4.12.1.4–4.12.1.5 - Partial application and function composition (C#)

One-, two- and three-argument types

A multi-argument calculation can be represented as a function that receives one argument and returns a function waiting for the next.

A one-argument function such as $\operatorname{square}:\mathrm{integer}\to\mathrm{integer}$ returns a value when fully applied: $\operatorname{square}\ 5=25$. Before application it is a function value; after its only required argument has been supplied, no argument remains.

Ordinary two-argument addition has type $\mathrm{integer}\times\mathrm{integer}\to\mathrm{integer}$. In the successive-argument representation it instead has type

Applying add to 4 returns an integer-to-integer function. Applying that result to 7 gives 11. Brackets in the type associate to the right, so $\mathrm{integer}\to\mathrm{integer}\to\mathrm{integer}$ is the same type. Application associates to the left: $\operatorname{add}\ 4\ 7$ means $(\operatorname{add}\ 4)\ 7$. The arrows describe returned values, not repeated assignments.

A three-argument total can be represented as

After one argument the remaining type is $\mathrm{integer}\to\mathrm{integer}\to\mathrm{integer}$; after two it is $\mathrm{integer}\to\mathrm{integer}$; after three it is an integer result. For $\operatorname{total}\ 2\ 3\ 8$, the stages are a function retaining 2, then a function retaining 2 and 3, then the value 13. A product-domain function and a successive-argument function implement corresponding calculations, but they are not identical types.

Partial application and returned functions

Partial application supplies fewer than all required arguments and produces a function for the arguments still missing.

For $\operatorname{add}:\mathrm{integer}\to\mathrm{integer}\to\mathrm{integer}$, $\operatorname{add}\ 4$ is a function that adds 4, not the value 4. This makes it reusable: $(\operatorname{add}\ 4)\ 6=10$, while $(\operatorname{add}\ 4)\ 9=13$. A one-argument function whose result is an ordinary value has no non-empty proper subset of arguments to fix: supplying its one argument is full application. A one-argument function can, however, itself return a function, as MakeShift did earlier.

C# does not automatically partially apply an ordinary two-parameter method when one parameter is omitted. Explicit nested functions implement the successive representation: