Iowa Type Theory Commute Aaron Stump

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).

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 strictly higher order than the first argument. This has the fascinating con...

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, things change with exponentiation, which we will consider in the next episode.

I review the typing rules and some basic examples for STLC. I also remind listeners of the Curry-Howard isomorphism for STLC.

In this episode, after a pretty long hiatus, I start a new chapter on simply typed lambda calculus. I present the typing rules and give some basic examples. Subsequent episodes will discuss various interesting nuances...