Date of Award
Doctor of Philosophy (PhD)
Computer and Information Science
Dependently typed languages allow us to develop programs and write
proofs quickly and without errors, and the last decade has seen many
success stories for verified programming with dependent types.
Despite these successes, dependently typed languages are rarely used
for day-to-day programming tasks. There are many reasons why these
languages have not been more widely adopted. This thesis addresses
two of them: First, existing dependently typed languages restrict
recursion and require programmers to prove that every function
terminates. Second, traditional representations of equality are
inconvenient to work with because they require too much typing
information and because their eliminations clutter terms.
This thesis introduces PCC-Theta, a new dependently typed core
language that addresses these problems. To handle potentially
non-terminating computations, PCC-Theta is split into two fragments: a
programmatic fragment with support for general recursion, and a
logical fragment that is restricted for consistency. Crucially,
while the logical fragment is consistent, it can reason about programs
written in the inconsistent programmatic fragment. To make equality
reasoning easier, PCC-Theta includes a novel heterogeneous notion of
equality whose eliminations are not marked in terms.
The metatheory of PCC-Theta is studied in detail, including a complete
proof of normalization and consistency for its logical fragment. The
normalization proof required the development of a novel technique,
partially step-indexed logical relations, which is motivated and
explained. Additionally, to demonstrate that PCC-Theta addresses the
problems described above, we have extended it to a complete core
language Theta, adding features like user-defined datatypes and an
infinite hierarchy of universes. Several examples are carried out in
Theta, and an implementation is available.
Casinghino, Chris, "Combining Proofs and Programs" (2014). Publicly Accessible Penn Dissertations. 1227.