6 research outputs found
Denotation of contextual modal type theory (CMTT):syntax and meta-programming
The modal logic S4 can be used via a Curry-Howard style correspondence to
obtain a lambda-calculus. Modal (boxed) types are intuitively interpreted as
`closed syntax of the calculus'. This lambda-calculus is called modal type
theory --- this is the basic case of a more general contextual modal type
theory, or CMTT.
CMTT has never been given a denotational semantics in which modal types are
given denotation as closed syntax. We show how this can indeed be done, with a
twist. We also use the denotation to prove some properties of the system
On the Semantics of Intensionality and Intensional Recursion
Intensionality is a phenomenon that occurs in logic and computation. In the
most general sense, a function is intensional if it operates at a level finer
than (extensional) equality. This is a familiar setting for computer
scientists, who often study different programs or processes that are
interchangeable, i.e. extensionally equal, even though they are not implemented
in the same way, so intensionally distinct. Concomitant with intensionality is
the phenomenon of intensional recursion, which refers to the ability of a
program to have access to its own code. In computability theory, intensional
recursion is enabled by Kleene's Second Recursion Theorem. This thesis is
concerned with the crafting of a logical toolkit through which these phenomena
can be studied. Our main contribution is a framework in which mathematical and
computational constructions can be considered either extensionally, i.e. as
abstract values, or intensionally, i.e. as fine-grained descriptions of their
construction. Once this is achieved, it may be used to analyse intensional
recursion.Comment: DPhil thesis, Department of Computer Science & St John's College,
University of Oxfor