Advanced Lambda Calculus Lecture 5

Transcription

1 Advanced Lambda Calculus Lecture 5

2 The fathers Alonzo Church ( ) as mathematics student at Princeton University (1922 or 1924) Haskell B. Curry ( ) as BA in mathematics at Harvard (1920)

3 Untyped lambda terms λ Λ, Simple types Free type algebras λ A A, Recursive types Type algebras λ A = A,, = Subtyping Type structures λ S S,, Intersection types Intersection type structures λ S S,,,,

4 Curry s contributions Curry: Russell paradox as fixed point of the operator Not Idea: For a A write Aa Then as {x P[x]} we can take λx.p[x] Indeed, we get the intended interpretation a {x P[x]} becomes (λx.p[x])a = P[a] Taking R = {x x / x} = λx. (xx) we get hence RR (RR) r.[rr (rr)] Note that RR (λx. (xx))(λx. (xx)) = Y( ) Typing of Whitehead-Russell was transformed into functionality Γ FABM Γ B(MN) Γ AN Γ,Ax BM Γ FAB(λx.M)

5 Curry s contributions Curry: Russell paradox as fixed point of the operator Not Idea: For a A write Aa Then as {x P[x]} we can take λx.p[x] Indeed, we get the intended interpretation a {x P[x]} becomes (λx.p[x])a = P[a] Taking R = {x x / x} = λx. (xx) we get hence RR (RR) r.[rr (rr)] Note that RR (λx. (xx))(λx. (xx)) = Y( ) Typing of Whitehead-Russell was transformed into functionality Γ M : A B Γ MN : B Γ N : A Γ,x : A M : B Γ λx.m : (A B)

6 An increasing chain of systems λ A λ A = λ S λ S Γ,x : A x : A λ A Γ M : (A B) Γ (MN) : B Γ N : A Γ,x : A M : B Γ (λx.m) : (A B) λ A = λ S Γ M : A A = B Γ M : B Γ M : A A B Γ M : B Γ M : A B Γ M : A B Γ M : A Γ M : B λ S Γ M : A Γ M : B Γ M : A B Γ M :

7 An increasing chain of systems λ A λ A = λ S λ S λ S 8 λ S 8 >< >< >: λ A = 8 >< >: λ A 8 >< >: Γ, x : A x : A Γ M : (A B) Γ N : A Γ, x : A M : B Γ (MN) : B Γ (λx.m) : (A B) Γ M : A A = B Γ M : B Γ M : A A B Γ M : B Γ M : A B Γ M : A B Γ M : A Γ M : B >: Γ M : A Γ M : Γ M : B Γ M : A B

8 Examples λ A λxy.xyy : (A A B) A B λ A = λx.xx : A if A = A B in A (λx.xx)(λx.xx) : B λ S λx.xx : A B only; if A A B in S λx.xx : (A B) B if A B A λ S λx.xx : A (A B) B KIΩ : A A where Ω (λx.xx)(λx.xx) as Ω :

9 Simply typed λ-calculus: λ o Simple types from ground type 0 T = 0 T T Λ(A): λ-terms of type A. Write Λ = A T Λ(A) x A Λ(A) M Λ(A B),N Λ(A) (MN) Λ(B) M Λ(B) (λx A.M) Λ(A B) Default equality = βη preserves types (λx A.M)N = M[x A := N] β-conversion λx A.Mx A = M η-conversion

10 Standard terms I A λx A.x Λ(A A) K A,B λx A y B.x Λ(A B A) S A,B,C λx A B C y A B z A.xz(yz) Λ((A B C) (A B) A C) Curry: Hey, these are tautologies Curry-Howard correspondence From these ( many) terms all others can be defined applicatively Prop. (i) For M Λ (A) one has M : A (ii) For M Λ in β-nf such that M : A there is a unique M A Λ(A) such that M A M (iii) For open M not in β-nf (ii) fails: KIy : A A (iv) Even for closed M not in β-nf (ii) fails: (λx.xi)(λy.i) : A A The counter-examples in (iii), (iv) are due to the presence or creation of a K-redex For normal M one can identify M : A and M A Λ(A)

11 Type structures Def. Let M = {M(A)} A T be a family of non-empty sets (i) M is called a type structure for λ o if M(A B) M(B) M(A) Here Y X denotes the collection of set-theoretic functions {f f : X Y } (ii) Let M be provided with application operators (M, ) = ({M(A)} A T, { A,B } A,B T ) A,B : M(A B) M(A) M(B). A typed applicative structure is such an (M, ) satisfying extensionality: f,g M(A B) [[ a M(A) f A,B a = g A,B a] f = g]. Prop. The notions type structure and typed applicative structure are equivalent

12 Full type structures Def. Given a set X. The full type structure over X M X = {X(A)} A T where X(A) is defined inductively as follows X(0) X; X(A B) X(B) X(A), the set of functions from X(A) into X(B) Def. M n M {1,,n} M(A B) M(A) M(B) M(2) M(1) = X X M(0) = X F M FM Partial view of M = M X F, M Λ ch finite at each level A if X is finite

13 Full type structures Def. Given a set X. The full type structure over X M X = {X(A)} A T where X(A) is defined inductively as follows X(0) X; X(A B) X(B) X(A), the set of functions from X(A) into X(B) Def. M n M {1,,n} M(A B) [F ] M(A) [M ] M(B) [FM ] M(2) M(1) = X X M(0) = X Partial view of M = M X F, M Λ ch finite at each level A if X is finite

14 Semantics in full type structures Let ρ : Var T M(A) be a valuation in M X : we require ρ(x A ) M(A) For M Λ (A) we define [M ] ρ M(A) [x A ] ρ ρ(x A ) [MN ] ρ [λx A.M ] ρ [M ] ρ [N ] ρ λd X(A).[M ] ρ(x A :=d) where ρ(x A : = d) = ρ with ρ (x A ) d ρ (y B ) ρ(y B ) if y B x A Define M X = M = N ρ [M ] ρ = [N ] ρ [M ] = [N ] if M,N Λ

15 Application We know that N = N, +,, 0, 1 can be λ-defined This is not so for the rational numbers Q = Q, +,,, :, 0, 1 1. Formulate this precisely. 2. Prove this. [Hint. Use that the homomorphic image of a field is the field itself; each M X (A) is finite for X finite.]

16 M 1, M 2 Def. Th(M)= {M = N M,N Λ & M = M = N} Th(M 1 ) is inconsistent: all terms of the same type are equated M 2 = c 1 = c 3 : M 2 = c 1 = c 2 : Exercises M 5 = c 4 = c 64 M 6 = c 4 = c 64 M 6 = c 5 = c 65 [Hint. Show that for i, j N one has M n = c i = c j i = j [i, j n 1 & i j(mod lcm{1,, n})]. For a M n (0), f M n (1) define the trace of a under f as {f i (a) i N}, directed by G f = {(a, b) f(a) = b}, which by the pigeonhole principle is lasso-shaped. Consider the traces of 1 under the functions f n, g m with 1 m n, where f n (k) = k + 1, if k < n, = n, if k = n, and g m (k) = k + 1, if k < m, = 1, if k = m, = k, else.]

17 Partial semantics (Friedman [1975]) Let M be a typed applicative structure A partial valuation in M is a family ρ = {ρ A } A T of partial maps ρ A : Var(A) M(A) The partial semantics [ ] M ρ : Λ (A) M(A) under ρ is [x A ] M ρ ρ A (x) [PQ] M ρ [λx A.P ] M ρ [P ] M ρ [Q]M ρ λd M(A).[P ] M ρ[x:=d] Often we write [M ] ρ for [M ] M ρ The expression [M ] ρ may not always be defined, even if ρ is total The problem arises with [λx.p ] ρ when λd M(A).[P ] M ρ[x:=d] M(B)M(A) M(A B) If [λx.p ] ρ exists it is uniquely defined

18 Typed λ-models A typed λ-model is a type structure M such that [M ] ρ is defined for all A T, M Λ(A), and ρ such that FV(M) dom(ρ) Examples of typed λ-models M X M βη : full type structures : open typed terms modulo βη-equality M[C] : closed typed term models modulo extensionality where C is a set of typed constants such that M[C](0) In M[C] one can define and show it works [M ] ρ = [M[ x := ρ( x)]]

19 Five Easy Pieces (Statman) There are only five Th(M[C]) coming from C 1 = {c 0,d 0 } C 2 = {c 0,f 1 } C 3 = {c 0,f 1,g 1 } C 4 = {c 0,Φ } C 5 = {c 0,b } There is a sixth, the inconsistent theory, coming from C 0 = {c 0 } One has Th(M[C 5 ]) Th(M[C 4 ]) Th(M[C 3 ]) Th(M[C 2 ]) Th(M[C 1 ]) Th(M[C 0 ]) Th(M[C 5 ]) = {M = N M = βη N} minimal theory Th(M[C 1 ]) is the unique maximally consistent theory consisting of all consistent equations together M[C 1 ] is the minimal model, with decidable equality (Loader)

20 M[C] Let C be a set of typed constants Examples C 0 = {c 0 }, C 1 = {c 0,d 0 } Define Λ [C] as the set of closed typed λ-terms built up from C Now Λ [C] modulo extensionality will be considered as a term model For M,N M[C](A) define M ext C N M = βη N if A = 0 P M[C](B).[MP ext C NP] if A = B C Then M ext C N P M[C].[M P = βη N P] Thm. M[C] = Λ [C]/ ext C is an extensional λ-model not trivial at all, we need M[C] = M = N M[C] = FM = FN & M[C] = λx.m = λx.n Exercises 1. M(c 0 ) = M 1, where M[c 0 ] = M[{c 0 }] 2. M[c 0, d 0 ] = c 1 = c 2 3. M[c 0, d 0 ] = M 2

21 Observational equality For M,N Λ [C](A) define M obs C N F : (A 0).FM = βη FN Remark M obs C N FM obs C FN M obs M ext Thm. M,N.[M obs C C N M ext C N C N Z.MZ ext C NZ N M ext Cor. M[C] is an (extensional) λ-model C N] non-trivial

22 Logical relations on M[C] A relation on M[C] is a family R = {R A } A T with R A M[C](A) n A relation is logical if for all A,B T and all M M[C](A B) n R A B (M 1,,M n ) N 1 M[C](A) N n M[C](A) Thus a logical relation is fully determined by R 0 [R A (N 1,,N n ) R B (M 1 N 1,,M n N n ) Prop. Suppose ext C is logical on M[C]. Then for all M,N M[C] M ext C Proof. Only ( ) is interesting N M C obs N Assume M C extn and F M[C](A 0) towards FM = βη FN Trivially F ext C F FM C ext FN, as ext C is logical FM = βη FN, as the type is 0

23 Analysis of ext C i (1) It remains to show that the ext C i are logical Def. BE is the logical relation on M[C] determined by BE 0 (M,N) M = βη N Lemma 1. Suppose BE(c, c) for c C. Then M Λ[C].BE(M, M) Proof. By the usual arguments for logical relations.

24 Analysis of C ext i (2) Lemma 2. Suppose BE(c, c) for all c C. Then ext C Proof. By Lemma 1 one has for all M M[C] is BE and hence logical BE(M, M) (0) It follows that BE is an equivalence relation on M[C]. We claim that for all F, G M[C](A) BE A (F, G) F ext C G, By induction on the structure of A. Case A = 0. By definition. Case A = B C, then ( ) BE B C (F, G) BE C (FP, GP), for all P M[C](B), ( ) since P ext C P and hence by the IH BE B (P, P) FP ext C GP, for all P M[C] by the IH F ext C G, by definition. F ext C G FP ext C GP, for all P M[C], BE C (FP, GP) (1) by the induction hypothesis. In order to prove BE B C (F, G), assume BE B (P, Q) towards BE C (FP, GQ). Well, since also BE B C (G, G), by (0), we have BE C (GP, GQ). (2) It follows from (1) and (2) and the transitivity of BE (which on this type is the same as by the IH) that BE C (FP, GQ) indeed. ext C By the claim ext is BE and therefore ext is logical. C C

25 Analysis of ext C i (3) Lemma 3. BE(M, M) holds for M M[C] of types 0, 1, Proof. Easy. Lemma 4. Let c = c 3 M[C]. Suppose Then BE A 0 (c, c) F, G M[C](2)[F ext C G F = βη G] Proof. Let c be given. Then for F, G M[C](2), P M[C](1) one has BE(F, G) FP = βη GP by Lemma 3 F ext C G F = βη G by assumption Therefore we have by definition BE(c, c) Last morgage For every F, G M[C](2) one has cf = βη cg F ext C 4 G F = βη G. We must show [ h M[C](1).Fh = βη Gh] F = βη G. (1)

26 Analysis of terms of given type λφx.x, λφx.φ(λf.x), λφx.φ(λf.f x), λφx.φ(λf.f(φ(λg.g(f x)))), λφx.φ(λf 1.w {f1 }x), λφx.φ(λf 1.w {f1 }Φ(λf 2.w {f1,f 2 }x)), ; λφx.φ(λf 1.w {f1 }Φ(λf 2.w {f1,f 2 } Φ(λf n.w {f1,,f n }x) )) w {f1 }, w {f1,f 2 },, w {f1,,f n } 3 o o f λφ 3 λx o o Φ λf 1 2 x Let h m λx.φ(λf.f m x) = f m : M[C](1) Claim F,G M[C](2) m N.[Fh m = Gh m F = βη G] See book p