Computation on Metric Algebras
Functions on data such as real and complex numbers, discrete
and continuous data streams, and scalar and vector fields,
are fundamental for many kinds of computation. Such data types
are modelled using topological, or metric, many-sorted algebras
and continuous mappings. Some of the issues that arise are:
(1) Abstract vs. Concrete models:
In concrete (unlike abstract) models, the computations depend
on the representations of the data. Examples of abstract
models are high level programming languages. In order to
derive an equivalence result between these two classes of
models, we have to consider issues of multivaluedness,
continuity and approximable computability.
We also apply this theory to the special case of discrete
data on countable algebras.
(2) The relationship between computation and specification:
Our theory shows the existence of finite universal algebraic
specifications for all the classically computable functions
on the reals. This provides finite universal algebraic
specifications for computable finite dimensional
deterministic dynamical systems.
Joint work with J.V. Tucker (Swansea)