Room 218

School of Computer Science

University of Birmingham

Edgbaston

B15 2TT

Birmingham

United Kingdom

My email address is: `axs1431[symbol]cs.bham.ac.uk`

I am interested in pointfree representations of spaces and their connections with logic. In particular I am interested in how pointfree topology (locale theory) can be extended to the bitopological setting.

I completed the Logic Year postgraduate programme at the Institute of Language, Logic and Computation (Amsterdam) in 2017.

I completed my Bachelor's degree in Mathematics at Heriot-Watt University (Edinburgh) in 2016.

- The coproduct of frames as encoding d-frame structure,
*TACL*, Nice, 2019. - Bitopological sublocales,
*Workshop in Algebra, Logic and Topology*Coimbra, 2018. Slides.

I am studying objects called *d-frames*. We may interpret these in various ways.

(1) They act as pointfree bitopological spaces, in virtue of a Stone-type adjunction between the two categories at play.
Bitopological spaces are sets of points equipped with two topologies. They arise naturally when we are dealing with topologies
that are described by two coarser topologies. For instance, the opens in the Euclidian topology on the real line have a very
complicated order structure. But this topology is generated by upper open intervals and lower ones, and both these
structures are much simpler order-theoretically.

(2) They act as representations of *four-valued* logical theories. The four truth values of these theories are
True, False, Neither, and Both. They refine two valued logic in that they take into account not only the truhhood
or falsehood of assertions, but also our ability to verify them or refute them. What this means is that in these
settings our access to information is imperfect. We may have not enough information (we may be, that is, unable to
verify or refute an assertion); or we may have too much information (we may receive false information which both
refutes and verifies the same assertion). Hence the two extra truth values. In particular these logical theories
give an account of the notion, important in Computer Science, of undecidability.

(3) They act as canonical presentations of certain frames in terms of two generating frames and relations.
To obtain a frame from such a presentation in general one takes the free frame generated by the
two frames (i.e. their coproduct), and quotients it by the relations. The frames which are
presented by d-frames are exactly those such that these relations are in a certain sense *finitary *.
By this we mean that these are relations that only involve elements of the coproduct expressible as finite
joins of finite meets of the generators.