  Using nonstandard analysis, we will extend the classical Turing machines into
the internal Turing machines. The internal Turing machines have the capability
to work with infinite ($*$-finite) number of bits while keeping the finite
combinatoric structures of the classical Turing machines. We will show the
following. The internal deterministic Turing machines can do in $*$-polynomial
time what a classical deterministic Turing machine can do in an arbitrary
finite amount of time. Given an element of $<M;x>\in HALT$ (more precisely, the
$*$-embedding of $HALT$), there is an internal deterministic Turing machine
which will take $<M;x>$ as input and halt in the $"yes"$ state. The language
${}^*Halt$ can not be decided by the internal deterministic Turing machines.
The internal deterministic Turing machines can be viewed as the asymptotic
behavior of finite precision approximation to real number computations. It is
possible to use the internal probabilistic Turing machines to simulate finite
state quantum mechanics with infinite precision. This simulation suggests that
no information can be transmitted instantaneously and at the same time, the
Turing machine model can simulate instantaneous collapse of the wave function.
The internal deterministic Turing machines are powerful, but if $P \neq NP$,
then there are internal problems which the internal deterministic Turing
machines can solve but not in $*$-polynomial time.
