Title:

Computational problems in matrix semigroups

This thesis deals with computational problems that are defined on matrix
semigroups, which playa pivotal role in Mathematics and Computer Science
in such areas as control theory, dynamical systems, hybrid systems, computational
geometry and both classical and quantum computing to name but
a few. Properties that researchers wish to study in such fields often turn out
to be questions regarding the structure of the underlying matrix semigroup
and thus the study of computational problems on such algebraic structures
in linear algebra is of intrinsic importance.
Many natural problems concerning matrix semigroups can be proven
to be intractable or indeed even unsolvable in a formal mathematical sense.
Thus, related problems concerning physical, chemical and biological systems
modelled by such structures have properties which are not amenable to
algorithmic procedures to determine their values.
With such recalcitrant problems we often find that there exists a tight
border between decidability and undecidability dependent upon particular
parameters of the system. Examining this border allows us to determine
which properties we can hope to derive algorithmically and those problems
which will forever be out of our reach, regardless of any future advances in
computational speed.
There are a plethora of open problems in the field related to dynamical
systems, control theory and number theory which we detail throughout
this thesis. We examine undecidability in matrix semigroups for a variety
of different problems such as membership and vector reachability problems,
semigroup intersection emptiness testing and freeness, all of which are well
known from the literature. We also formulate and survey decidability questions
for several new problems such as vector ambiguity, recurrent matrix
problems, the presence of any diagonal matrix and quaternion matrix semigroups,
all of which we feel give a broader perspective to the underlying
structure of matrix semigroups.
