About these lecture notes. Simply Typed λ-calculus. Types
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1 About these lecture notes Simply Typed λ-calculus Akim D le EPITA École Pour l Informatique et les Techniques Avancées Many of these slides are largely inspired from Andrew D. Ker s lecture notes [Ker, 2005a, Ker, 2005b]. Some slides are even straightforward copies. June 14, 2016 A. D le Simply Typed λ-calculus 2 / 46 Simply Typed λ-calculus Untyped λ-calculus Paradoxes Church vs. Curry A. D le Simply Typed λ-calculus 3 / 46 A. D le Simply Typed λ-calculus 4 / 46
2 first appeared with Curry (1934) for Combinatory Logic Church (1940) are syntactic objects assigned to terms: M : A M has type A For instance: I : A A Alonzo Church ( ) Haskell Curry ( ) A. D le Simply Typed λ-calculus 5 / 46 A. D le Simply Typed λ-calculus 6 / 46 Untyped λ-calculus λ-terms Untyped λ-calculus Paradoxes Church vs. Curry Λ, set of λ-terms x V x Λ M Λ N Λ (MN) Λ M Λ x V (λx M) Λ A. D le Simply Typed λ-calculus 7 / 46 A. D le Simply Typed λ-calculus 8 / 46
3 λβ Properties of λβ The λβ Formal System M = N M = N N = L M = M N = M M = L M = M N = N M = N MN = M N λx M = λx N (λx M)N = [N/x]M β-reduction is Church-Rosser. Any term has (at most) a unique NF. β-reduction is not normalizing. Some terms have no NF (Ω). A. D le Simply Typed λ-calculus 9 / 46 A. D le Simply Typed λ-calculus 10 / 46 Paradoxes Self application Untyped λ-calculus Paradoxes Church vs. Curry What is the computational meaning of λx xx? Stop considering anything can be applied to anything A function and its argument have different behaviors A. D le Simply Typed λ-calculus 11 / 46 A. D le Simply Typed λ-calculus 12 / 46
4 Church vs. Curry Simple Untyped λ-calculus Paradoxes Church vs. Curry A set of type variables α, β,... A symbol for functions α α, α (β γ), (α β) γ,... Possibly constants for primitive types ι for integers, etc. A. D le Simply Typed λ-calculus 13 / 46 A. D le Simply Typed λ-calculus 14 / 46 Simple Alonzo Style, or Haskell Way? By convention is right-associative: α β γ = α (β γ) This matches the right-associativity of λ: λx λy M = λx (λy M) Church: Typed λ-calculus x : α λx α x : α α Curry: λ-calculus with x : α λx x : α α A. D le Simply Typed λ-calculus 15 / 46 A. D le Simply Typed λ-calculus 16 / 46
5 λ : Type Assignments Reducibility Reducibility A. D le Simply Typed λ-calculus 17 / 46 A. D le Simply Typed λ-calculus 18 / 46 Simple Type Contexts T V a set of type variables α, β,... Simple The set T of types σ, τ,...: α T V α T σ T τ T (σ τ) T Statement A statement M : σ is a pair with M Λ, σ T. M is the subject, σ the predicate. Type Context, Basis A type context Γ is a finite set of statements over distinct variables {x 1 : σ 1,...}. Assignment The variable x is assigned the type σ in Γ iff x : σ Γ. A. D le Simply Typed λ-calculus 19 / 46 A. D le Simply Typed λ-calculus 20 / 46
6 Type Contexts Type Context Restrictions Γ x is the Γ with all assignment x : σ removed. Γ M is Γ FV(M). Reducibility A. D le Simply Typed λ-calculus 21 / 46 A. D le Simply Typed λ-calculus 22 / 46 A Natural Presentation Type Statement Type derivations are trees built from the following nodes. [x : σ] M : σ τ N : σ MN : τ M : τ λx M : σ τ Type Statement A statement M : σ is derivable from the type context Γ, Γ M : σ if there is a derivation of M : σ whose non-canceled assumptions are in Γ. A. D le Simply Typed λ-calculus 23 / 46 A. D le Simply Typed λ-calculus 24 / 46
7 Type Statements Type Statements Prove λfx f (fx) : (σ σ) σ σ Prove λxy x : σ τ σ (2) [f : σ σ] [f : σ σ] (2) f (fx) : σ (1) λx f (fx) : σ σ fx : σ λfx f (fx) : (σ σ) σ σ [x : σ] (1) (2) [x : σ] (1) λy x : τ σ (1) λxy x : σ τ σ A. D le Simply Typed λ-calculus 25 / 46 A. D le Simply Typed λ-calculus 26 / 46 Alternative Presentation of Type Derivations Type Statements Type Derivations {x : σ} x : σ Γ M : σ τ N : σ Γ MN : τ Γ M : τ Γ \ {x : σ} λx M : σ τ Γ M : τ Γ, M : τ Γ, consistent Γ, {x : σ} consistent Prove λxy x : σ τ σ {x : σ} x : σ [x : σ] (1) {x : σ} λy x : τ σ λy x : τ σ (1) λxy x : σ τ σ λxy x : σ τ σ A. D le Simply Typed λ-calculus 27 / 46 A. D le Simply Typed λ-calculus 28 / 46
8 Type Statements Type ω = λx xx. λx xx : σ {x : σ 1 } xx : σ 2 σ = σ1 σ 2 λx xx : σ A term M is typable if there exists a type σ such that M : σ. ω, Ω are not typable. S, K, I are typable. Y is not typable! {x : σ 1 } x : τ σ 2 {x : σ 1 } x : τ {x : σ 1 } xx : σ 2 σ = σ1 σ 2 λx xx : σ A. D le Simply Typed λ-calculus 29 / 46 A. D le Simply Typed λ-calculus 30 / 46 Subject Construction Lemma I Subject Construction Lemma I Consider the derivation π for M : σ. If M = x, then Γ = {x : σ} and π = {x : σ} x : σ If M = NL, then π = Γ N N : τ σ Γ L L : τ Γ NL : σ Consider the derivation π for M : σ. If M = λx N, then σ = σ 1 σ 2 and If x FV(N) If x FV(N) π = Γ {x : σ 1} N : σ 2 Γ λx N : σ 1 σ 2 Γ N : σ 2 π = Γ λx N : σ 1 σ 2 A. D le Simply Typed λ-calculus 31 / 46 A. D le Simply Typed λ-calculus 32 / 46
9 Derivations are not Unique They are for β-normal forms. Reducibility A. D le Simply Typed λ-calculus 33 / 46 A. D le Simply Typed λ-calculus 34 / 46 Conversions and α-invariance If Γ M : σ and M α N then Γ N : σ. Substitution If then Γ {x : τ} M : σ, N : τ, Γ, consistent, x FV(N) Γ [N/x]M : σ If Γ M : σ and M β N then Γ N : σ. What about the converse? Subject Expansion If Γ N : σ and M β N then Γ M : σ. Ω is not typable, but KI Ω I. A. D le Simply Typed λ-calculus 35 / 46 A. D le Simply Typed λ-calculus 36 / 46
10 Reducibility and Normalization Reducibility λ is Strongly Normalizing All typable terms are β-strongly normalizing. A. D le Simply Typed λ-calculus 37 / 46 A. D le Simply Typed λ-calculus 38 / 46 M : σ suffices Reducibility Note that with Γ = {x 1 : σ 1,..., x n : σ n } Γ M : τ is equivalent to λx 1... x n M : σ 1... σ n τ A. D le Simply Typed λ-calculus 39 / 46 A. D le Simply Typed λ-calculus 40 / 46
11 Decidability of Type Assignment Decidability of Type Assignment Type checking Given M, σ, does M : σ? M : σ? Given M, does there exist σ such that M : σ? M :? Inhabitation Given σ, does there exist M such that M : σ? Type Checking is Decidable It is decidable whether a statement of λ is provable. is Decidable It is decidable whether a term of λ has a type.? : σ A. D le Simply Typed λ-calculus 41 / 46 A. D le Simply Typed λ-calculus 42 / 46 are not Unique... are not Unique... but some are more Unique than others... Typable terms do not have unique types. {x : σ} x : σ {x : σ} λy x : τ σ K : σ τ σ is valid for any σ, τ, including σ = σ σ, τ = (τ τ ) τ etc. All the types of K are instances of σ τ σ. A. D le Simply Typed λ-calculus 43 / 46 A. D le Simply Typed λ-calculus 44 / 46
12 Bibliography Notes Bibliography I [Ker, 2005a] Complete and readable lecture notes on λ-calculus. Uses conventions different from ours. [Ker, 2005b] Additional information, including slides. [Barendregt and Barendsen, 2000] A classical introduction to λ-calculus. Barendregt, H. and Barendsen, E. (2000). Introduction to lambda calculus. http: // Ker, A. D. (2005a). Lambda calculus and types. lambda-calculus-notes-full-v3.pdf. Ker, A. D. (2005b). Lambda calculus notes. A. D le Simply Typed λ-calculus 45 / 46 A. D le Simply Typed λ-calculus 46 / 46
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