Turing Machines for Dummies: why representations do matter
Peter van Emde Boas

Abstract:
Various methods exists in the literature for denoting the
configuration of a Turing Machine. A key difference is whether the
head position is indicated by some integer (mathematical
representation) or is specified by writing the machine state next to
the scanned tape symbol (intrinsic representation).

From a mathematical perspective this will make no difference. However,
since Turing Machines are primarily used for proving undecidability
and/or hardness results these representations do matter. Based on a
number of applications we show that the intrinsic representation
should be preferred.