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.
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18:49
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18:49
The Stunner: Linear System T is Diverging!
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.
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21:03
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21:03
Terminating Computation First?
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.
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11:27
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11:27
Correction: the Correct Author of the Proof from Last Episode, and an AI flop
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.
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7:10
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7:10
Krivine's Proof of FD, Using Intersection Types
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.
France