FP (short for Function Programming) is a programming language created by John Backus to support the Function-level programming paradigm.

Overview

The values that FP programs map into one another comprise a set which is closed under sequence formation:

 if x1,...,xn are values, then the sequence 〈x1,...,xn〉 is also a value 

These values can be built from any set of atoms: booleans, integers, reals, characters, etc.:

 boolean : {T, F} integer : {0,1,2,...,∞} character : {'a','b','c',...} symbol : {x,y,...} 

⊥ is the undefined value, or bottom. Sequences are bottom-preserving:

 〈x1,...,⊥,...,xn〉 = ⊥ 

FP programs are functions f that each map a single value x into another:

 f:x represents the value that results from applying the function f to the value x 

Functions are either primitive (i.e., provided with the FP environment) or are built from the primitives by program-forming operations (also called functionals). An example of one such operation is constant, which transforms a value x into the constant-valued function x̄. Functions are strict:

 f:⊥ = ⊥ 

Some functions have a unit value, such as 0 for addition and 1 for multiplication. The functional unit produces such a value when applied to a function f that has one:

 unit + = 0 unit × = 1 unit foo = ⊥ 

Functionals

These are the core functionals of FP:

 constant x̄ where x̄:y = x 

 composition f°g where f°g:x = f:(g:x) 

 construction [f1,...fn] where [f1,...fn]:x = 〈f1:x,...,fn:x〉 

 condition (h ⇒ f;g) where (h ⇒ f;g):x = f:x if h:x = T = g:x if h:x = F = ⊥ otherwise 

 apply-to-all αf where αf:〈x1,...,xn〉 = 〈f:x1,...,f:xn〉 

 insert-right /f where /f:〈x〉 = x and /f:〈x1,x2,...,xn〉 = f:〈x1,/f:〈x2,...,xn〉〉 and /f:〈 〉 = unit f 

 insert-left  f where  f:〈x〉 = x and  f:〈x1,x2,...,xn〉 = f:〈 f:〈x1,...,xn-1〉,xn〉 and  f:〈 〉 = unit f 

Equational functions

In addition to being constructed from primitives by functionals, a function may be defined recursively by an equation, the simplest kind being:

 f ≡ Ef 

where E'f is an expression built from primitives, other defined functions, and the function symbol f itself, using functionals.

An example of a primitive function is the selector function family, denoted by 1,2,... where:

 1:〈x1,...,xn〉 = x1 i:〈x1,...,xn〉 = xi if 0 < i ≤ n = ⊥ otherwise 

See also

• FL programming language (Backus' successor to FP)
• Function-level programming
• Programs as mathematical objects
• J and NGL programming languages
• John Backus