Project description

There is a strong connection between programs and the proofs for certain logics. By finding mathematical models for either of these we increase our understanding of them. Because logical connectives relate closely to constructions in the model it is often mathematically cleaner to consider the former. (Categorical) proof theory is concerned with how proofs may be modelled, and which proofs can be considered equivalent to each other. While there are some logics whose proof theory is well understood these are fairly limited. The aim of this project is to extend what is known to more interesting logics.

As described above there are several projects that might fit under this description, and precise details are to be agreed between a successful applicant and the supervisor.

One possibility would be the following: We have a number of models that function well in the context of multiplicative linear logic, in the sense that every morphism in the model is the denotation of a derivation and and that there is an independently defined equivalence relation of those derivations which are interpreted by the same morphism in the model, but extending this result to the additive connectives has proved hard (there is only one published proof and that does not apply to the generally accepted notion of equivalence of proofs). A possible PhD project in this space could aim to look at this question, building on a new description of the model for the multiplicative fragment.

Other possibilities would be to look at theories of derivations that arise from `deep inference' systems, or the question of how we might use higher categorical structure to model more fine-grained connections between derivations than equivalence relations.