Afleveringen
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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.
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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 [email protected]).
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Zijn er afleveringen die ontbreken?
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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 consequence that we cannot define self-exponentiation, \ x . exp x x. That term would reduce to \ x . x x, which is provably not typable in STLC.
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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.
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I review the typing rules and some basic examples for STLC. I also remind listeners of the Curry-Howard isomorphism for STLC.
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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...
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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.
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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.
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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.
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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.
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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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I continue the discussion of Mitchell's paper Type Inference with Simple Subtypes. Coming soon: a discussion of semantics of subtyping.
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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.
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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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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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We begin a discussion of subtyping in functional programming. In this episode, I talk about how subtyping is a neglected feature in implemented functional programming languages (for example, not found in Haskell), and how it could be very useful for writing lighter, more elegant code. I also talk about how subtyping could help realize a new vision for practical strong functional programming, where the language has a pure, terminating core language, then a monad for pure but possibly diverging computation, and finally a monad for impurity and divergence.
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In this episode, I conclude my discussion of some (but hardly all!) points from Pujet and Tabareau's POPL 2022 paper, "Observational Equality -- Now for Good!". I talk a bit about the structure of the normalization proof in the paper, which uses induction recursion. See this paper by Peter Dybjer for more about that feature. Also, feel free to join the new Telegram group for the podcast if you want to discuss episodes.
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I continue discussing the Puject and Tabareau paper, "Observational Equality -- Now for Good", in particular discussing more about how the equality type simplifies based on its index (which is the type of the terms being equated by the equality type), and how proofs of equalities can be used to cast terms from one type to another.
Also, in exciting news, I created a Telegram group that you can join if you want to discuss topics related to the podcast or particularly podcast episodes. I will be monitoring the group. I believe you have to request to join, and then I approve (it might take me until later in the day to do that, just fyi). The invitation link is here. -
In this episode, I introduce an important paper by Pujet and Tabareau, titled "Observational Equality: Now for Good", that develops earlier work of McBride, Swierstra, and Altenkirch (which I will cover in a later episode) on a new approach to making a type theory extensional. The idea is to have equality types reduce, within the theory, to statements of extensional equality for the type of the values being equated.
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I pause the chapter on extensionality in type theory to talk about something very exciting that I just learned about (though the project was completed Summer 2022): the so-called Liquid Tensor Experiment, to formalize a recent very difficult proof by a mathematician named Peter Scholze, in Lean. This is the first time in history, that I know of, when a theorem was formalized in a theorem prover, in order to resolve doubts of the mathematician who proved it. An amazing achievement. This episode tells the story, as I have understood it on line. The result apparently sparked this recent workshop.
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