Advertise on podcast: Iowa Type Theory Commute

In this episode, I reflect on the recent announcement that Anthropic researchers have autoformalized the proof of Fermat's Last Theorem.  That is, they instructed an LLM to create a computer-checkable proof, in the Lean prover, of this theorem, following existing paper proofs in the literature.  The resulting proof weighs in at 13 million lines of Lean, a staggering amount.

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 companion blog post that talks about this, with a link to the repo with my Agda code so far.

A system of word equations is called quadratic if no variable occurs more than twice in it.  There is an interesting simple algorithm to solve quadratic systems of word equations, which I talk through in this episode.  My source is Chapter 12 of "Algebraic Combinatorics on Words" by Lothaire.

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.

In this episode, I give further arguments in favor of coercive subtyping from a software-engineering perspective.  I also explain the critical concept of coherence.

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 calculating the type checker.)

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 compilation technique.  For this episode, we just discuss the idea of double-negation translation: classical theorems can be translated to intuitionistic ones, by adding some double negations.  As an example, we talk through the intuitionistic proof of the double negation of the law of excluded middle: not not (p or not p).

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 else-branches are lambda abstractions, and you apply that if-then-else to an argument, you need commuting conversions to move the argument into the branches, so you can call the functions (in the then- and else-branches) with it. See Section 10.1 of Girard's Proofs and Types for more on the problem, and a nice paper by de Groote on strong normalization with commuting conversions.

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 termination for the simply typed lambda calculus using a measure function. Let's call this the start of season 7, because it seems about time for that.

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 construction").  But this prevents higher-order functions like map, which need to repeat a computation involving a variable f to be mapped over the elements of a list.  The solution is to allow schematic definition of terms, using schema variables ranging over closed 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 function to be iterated is closed (no free variables).  This extraordinary observation does not sink Victor's idea of basing type theory on a terminating untyped core language, but it does sink the specific language he and I were thinking about, namely affine lambda calculus plus structural recursion. My notes are here.