Iowa Type Theory Commute

By: Aaron Stump
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  • Summary

  • Aaron Stump talks about type theory, computational logic, and related topics in Computer Science on his short commute.
    © 2024 Iowa Type Theory Commute
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Mathematics Science
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Episodes
  • Turing's proof of normalization for STLC

    Turing's proof of normalization for STLC
    May 21 2024

    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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    18 mins
  • Introduction to normalization for STLC

    Introduction to normalization for STLC
    May 14 2024

    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).
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    10 mins
  • The curious case of exponentiation in simply typed lambda calculus

    The curious case of exponentiation in simply typed lambda calculus
    May 4 2024

    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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    7 mins

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