Primitive Functions

Combinators: functions without variables

2021 James B. Wilson, Colorado State University

\[\mathbb{SIKII}\rhd \mathbb{SKI}\]

Functions are made

of simpler functions,

made of simpler functions...

then even simpler functions...

 

 

...turtles all the way down or are some functions atoms?

 

mov dx, 0     
mov ax, 7048  ; load n = 7048
mov bx, 541   ; load m = 541
div bx        ; now bx=13 (q) AND dx=15 (r)

Machine Language/ Assembly as atomic functions

But x86 is an industry choice.

Math like this came before...and probably will out last x86.

Combinators

Atomic functions need

  • names, e.g. \(\mathbb{S}\), \(\mathbb{K}\), \(\mathbb{I}\), but also , \(\mathbb{C}\), \(\mathbb{D}\), \(\mathbb{Y}\)...
  • yes ``\(\mathbb{Y}\)-combinator'' is a math not business.  Thanks Turing (sorry  Graham).
  • Combined in ordered lists, e.g. \(\mathbb{SKI}\)

Combinators do not need input variables.

Hence have no domains and codomains.

 

TLDR: Variables are symbols outside the alphabet; so, the following are technically rubish

"\(x:=5\)",  "if \(y=2\) then ..."

Variables are never assigned, never equal any thing.  They only locate substitution.

Atomic Function \(\mathbb{I}\):

       \(\mathbb{I}\)dentity Functions

\(y=x\)

\(x\mapsto x\)

\(I(x)=x\)

What is missing?

  • No domain
  • No codomain
  • Only one identity function

The

x

Atomic Function I:

       Identity Functions

Characterize \(\mathbb{I}\) by

\[\mathbb{I}V\rhd V\]

That is our first (weak) reduction rule.

The

x

TMI

  • Why \(\rhd\) instead of \(=\) or `:=`?

  • It is neither equal nor assigned.  It is verb is reduce, so use reducing symbol.

Atomic Function \(\mathbb{K}\):

       \(\mathbb{K}\)onstant Functions

\(y=3\)

\(x\mapsto \alpha\)

\(\mathbb{K}_c(x)=c\)

Still no domain/codomain.

  • Still one function
  • Just Two "inputs"

 

The

x

Characterize \(\mathbb{K}\) by

\[\mathbb{K}UV\rhd U\]

 

That is our second (weak) reduction rule.

 

Again... not equal, not assigned, reduced.

Atomic Function \(\mathbb{K}\):

       \(\mathbb{K}\)onstant Functions

The

x

Can't \(\mathbb{SKI}\) without an \(\mathbb{S}\)

Identity and constants are interesting, but we need real programs to choose a path.

Atomic Function \(\mathbb{S}\):

       \(\mathbb{S}\)hoenfinkel Composition

Some logic needs to branch.

  • First, offer a path off the main, \(g(x)\).
  • Second, decide what to do with both paths, \(f(x,g(x))\).
  • E.g. pick a side.
def branch(P)(x) = {
  if P then 
	x
  else
	g(x)
}
f after branch

\(S_{f,g}(x)=f(x,g(x))\)

\(\mathbb{S}_{f,g}(x)=f(x,g(x))\)

\[x\mapsto (x,g(x))\mapsto f(x,g(x))\]

\(d/dx\)

\(x\mapsto x^2\)

Atomic Function \(\mathbb{S}\):

       \(\mathbb{S}\)hoenfinkel Composition

You don't have to branch, you might both paths!

Characterize \(\mathbb{S}\) by

\[\mathbb{S}UVW\rhd UW(VW)\]

 

That is our third (weak) reduction rule.

 

Atomic Function \(\mathbb{S}\):

       \(\mathbb{S}\)hoenfinkel Composition

Pure Combinators

Informally

  • \(\mathbb{I}(x)=x\)
  • \(\mathbb{K}_c(x)=c\)
  • \(\mathbb{S}_{f,g}(x)=f(x,g(x))\)

Pure

Combinators

Logically

  • \(\mathbb{I}U\rhd U\)
  • \(\mathbb{K}UV\rhd U\)
  • \(\mathbb{S}UVW\rhd UW(VW)\)

But wait there is more...

  • SKI-combinators are the pure combinators.  Enough to describe all computable functions.  
  • Other combinators offer short cuts, e.g. to recursion (Barney R or Turing Y combinators).
  • Applied combinators, such as most instructions in x86 are not necessary for math but surely make functions shorter, faster, and easier to understand.

Further Reading

 

  • Hindley-Seldin Lambda-Calculus and Combinators, Cambridge U. Press, Chapter 1.