Program (mathematical object)

A program is a mathematical space suggested by John Backus. He wished to turn computer programming into a formal science by strictly defining it.

Informally, a mathematical space S is a set S_obj of the objects themselves and a set S_opr of operations on S_obj which map (S_obj)^{n into S_obj and are interrelated by algebraic laws.}

Backus argued that if computer programming is to be a science (let alone an engineering discipline) and less of an art, it must become a truly mathematical discipline—and to become a mathematical discipline, he claimed, it is important to find a way to make the space of programs a mathematical space with respect to program-forming operations (functionals) over that space, as distinguished from the space of values and the value-forming operations over it (functions).

The most useful laws are those that display some kind of symmetry, such as the distributive law that relates two operations O_A and O_B in a manner which O_A, combining values formed by O_B, is expressed as O_B combining values formed by O_A. Other, somewhat less useful, laws may display a symmetry that relates more than two operations.

The more and more useful laws present on S_opr, the "stronger" is their algebraic structure and the mathematical structure of the space S.

See also

• Function-level programming
• Value-level programming (contrast)
• FP programming language

External links

• J. Backus, Function level programs as mathematical objects

es:Programas como objetos matemáticos