Simply Typed λ-calculus

Transcription

1 Simply Typed λ-calculus Lecture 1 Jeremy Dawson The Australian National University Semester 2, 2017 Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

2 A Brief History of Type Theory First developed by Bertrand Russell (around 1903), to avoid certain paradoxes in the foundations of math, e.g., Russell s paradox. A hierarchy of types introduced to solve this. Typically, a class of all sets is of a different type than a set, etc. Formalised by Alonzo Church in the 1930 s in his simple type theory, which introduces types to (untyped) λ-calculus. The full simple type theory deals with logical concepts as well (also known as higher-order logic). We shall deal only with the functional part, i.e., we only consider the β-equality part of Church s simple theory of types. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

3 Types in computation Types are used to restrict the behaviours of a program. Maxim: Well-typed programs don t go wrong. In the case of λ-calculus, types can be used to enforce termination. Think of it as a light-weight theorem proving: types correspond to the property we want the program to satisfy. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

4 Outline of the course Lecture 1: Introduction, Church & Curry type systems. Lecture 2: The type preservation theorem. Lecture 3: Type inference. Lecture 4: The normalisation theorem (typed terms always have normal forms); Kripke semantics. Lecture 5: Curry-Howard correspondence, relating intuitionistic logic and simply typed lambda calculus. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

5 References Henk Barendregt. Lambda calculi with types. Handbook of Logic in Computer Science, Vol. II, Editors: S. Abramsky, D.M. Gabbay, T.S.E. Maibaum, Oxford University Press, J. Roger Hindley. Basic Simple Type Theory. Cambridge University Press, Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

6 Course website Course materials will be made available on Wattle and/or the COMP4630 website: me at phone (612)57778 I am in the RSISE building, bldg 115, Tuesdays, Thursdays and alternate Mondays. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

7 Overview of untyped λ-calculus Syntax: M,N ::= x M N λx.m α-equivalence: syntactic equality modulo renaming of bound variables, e.g., λxλy.x λuλv.u. Substitution. Notation: M[x := N]. (Other notations exist also). Capture avoiding: no free variables in N got captured in M after substitution: use α-equivalence first to avoid the problem (λv. u v)[u := v] λv. v v but (λv. u v)[u := v] λw. v w. Substitute for free variables only (λv. u v)[v := w] λv. u w but (λv. u v)[v := w] λv. u v Computation via β-reduction: (λx.m) N β M[x := N] Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

8 Two type systems There are two ways to define typed λ-calculus: Curry s style: type assignment (also, implicit typing) Terms are as untyped λ-calculus. Given a term M, what types can be assigned to M? There can be zero or many. Church s style: type annotations (also, explicit typing). Terms are explicitly annotated with type expressions. Given a term M, if it is typeable, it has a unique type. We shall see that the two systems are essentially equivalent. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

9 Some questions Some typical questions one asks about a type system: What are the properties of well-typed terms? Termination: A well-typed term has a normal form. Type preservation: If M has type τ and M β N, then N has type τ. Type checking: Given a term M and a type τ, is it decidable to check whether M has type τ? Type inference: Given a term M, is there a type τ such that M has type τ? Type inhabitant: Given a type τ, is there a term M such that M has type τ? We ll see answers to some of these questions for simple type systems. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

10 Simple types Assume an infinite set of type variables. We use α,β,γ to denote type variables. The set T of simple types is the smallest set such that: Type variables: α T, for every type variable α. Function types: If τ1,τ 2 T, then τ 1 τ 2 T. We write τ τ if τ and τ are syntactically equal. Notational convention: we assume associate to the right, so α 1 α 2 α 3 means α 1 (α 2 α 3 ). Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

11 Typing statements, type declarations and typing contexts When assigning a type to a term (in both Church & Curry type systems), we to assign types to free variables in the term. A type declaration is a pair x : τ of a (term) variable x and a type τ. A typing statement is a pair M : τ of a term and a type (read M has type τ). A typing context Γ is a finite set of typing declarations {x 1 : τ 1,...,x n : τ n } where x 1,...,x n are pairwise distinct. Typing judgments: Γ M : τ Read: the statement M : τ is derivable (in a type system) from the typing context Γ Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

12 λ Cu : Curry s type system The typing judgment Γ λcu M : τ is derivable in λ Cu if Γ λcu M : τ can be produced by the following rules: Γ λcu x : τ axiom,if (x : τ) Γ Γ,x : τ 1 λcu M : τ 2 Γ λcu (λx.m) : τ 1 τ 2 intro Γ λcu M : τ τ Γ λcu N : τ Γ λcu (M N) : τ elim Here Γ,x : τ 1 means Γ {x : τ 1 }. Note that in intro, x has to be fresh w.r.t. Γ (ie, x not in Γ) Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

13 Example of a typing deduction (or typing derivation) For simplicity, we omit the subscript λ Cu in λcu : x : α β,y : α x : α β x : α β,y : α y : α x : α β,y : α (x y) : β x : α β (λy.x y) : α β (λxλy.x y) : (α β) α β Exercises: prove the following typing judgments. (λx.x) : α α. (λx.x) : (β β) (β β). λxλyλz. x z (y z) : (α β γ) (α β) α γ. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

14 Example: an untypable term The term λx.x x is untypeable in λ Cu, i.e., for every type τ, there does not exist a derivation of (λx.x x) : τ. Exercise: Prove this. (We ll see a more systematic way to prove this in Lecture 3). Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

15 Church s type system λ Ch : annotated terms In λ Ch, types are incorporated explicitly into terms. The set of T-annotated terms Λ T is defined inductively as follows: x ΛT for every term variable x. If M,N ΛT then (M N) Λ T. If M ΛT and τ T then (λx : τ.m) Λ T. The notions of type declarations, typing statement, typing contexts and typing judgments are as in Curry s system. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

16 Reductions in λ Ch -terms Strictly speaking, terms in Λ T are not λ-terms, so one needs to define its equality theory, reductions, etc. The notions of β-reduction β, multi-step reduction β are defined as in untyped λ-calculus. (λx : τ.m) N β M[x := N]. Church-Rosser property also holds for T-annotated terms. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

17 Inference rules of λ Ch Γ λch x : τ axiom, if (x : τ) Γ Γ,x : τ 1 λch M : τ 2 Γ λch (λx : τ 1.M) : (τ 1 τ 2 ) intro Note that x has to be fresh w.r.t. Γ. Γ λch M : τ τ Γ λch N : τ Γ λch (M N) : τ elim Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

18 Example The following are derivable in λ Ch : (λx : τ.x) : (τ τ). (λx : σλy : τ.x) : (σ τ σ). x : σ (λy : τ.x) : (τ σ). Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

19 Church s system: uniqueness of types Theorem Suppose Γ λch M : τ and Γ M : τ. Then τ τ. Proof. By induction on the structure of M. Exercise: complete the proof. Type uniqueness does not hold for Curry s system, but we ll see that every typeable term in λ Cu has a unique principal type. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

20 Relating the Curry and Church systems Church s system can be simulated by Curry s system by forgetting the type annotations. Let Λ denote the set of unannotated λ-terms. Define a forgetful map from Λ T to Λ as follows: x x M N M N λx : τ.m λx. M Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

21 Relating the Curry and Church systems Theorem (Church to Curry) Let M Λ T. Then Γ λch M : τ implies Γ λcu M : τ. Proof: (exercise) Theorem (Curry to Church) Let M Λ. Then Γ λcu M : τ implies that there exists M Λ T such that Γ λch M : τ and M M. Proof: (exercise) Corollary A type τ T is inhabited in λ Ch if and only if it is inhabited in λ Cu. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

22 On-line type checking and inferencing - Haskell Haskell: a functional language with types based on simple type theory ghci on the departmental systems To download it yourself, can use Hugs (adequate and much smaller) λxλyλz. x z (y z) is entered as \x -> \y -> \z -> x z (y z) to get the type of this, > :t \x -> \y -> \z -> x z (y z) Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23

23 Summary We have covered two styles of type systems for λ-calculus. Church s system annotates the λ-terms with types explicitly. Curry s system leaves types implicit. Both are essentially the same. For next lecture, we shall be studying mainly properties of Curry s system. Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, / 23