Lecture 2:

Simply-typed $λ$-calculus

Frege ⟶ type of natural numbers.

Let $A$ be a type.

  • $f: A ⟹ A$
  • $a: A$

so that:

\[\overline{n} ≝ λf, a. f^n(a) : (A ⟹ A) ⟹ (A ⟹ A)\]
Sequent:

a triple of the form: \(x_1:A_1, \ldots, x_n:A_n \vdash P: B\)

  • Axiomatic rule: Variable
  • Logical rules: Abstraction, Application
  • Structural rules: Weakening, Contraction, Exchange

NB: Multiplicative (with $Γ$ and $Δ$) application rule VS additive (with only $Γ$) application rule ⟶ equivalent in intuitionistic logic

  • Multiplicative one: used in linear logic
  • Additive one: used in dependent type theory

Dana Scott (1968) ⟶ mathematical model of pure $λ$-calculus: Domain theory

Curry-Howard correspondence ⟶ related to the Brouwer-Heyting-Kolmogorov interpretation

Ex: a proof of $A ⟹ B$: impossible to define without a language: “it’s a mechanism which turns any proof of $A$ into a proof of $B$” ⟶ but what on earth is a “mechanism”? ⟹ $λ$-calculus aims at defining it (creativity to come up with such a mechanism is up to us)

Algebraic Church-Rosser theorem: from a categorical standpoint

If

  • $M \overset{f}{⟶} P$
  • $M \overset{g}{⟶} Q$

then there exist $P \overset{g’}{⟶} N$ and $Q \overset{f’}{⟶} N$ s.t.

\[M \overset{f}{⟶} P \overset{g'}{⟶} N \sim M \overset{g}{⟶} Q \overset{f'}{⟶} N\]

$λ$-calculus reduction paths defines a graph where nodes are the $λ$-terms, arrows are the rewriting rules, but two paths are permutation equivalent is you can permute the order of computations between them.

Ex:

  1. If $M ⟶ P$, $N ⟶ Q$, then
\[MN ⟶ PN ⟶ PQ \sim MN ⟶ MQ ⟶ PQ\]
  1. Inside-out $\sim$ Outside-in

$λσ$-calculus: $λ$-calculus with explicit substitutions

\[(λx.M)P ⟶ M[x := P]\]

but $x$ is not duplicated in $M$’s tree: there’s a pointer only (until we reach a variable).

Critical pairs: superposition de deux motifs de réécriture dans un terme ⟶ pas confluence en général

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