Lecture 3: Typed Lambda Calculus and Curry-Howard

Transcription

1 Lecture 3: Typed Lambda Calculus and Curry-Howard H. Geuvers Radboud University Nijmegen, NL 21st Estonian Winter School in Computer Science Winter 2016 H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 1 / 65

2 Outline H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 2 / 65

3 Typed λ calculus as a basis for logic Aim: λ-term : type M : A program : data type proof : formula program : (full) specification Type Theory as an integrated system for proving and programming. Type Theory as a basis for proof assistants and interactive theorem proving. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 3 / 65

4 Simple type theory Simplest system: λ or simple type theory, STT. Just arrow types Typ := TVar (Typ Typ) Examples: (α β) α, (α β) ((β γ) (α γ)) Brackets associate to the right and outside brackets are omitted: (α β) (β γ) α γ Types are denoted by A, B,.... H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 4 / 65

5 Simple type theory à la Church Formulation with contexts to declare the free variables: x 1 : A 1, x 2 : A 2,..., x n : A n is a context, usually denoted by Γ. Derivation rules of λ (à la Church): x:a Γ Γ x : A Γ M : A B Γ M N : B Γ N : A Γ, x:a P : B Γ λx:a.p : A B Γ λ M : A if there is a derivation using these rules with conclusion Γ M : A H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 5 / 65

6 Examples λx : A.λy : B.x : A B A λx : A B.λy : B C.λz : A.y (x z) : (A B) (B C) A C λx : A.λy : (B A) A.y(λz : B.x) : A ((B A) A) A Not for every type there is a closed term of that type: (A A) A is not inhabited That is: there is no term M such that M : (A A) A. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 6 / 65

7 Typed Terms versus Type Assignment With typed terms also called typing à la Church, we have terms with type information in the λ-abstraction λx : A.x : A A Terms have unique types, The type is directly computed from the type info in the variables. With typed assignment also called typing à la Curry, we assign types to untyped λ-terms λx.x : A A Terms do not have unique types, A principal type can be computed using unification. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 7 / 65

8 Church vs. Curry typing The Curry formulation is especially interesting for programming: you want to write as little type information as possible; let the compiler infer the types for you. The Church formulation is especially interesting for proof checking: terms are created interactively; type structure is so intricate that type inference is undecidable (if you start from an untyped term). [ This lecture] H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 8 / 65

9 Formulas-as-Types (Curry, Howard) Recall: there are two readings of a judgement M : A 1 term as algorithm/program, type as specification: M is a function of type A 2 type as a proposition, term as its proof: M is a proof of the proposition A There is a one-to-one correspondence: typable terms in λ derivations in minimal proposition logic x 1 : B 1, x 2 : B 2,..., x n : B n M : A can be read as M is a proof of A from the assumptions B 1, B 2,..., B n. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 9 / 65

10 Example [A B C] 3 [A] 1 [A B] 2 [A] 1 B C B C 1 A C 2 (A B) A C 3 (A B C) (A B) A C λx:a B C.λy:A B.λz:A.x z (y z) : (A B C) (A B) A C H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 10 / 65

11 Example [x : A B C] 3 [z : A] 1 [y : A B] 2 [z : A] 1 x z : B C y z : B x z (y z) : C 1 λz:a.x z (y z) : A C 2 λy:a B.λz:A.x z (y z) : (A B) A C λx:a B C.λy:A B.λz:A.x z (y z) : (A B C) (A B) A C 3 Exercise: Give the derivation that corresponds to λx:c E.λy:(C E) E.y(λz.y x) : (C E) ((C E) E) E H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 11 / 65

12 Typed Combinatory Logic We have seen Combinatory Logic with the axioms for I, K and S. We now know their typed definition in λ : I := λx : A.x : A A K := λx : A.λy : B.x : A B A S := λx:a B C.λy:A B.λz:A.x z (y z) : (A B C) (A B) A C The three axiom schemes A A, A B A and (A B C) (A B) A C together with the derivation rule Modus Ponens is exactly Hilbert style minimal proposition logic. The typed CL terms are exactly the derivations in this logic. Modus Ponens corresponds with Application in CL Exercise: Show that the scheme A A is derivable. Cast in CL terminology: I can be defined in terms of S and K. To be precise: I = S K K. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 12 / 65

13 Computation = Cut-elimination β-reduction: (λx:a.m)p β M[x := P] Cut-elimination in minimal logic = β-reduction in λ. [A] 1 D 1 B 1 A B B [x : A] 1 D 1 M : B 1 λx:a.m : A B (λx:a.m)p : B D 2 A D 2 P : A D 2 A D 1 B β D 2 P : A D 1 M[x := P] : B H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 13 / 65

14 Example Proof of A A B, (A B) A B with a cut. [A] 1 A A B [A] 1 A B B A B (A B) A It contains a cut: a -i directly followed by an -e. B [A] 1 A A B [A] 1 A B A B A B H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 14 / 65

15 Example proof with term information [y : A] 1 p : A A B [y : A] 1 p y : A B [x : A] 1 p : A A [x : A] 1 p x : A B p x x : B p y y : B q : (A B) A λx:a.p x x : A B λy:a.p y y : A B q(λx:a.p x x) : A (λy:a.p y y)(q(λx:a.p x x)) : B Term contains a β-redex: (λx:a.p x x) (q(λx:a.p x x)) H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 15 / 65

16 Extension with other connectives Adding product types to λ. (Proposition logic with conjunction.) Γ M : A B Γ π 1 M : A With reduction rules Γ M : A B Γ π 2 M : B π 1 P, Q P π 2 P, Q Q Γ P : A Γ Q : B Γ P, Q : A B Similar rules can be given for sum-types A + B, corresponding to disjunction A B. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 16 / 65

17 Extension to predicate logic First order language: domain D, with variables x, y, z : D and possibly functions over D, e.g. f : D D, g : D D D. Rules for x:d.φ and x:d.φ. NB There are two kinds of variables: the first order variables (ranging over the domain D) and the proof variables (used as [local] assumptions of formulas). Formulas and domain are both types. What is the type of a predicate or relation? A predicate P is a map from D to the collection of types, P : D for P a predicate and R : D D for R a binary relation on D. We will have to make this more precise... H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 17 / 65

18 Idea of extending to Term rules for the -quantifier in predicate logic. Γ M : x:d.a if t : D Γ M t : A[x := t] With the usual β-reduction rule Γ M : A x not free in Γ Γ λx:d.m : x:d.a (λx:d.m)t M[x := t]. This conforms with cut-elimination (or detour elimination ) on logical derivations. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 18 / 65

19 Example Deriving irreflexivity from anti-symmetry AntiSym R := x, y:d.(rxy) (Ryx) Irrefl R := x:d.(rxx) Derivation in predicate logic: x, y:d.r x y R y x y:d.r x y R y x R x x R x x [R x x] 1 R x x [R x x] 1 1 R x x x:d.r x x H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 19 / 65

20 Example derivation in type theory, with terms H : x, y:d.r x y R y x H x : y:d.r x y R y x H x x : R x x R x x [H : R x x] 1 H x x H : R x x [H : R x x] 1 H x x H H : 1 λh :(R x x).h x x H H : R x x λx:a.λh :(R x x).h x x H H : x:d.r x x H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 20 / 65

21 Dependent Type Theory We have seen informally dependent types at work in the predicate logic example. Now: the rules With dependent types: everything depends on everything we can t first define the types, then the terms two universes: and is the universe of types We can t have :, so we have another universe: :. NB The Coq system uses Set and Prop for what I call and Type for what I call. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 21 / 65

22 First order Dependent Type theory, λp Derive judgements of the form Γ M : B Γ is a context x 1 : B 1, x 2 : B 2,..., x n : B n M and B are terms taken from the set of pseudoterms T ::= Var (T T) (λx:t.t) Πx:T.T Auxiliary judgement Γ denoting that Γ is a correct context. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 22 / 65

23 Derivation rules of λp s ranges over {, }. (base) (ctxt) Γ A : s Γ, x:a if x not in Γ (ax) Γ Γ : Γ (proj) Γ x : A if x:a Γ Γ A : Γ, x:a B : s (Π) Γ Πx:A.B : s Γ, x:a M : B Γ Πx:A.B : s (λ) Γ λx:a.m : Πx:A.B (conv) Γ M : B Γ A : s A = βη B Γ M : A Notation: write A B for Πx:A.B if x / FV(B). (app) Γ M : Πx:A.B Γ N : A Γ MN : B[x := N] H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 23 / 65

24 The use of the Π-type The Π rule allows to form two forms of function types. Γ, x:a B : s Γ A : (Π) Γ Πx:A.B : s Πx:A.B {f a : A(f a : B[x := a])} Write A B if x / FV(B) With s =, we can form D D and Πx:D.x = x, etc. With s =, we can form D D and D. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 24 / 65

25 Representation of PRED (minimal predicate logic) into λp Represent both the domains of the logic and the formulas as types. A :, P : A, R : A A, Now implication is represented as and is represented as Π: x:a.p x Πx:A.P x Intro and elim rules are just λ-abstraction and application H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 25 / 65

26 Example A :, R : A A λz:a.λh:(πx, y:a.r x y).h z z : Πz:A.(Πx, y:a.r x y) R z z This term is a proof of z:a.( x, y:a.r(x, y)) R(z, z) Exercise: Find terms of the following types (NB binds strongest) and (Πx:A.P x Q x) (Πx:A.P x) Πx:A.Q x (Πx:A.P x Πz.R z z) (Πx:A.P x) Πz:A.R z z). Also write down the contexts in which these terms are typed. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 26 / 65

27 Direct embedding of logic in type theory For λ and λp we have seen Direct representations of logic in type theory. Connectives each have a counterpart in the type theory: implication -type universal quantification -type Logical rules have their direct counterpart in type theory λ-abstraction -introduction application - elimination λ-abstraction -introduction application -elimination Context declares signature, local varibales and assumptions. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 27 / 65

28 LF embedding of logic in type theory Second way of interpreting logic in type theory De Bruijn: Logical framework encoding of logic in type theory. Type theory used as a meta system for encoding ones own logic. Choose an appropriate context Γ L, in which the logic L (including its proof rules) is declared. Context used as a signature for the logic. Use the type system as the meta calculus for dealing with substitution and binding. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 28 / 65

29 Direct and LF embedding proof formula direct embedding λx:a.x A A LF embedding imp intr A A λx:t A.x T (A A) Direct representation: One type system : One logic, Logical rules type theoretic rules LF encoding One type system : Many logics, Logical rules context declarations H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 29 / 65

30 Examples of the Deep embedding The encoding of logics in a logical framework is shown by three examples: 1 Minimal proposition logic 2 Minimal predicate logic (just {, }) 3 Untyped λ-calculus H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 30 / 65

31 Minimal propositional logic Fix the signature (context) of minimal propositional logic. prop : imp : prop prop prop Notation: A B for imp A B The type prop is the type of names of propositions. NB : A term of type propcan not be inhabited (proved), as it is not a type. We lift a name p : prop to the type of its proofs by introducing the following map: T : prop. Intended meaning of Tp is the type of proofs of p. We interpret p is valid by Tp is inhabited. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 31 / 65

32 Encoding of derivations To derive Tp we also encode the logical derivation rules imp intr : Πp, q : prop.(tp Tq) T(p q), imp el : Πp, q : prop.t(p q) Tp Tq. New phenomenon: Π-type: Πx:A.B(x) the type of functions f such that f a : B(a) for all a:a imp intr takes two (names of) propositions p and q and a term f : T p T q and returns a term of type T(p q) Indeed A A, becomes valid: imp intra A(λx:T A.x) : T(A A) Exercise: Construct a term of type T(A (B A)) H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 32 / 65

33 Signature of PROP in LF To encode proposition logic in LF we need a context (signature) Σ PROP : prop : : prop prop prop T : prop imp intr : (A, B : prop)(t A T B) T(A B) imp el : (A, B : prop)t(a B) T A T B. Desired properties of the encoding: Adequacy (soundness) of the encoding: PROP A = Σ PROP, a 1 :prop,..., a n :prop p : T A for some {a,..., a n } is the set of proposition variables in A. Faithfulness (or completeness) is the converse. It also holds, but more involved to prove. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 33 / 65

34 Minimal predicate logic over one domain A Signature: prop :, A :, T : prop f : A A, R : A A prop, : prop prop prop, imp intr : Πp, q : prop.(tp Tq) T(p q), imp el : Πp, q : prop.t(p q) Tp Tq. Now encode : takes a P : A prop and returns a proposition, so we add: : (A prop) prop H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 34 / 65

35 Minimal predicate logic over one domain A Signature: Σ PRED prop :, A :,. imp intr : Πp, q : prop.(tp Tq) T(p q), imp el : Πp, q : prop.t(p q) Tp Tq. Now encode : takes a P : A prop and returns a proposition, so: : (A prop) prop Universal quantification is translated as follows. x:a.(px) (λx:a.(px)) H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 35 / 65

36 Intro and Elim rules for : (A prop) prop, intr : ΠP:A prop.(πx:a.t(px)) T( P), elim : ΠP:A prop.t( P) Πx:A.T(Px). The proof of z:a( x, y:a.rxy) Rzz is now mirrored by the proof-term intr[ ]( λz:a.imp intr[ ][ ](λh:t( x, y:a.rxy). elim[ ]( elim[ ]hz)z) ) We have replaced the instantiations of the Π-type by [ ]. This term is of type T( (λz:a.imp( (λx:a.( (λy:a.rxy))))(rzz))) H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 36 / 65

37 Intro and Elim rules for : (A prop) prop, intr : ΠP:A prop.(πx:a.t(px)) T( P), elim : ΠP:A prop.t( P) Πx:A.T(Px). The proof of z:a( x, y:a.rxy) Rzz is now mirrored by the proof-term intr[ ]( λz:a.imp intr[ ][ ](λh:t( x, y:a.rxy). elim[ ]( elim[ ]hz)z) ) Exercise: Construct a proof-term that mirrors the (obvious) proof of x(p x Q x) x.p x x.q x H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 37 / 65

38 Untyped λ-calculus Signature Σ lambda : D : ; app : D (D D); abs : (D D) D. A variable x in λ-calculus becomes x : D in the type system. The translation [ ] : Λ Term(D) is defined as follows. [x] = x; [PQ] = app [P] [Q]; [λx.p] = abs (λx:d.[p]). Examples: [λx.xx] := abs(λx:d.app x x) [(λx.xx)(λy.y)] := app(abs(λx:d.app x x))(abs(λy:d.y)). H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 38 / 65

39 Introducing β-equality Notation P = Q for eq P Q. Rules for proving equalities. refl : Πx:D.x = x, eq:d D. sym : Πx, y:d.x = y y = x, trans : Πx, y, z:d.x = y y = z x = z, mon : Πx, x, z, z :D.x = x z = z (app z x) = (app z x ), xi : Πf, g:d D.(Πx:D.(fx) = (gx)) (abs f ) = (abs g), beta : Πf :D D.Πx:D.(app(abs f )x) = (fx). H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 39 / 65

40 Properties of λp Uniqueness of types If Γ M : σ and Γ M : τ, then σ= βη τ. Subject Reduction If Γ M : σ and M βη N, then Γ N : σ. Strong Normalization If Γ M : σ, then all βη-reductions from M terminate. Proof of SN is by defining a reduction preserving map from λp to λ. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 40 / 65

41 Decidability Questions Γ M : σ? TCP Γ M :? TSP Γ? : σ TIP For λp: TIP is undecidable TCP/TSP: simultaneously with Context checking H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 41 / 65

42 Curry-Howard-de Bruijn logic type theory formula type proof term detour elimination β-reduction proposition logic simply typed λ-calculus predicate logic dependently typed λ-calculus λp intuitionistic logic... + inductive types higher order logic... + higher types and polymorphism classical logic... + exceptions H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 42 / 65