Logical Refinements of Church's Problem

Type Refinements for Compiler Correctness

Type refinements, introduced by Freeman and Pfenning and explored by Davies and Dunfield, unify the ontological and epistemic views of typing. Types tell us what programming language constructs exist, whereas refinements express properties of the values of a type. Here we show that refinements are very useful in compiler correctness proofs, wherein it often arises that two expressions that are ...

Logical s-t Min-Cut Problem: An Extension to the Classic s-t Min-Cut Problem

Let $G$ be a weighted digraph, $s$ and $t$ be two vertices of $G$, and $t$ is reachable from $s$. The logical $s$-$t$ min-cut (LSTMC) problem states how $t$ can be made unreachable from $s$ by removal of some edges of $G$ where (a) the sum of weights of the removed edges is minimum and (b) all outgoing edges of any vertex of $G$ cannot be removed together. If we ignore the second constraint, ca...

Plantinga and the Logical Problem of Evil

On the Expressivity of Two Refinements of Multiplicative Exponential Linear Logic

The decidability of multiplicative exponential linear logic (MELL) is currently open. I show that two independently interesting refinements of MELL that alter only the syntax of proofs—leaving the underlying truth untouched— are undecidable. The first refinement uses new modal connectives between the linear and the unrestricted judgments, and the second is based on focusing with priority assign...

Refining Inductive Types

Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the N-indexed type of vectors refines lists by their lengths. Other data types may be refined in similar ways, but programmers must produce ...