Iowa Type Theory Commute

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...

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 (CP...

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-...

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 termi...

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 recu...

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...

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.

I correct what I said in the last episode about the author of the proof of FD from last episode based on intersection types.  I also describe AI flopping when I ask it a question about this.

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.

I discuss the paper "A Direct Proof of the Finite Developments Theorem" , by Roel de Vrijer.  See also the write-up at my blog.

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.

In this episode, I discuss the paper Nominal Techniques in Isabelle/HOL , by Christian Urban.  This paper shows how to reason with terms modulo alpha-equivalence, using ideas from nominal logic.  The basic idea is that instead of renamings, one works with permutations of names. 

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 lam...

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

I continue the discussion of POPLmark Reloaded , discussing the solutions proposed to the benchmark problem.  The solutions are in the Beluga, Coq (recently renamed Rocq), and Agda provers.

In this episode, I begin discussing the question and history of formalizing results in Programming Languages Theory using interactive theorem provers like Rocq (formerly Coq) and Agda.

In this episode, I describe the first proof of normalization for STLC, written by Alan Turing in the 1940s.  See this short note for Turing's original proof and some historical comments.

In this episode, after a quick review of the preceding couple, I discuss the property of normalization for STLC, and talk a bit about proof methods.  We will look at proofs in more detail in the coming episodes.  Feel free to join the Telegram group for the podcast if you want to discuss anything (or just email me at aaron.stump@gmail.com).

It is maybe not so well known that arithmetic operations -- at least some of them -- can be implemented in simply typed lambda calculus (STLC).  Church-encoded numbers can be given the simple type (A -> A) -> A -> A, for any simple type A.  If we abbreviate that type as Nat_A, then addition and multiplication can both be typed in STLC, at type Nat_A -> Nat_A -> Nat_A.  Interestingly...

Like addition and multiplication on Church-encoded numbers, exponentiation can be assigned a type in simply typed lambda calculus (STLC).  But surprisingly, the type is non-uniform.  If we abbreviate (A -> A) -> A -> A as Nat_A, then exponentiation, which is defined as \ x . \ y . y x, can be assigned type Nat_A -> Nat_(A -> A) -> Nat_A.  The second argument needs to have type at...