by Some people may find the following of (purely theoretical) interest. It
appears that the latest C++ standard defines templates in such a way
that they now form a universal computer. In principle we can now push as
much computation as we like to compile time lightening the load at run
time. Of course the example below is a little extreme but it
demonstrates the point. It's a primality tester that does all of its
work at compile time and yet makes no use of the usual C++ compile time
expression evaluator. It does so by defining integers from basic axioms
(cf. Godel, Escher, Bach by D Hofstadter). Clearly this isn't directly
useful but I found that writing it opened my eyes to some potentially
useful techniques.
//
// A C++ program to test for the primality of the number 13
//
// It has the curious property that it does no arithmetic
// but uses only the template mechanism and class derivation
// to achieve its result. It starts at the most basic axioms
// of arithmetic starting with the definition of zero and
// its successors...
//
// You'll need a good C++ compiler.
//
// (c) D Piponi (But copy it as much as you like if you credit me)
//
//
// First a little trick to find the base class of our
// classes because the template mechanism requires
// exact matches.
//
// Two values are considered equivalent if they have
// the same baseclass. For example 1+1 isn't
// identical to 2 but 1+1 *is* derived from 2.
//
template<class V> struct Value { typedef V value; };
//
// Define zero
//
struct zero
: public Value<zero> { };
//
// Define the successor of an integer
//
template<class C> struct S
: public Value<S<C> > { typedef C predecessor; };
//
// Now we have the integers. So introduce some convenience
// types.
//
typedef S<zero> one;
typedef S<one> two;
typedef S<two> three;
typedef S<three> four;
typedef S<four> five;
typedef S<five> six;
typedef S<six> seven;
typedef S<seven> eight;
typedef S<eight> nine;
typedef S<nine> ten;
//
// Define plus,...
//
template<class C,class D> struct plus
: public S<plus<C,D::predecessor> > { };
template<class C> struct plus<C,zero>
: public C { };
//
// ...minus...
//
template<class C,class D> struct minus
: public minus<C,D::predecessor>::predecessor { };
template<class C> struct minus<C,zero>
: public C { };
//
// ...and times.
//
template<class C,class D> struct times
: public plus<C,times<C,D::predecessor>::value> { };
template<class C> struct times<C,zero>
: public zero { };
//
// Define the usual ordering on the integers.
//
template<class C,class D> struct ge
: public ge<C::predecessor,D::predecessor> { };
template<class C> struct ge<C,zero>
: public one { };
template<class C> struct ge<zero,C>
: public zero { };
template<> struct ge<zero,zero>
: public one { };
//
// Divisibility testing
//
template<class C,class D,class E = S<S<zero> > > struct Divisible { };
template<class C,class D> struct Divisible<C,D,S<S<zero> > >
: public Divisible<C,D,ge<C,D>::value> { };
//
// The case C<D:
// In this case D divides C iff C is zero.
//
template<class C,class D> struct Divisible<C,D,zero>
: public zero { };
template<class C> struct Divisible<zero,C,zero>
: public one { };
//
// The case C>=D:
// D divides C iff D divides C-D.
//
template<class C,class D> struct Divisible<C,D,S<zero> >
: public Divisible<minus<C,D>::value,D> { };
//
// Primality testing
//
template<class C,class D = two,class S = zero,class E = zero,class F =
zero> struct Prime { };
//
// We use recursion to set up a loop from 2 to one less
// than the integer we are testing. Essentially this is a state machine
// using template parameters to record the current state.
//
// Are we at the end of the recursion?
//
template<class C,class D> struct Prime<C,D,zero,zero,zero>
: public Prime<C,D,one,zero,ge<D,C>::value> { };
//
// Test potential factor D, trying successor on failure.
//
template<class C,class D> struct Prime<C,D,one,zero,zero>
: public Prime<C,S<D>,zero,Divisible<C,D>::value,zero> { };
//
// Found a factor.
//
template<class C,class D> struct Prime<C,D,zero,one,zero>
: public zero { };
//
// Reached the end of the loop without finding divisor.
//
template<class C,class D> struct Prime<C,D,one,zero,one>
: public one { };
//
// Convenience method for humans allowing input of
// numbers in decimal format.
//
template<class C,class D> struct Decimal
: public plus<times<ten,C>::value,D> { };
//
// Unfortunately the I/O interface spoils the purity of it all.
//
#include <stdio.h>
template<class C> char *output(C);
template<> char *output(zero) { return "No"; }
template<> char *output(one) { return "Yes"; }
main() {
//
// Is 13 prime?
//
puts(output(Prime<Decimal<one,three>::value>::value()));
--Quote:}
Dan Piponi
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