnbaz(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;

See also

• Combinatory logic
• Concatenative programming language
• Function-level programming
• Joy (programming language),modern highly tacit language

External links

• From Function-Level Programming to Pointfree Style
• Pure Functions in APL and JHow to use tacit programming in any APL-like language
• Closed applicative languages 1971 - 1976 ff,in John W. Backus (Publications)