Iowa Type Theory Commute

I talk about my efforts to formalize lambda-calculus with named variables and explicit alpha-equivalence, as originally proposed by Church. One reason to do that, besides just a love of being ornery, is to be able to state and prove theorems about alpha-equivalence. One example class of such theorems concern when alpha-equivalence can be avoided, in the sense that beta-reduction can proceed without any variable capture, while not requiring renaming variables. I have a compan...

The problem of word equations is a rather storied one, including frustrated connections to Hilbert's Tenth problem. Word equations relate expressions consisting of concatenations of variables and constant symbols. An example is a X = X a, where X is a variable and a is a constant. A solution maps variables to strings of constant symbols making the two sides identical. In this episode, I discuss the problem a little, and what I learned so far about how it is solved.

I explain the story from last episode.

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 calculating the type checker.)

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 compilation technique. For this episode, we just discuss the idea of double-negation ...

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.

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 are disallowed entirely unless the function to be iterated is closed ("closed at cons...

In this episode, I discuss an intriguing idea proposed by Victor Taelin, to base a logically sound type theory on an untyped but terminating language, upon which one may then erect as exotic a type system as one wishes. By enforcing termination already for the untyped language, we no longer have to make the type system do the heavy work of enforcing termination.

Krivine's book (Section 4.2) has a proof of the Finite Developments Theorem, based on intersection types. I discuss this proof in this episode.

The finite developments theorem in pure lambda calculus says that if you select as set of redexes in a lambda term and reduce only those and their residuals (redexes that can be traced back as existing in the original set), then this process will always terminate. In this episode, I discuss the theorem and why I got interested in it.

I discuss what is called the locally nameless representation of syntax with binders, following the first couple of sections of the very nicely written paper "The Locally Nameless Representation," by Charguéraud. I complain due to the statement in the paper that "the theory of λ-calculus identifies terms that are α-equivalent," which is simply not true if one is considering lambda calculus as defined by Church, where renaming is an explicit reduction step, on a par with beta-reduction.&n...

I discuss the paper POPLmark Reloaded: Mechanizing Proofs by Logical Relations, which proposes a benchmark problem for mechanizing Programming Language theory.