3 research outputs found
Categories and Types for Axiomatic Domain Theory
Submitted for the degree of Doctor of Philosophy, University of londo
LNL-FPC: The Linear/Non-linear Fixpoint Calculus
We describe a type system with mixed linear and non-linear recursive types
called LNL-FPC (the linear/non-linear fixpoint calculus). The type system
supports linear typing, which enhances the safety properties of programs, but
also supports non-linear typing as well, which makes the type system more
convenient for programming. Just as in FPC, we show that LNL-FPC supports
type-level recursion, which in turn induces term-level recursion. We also
provide sound and computationally adequate categorical models for LNL-FPC that
describe the categorical structure of the substructural operations of
Intuitionistic Linear Logic at all non-linear types, including the recursive
ones. In order to do so, we describe a new technique for solving recursive
domain equations within cartesian categories by constructing the solutions over
pre-embeddings. The type system also enjoys implicit weakening and contraction
rules that we are able to model by identifying the canonical comonoid structure
of all non-linear types. We also show that the requirements of our abstract
model are reasonable by constructing a large class of concrete models that have
found applications not only in classical functional programming, but also in
emerging programming paradigms that incorporate linear types, such as quantum
programming and circuit description programming languages
Categories and Types for Axiomatic Domain Theory
Domain Theory provides a denotational semantics for programming languages and calculi containing fixed point combinators and other so-called paradoxical combinators. This dissertation presents results in the category theory and type theory of Axiomatic Domain Theory. Prompted by the adjunctions of Domain Theory, we extend Benton’s linear/nonlinear dualsequent calculus to include recursive linear types and define a class of models by adding Freyd’s notion of algebraic compactness to the monoidal adjunctions that model Benton’s calculus. We observe that algebraic compactness is better behaved in the context of categories with structural actions than in the usual context of enriched categories. We establish a theory of structural algebraic compactness that allows us to describe our models without reference to enrichment. We develop a 2-categorical perspective on structural actions, including a presentation of monoidal categories that leads directly to Kelly’s reduced coherence conditions. We observe that Benton’s adjoint type constructors can be treated individually, semantically as well as syntactically, using free representations of distributors. We type various of fixed point combinators using recursive types and function types, whic