Iowa Type Theory Commute

Latest Episodes:

Begin Chapter on Simple Type Theory Apr 19, 2024

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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...

Some advanced examples in DCS Sep 24, 2023

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This episode presents two somewhat more advanced examples in DCS. They are Harper's continuation-based regular-expression matcher, and Bird's quickmin, which finds the least natural number not in a given list of distinct natural numbers, in linear time. I explain these examples in detail and then discuss how they are implemented in DCS, which ensures that they are terminating on all inputs.

DCS compared to termination checkers for type theories Sep 18, 2023

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In this episode, I continue introducing DCS by comparing it to termination checkers in constructive type theories like Coq, Agda, and Lean. I warmly invite ITTC listeners to experiment with the tool themselves. The repo is here.

Getting started with DCS Sep 09, 2023

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In this episode, I talk more about the DCS tool, and invite listeners to check it out and possibly contribute! The repo is here.

Introduction to DCS Sep 03, 2023

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DCS is a new functional programming language I am designing and implementing with Stefan Monnier. DCS has a pure, terminating core, around which monads will be layered for possibly diverging, impure computation. In this episode, I talk about this basic design, and its rationale.

Semantics of subtyping Jul 23, 2023

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I answer a listener's question about the semantics of subtyping, by discussing two different semantics: coercive subtyping and subsumptive subtyping. The terminology I found in this paper by Zhaohui Luo; see Section 4 of the paper for a comparison of the two kinds of subtyping. With coercive subtyping, we have subtyping axioms "A <: B by c", where c is a function from A to B. The idea is that the compiler should automatically insert calls to c whenever an expression of type A needs to be converted to one of type B. Subsumptive subtyping says that A <: B means that the meaning of A is a subset of the meaning of B. So this kind of subtyping depends on a semantics for types. A simple choice is to interpret a type A as (as least roughly) the set of its inhabitants. So a type like Integer might be interpreted as the set of all integers, etc. Luo argues that subsumptive subtyping does not work for Martin-Loef type theory, where type annotations are inherent parts of terms. For in that situation, A <: B does not imply List A <: List B, because Nil A is an inhabitant of List A but not of List B (which requires instead Nil B).
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More on type inference for simple subtypes Jul 15, 2023

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I continue the discussion of Mitchell's paper Type Inference with Simple Subtypes. Coming soon: a discussion of semantics of subtyping.

Subtyping, the golden key Jul 08, 2023

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In this episode, I wax rhapsodic for the potential of subtyping to improve the practice of pure functional programming, in particular by allowing functional programmers to drop various irritating function calls that are needed just to make types work out. Examples are lifting functions with monad transformers, or even just the pure/return functions for applicative functors/monads.

Type inference with simple subtypes Jun 29, 2023

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In this episode, I begin discussing a paper titled "Type Inference with Simple Subtypes," by John C. Mitchell. The paper presents algorithms for computing a type and set of subtype constraints for any term of the pure lambda calculus. I mostly focus here on how subtype constraints allow typing any term (which seems surprising).
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Basics of subtyping Jun 21, 2023

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In this episode, I discuss a few of the basics for what we expect from a subtyping relation on types: reflexivity, transitivity, and the variances for arrow types.

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