Iowa Type Theory Commute

I explain the story from last episode.

The Curry-Howard isomorphism for the law of excluded middle, as a radio drama. I first saw a version of this story performed by Phil Wadler and Frank Pfenning (wearing fake horns!) at RTA in Nara, Japan in 2005. This is my take on it. In a subsequent episode, I will explain how the story illustrates the computational interpretation of the law of excluded middle.

I discuss a nice paper I quite enjoyed reading, called The Calculated Typer, by Garby, Bahr, and Hutton. The authors take a very nice general look at the specification of a type checker, for a very simple expression language. They then manually derive the actual code for the type checker by effectively trying to prove that this as yet unknown code satisfies its spec. (This is what is meant by...

In this episode, I talk about the control operator callcc, and how it is implemented during compilation using continuation-passing style (CPS). I sketch how CPS conversion (transforming a program with callcc into one in CPS that does not need callcc any more) corresponds to double-negation translation from classical to intuitionistic logic. The paper I am referencing is here.

In this episode, I talk about a somewhat more advanced case of the Curry-Howard isomorphism (the connection between logic and programming languages where formulas in logic are identified with types, and proofs with programs). This is the identification of double-negation translations in logic, which go back to a paper of Kolmogorov's in 1925, with conversion to continuation-passing style (CPS), a...

Commuting conversions are transformations on proofs in natural deduction, that move certain stuck inferences out of the way, so that the normal detour reductions (which correspond to beta-reduction under Curry-Howard) are enabled. The stuck inferences are uses of disjunction elimination. In programming terms, if you have an if-then-else (a simple case of or-elimination) where the then- and...

I am currently on a frolic into the literature on Control Flow Analysis (CFA), and discuss what this is, for pure lambda calculus. A wonderful reference for this is this paper by Palsberg.

In this episode, I talk about what we should consider to be a measure function. Such functions can be used to show termination of some process or program, by assigning a measure to each program, and showing that as the program computes, the measure decreases in some well-founded ordering. But what should count as a measure function? The context for this is RTA Open Problem 19, on showing...

To solve the problem raised in the last episode, I propose schematic affine recursion. We saw that affine lambda calculus (where lambda-bound variables are used at most once) plus structural recursion does not enforce termination, even if you restrict the recursor so that the function to be iterated is closed when you reduce ("closed at reduction"). You have to restrict it so that recursion terms...

In this episode, I shoot down last episode's proposal -- at least in the version I discussed -- based on an amazing observation from an astonishing paper, "Gödel’s system T revisited", by Alves, Fernández, Florido, and Mackie. Linear System T is diverging, as they reveal through a short but clever example. It is even diverging if one requires that the iterator can only be reduced when the...