  According to the Church-Turing Thesis (CTT), effective formal behaviours can
be simulated by Turing machines; this has naturally led to speculation that
physical systems can also be simulated computationally. But is this wider claim
true, or do behaviours exist which are strictly hypercomputational? Several
idealised computational models are known which suggest the possibility of
hypercomputation, some Newtonian, some based on cosmology, some on quantum
theory. While these models' physicality is debatable, they nonetheless throw
into question the validity of extending CTT to include all physical systems. We
consider the physicality of hypercomputational behaviour from first principles,
by showing that quantum theory can be reformulated in a way that explains why
physical behaviours can be regarded as 'computing something' in the standard
computational state-machine sense. While this does not rule out the physicality
of hypercomputation, it strongly limits the forms it can take. Our model also
has physical consequences; in particular, the continuity of motion and arrow of
time become theorems within the basic model.
