FP programming language



         


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

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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 = ⊥
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Functionals

These are the core functionals of FP:

constant x̄ where x̄:y = x
composition f&#x°g where f&#x°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
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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
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See also







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