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FP
Paradigm	function-level
Appeared in	1977
Designed by	John Backus
Influenced by	APL
Influenced	FL, J

FP (short for Function Programming) is a programming language created by John Backus to support the function-level programming paradigm. This allows for the elimination of named variables.

Contents 1 Overview 2 Functionals 3 Equational functions 4 See also 5 References

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 primitive function is constant, which transforms a value x into the constant-valued function x̄. Functions are strict:

f:⊥ = ⊥

Another 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

In contrast to primitive functions, functionals operate on other functions. For example, 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 =  ⊥

These are the core functionals of FP:

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

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.

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

• Can Programming Be Liberated from the von Neumann Style? Backus' Turing award lecture.