Lambda Calculus. Andrés Sicard-Ramírez. Semester Universidad EAFIT

Transcription

1 Lambda Calculus Andrés Sicard-Ramírez Universidad EAFIT Semester

2 Bibliography Textbook: Hindley, J. R. and Seldin, J. (2008). Lambda-Calculus and Combinators. An Introduction. Cambridge University Press. Barendregt, Henk and Barendsen, Erik (2000). Introduction to Lambda Calculus. Revisited edition, Mar Barendregt, H. P. (2004). The Lambda Calculus. Its Syntax and Semantics. Revised edition, 6th impression. Vol Studies in Logic and the Foundations of Mathematics. Elsevier. Paulson, Lawrence C. (2000). Foundations of Functional Programming. Lecture notes. url: (visited on 09/12/2016). Introduction 2/49

3 What is the Lambda Calculus? Invented by Alonzo Church (around 1930s). The goal was to use it in the foundation of mathematics. Intended for studying functions and recursion. Computability model. Model of untyped functional programming languages. Lambda Calculus 3/49

4 Introduction λ-calculus is a collection of several formal systems λ-notation Anonymous functions Currying Lambda Calculus 4/49

5 Introduction Definition (λ-terms) The set of λ-terms is inductively defined by v V v λ-terms (atom) c C c λ-terms (atom) M, N λ-terms (MN) λ-terms (application) M λ-terms, x V (λx.m) λ-terms (abstraction) where V/C is a set of variables/constants. Lambda Calculus 5/49

6 Introduction Conventions and syntactic sugar Example M N means the syntactic identity Application associates to the left MN 1... N k means (...((MN 1 )N 2 )...N k ) Application has higher precedence λx.p Q means (λx.(p Q)) λx 1 x 2... x n.m means (λx 1.(λx 2.(... (λx n.m)... ))) (λxyz.xz(yz))uvw ((((λx.(λy.(λz.((xz)(yz)))))u)v)w). Lambda Calculus 6/49

7 Term-Structure and Substitution Substitution ([N/x]M) The result of substituting N for every free occurrence of x in M, and changing bound variables to avoid clashes. [N/x]x N (1) [N/x]a a for all atoms a x (2) [N/x](P Q) ([N/x]P [N/x]Q) (3) [N/x](λx.P ) λx.p [N/x](λy.P ) λy.p y x, x F V (P ) (5) [N/x](λy.P ) λy.[n/x]p y x, x F V (P ), (6) y F V (N) [N/x](λy.P ) λz.[n/x][z/y]p y x, x F V (P ), (7) y F V (N) where in the last equation, z is chosen to be a variable F V (NP ). Lambda Calculus 7/49 (4)

8 Term-Structure and Substitution Example [(λy.vy)/x](y(λv.xv)) y(λz.(λy.vy)z) (with z v, y, x). Lambda Calculus 8/49

9 Term-Structure and Substitution α-conversion or changed of bound variables Replace λx.m by λy.[y/x]m (y F V (M)). α-congruence (P α Q) P is changed to Q by a finite (perhaps empty) series of α-conversions. Example Whiteboard. Theorem The relation α is an equivalence relation. Lambda Calculus 9/49

10 Beta-Reduction β-contraction ( 1β ) (λx.m)n: β-redex [N/x]M: contractum (λx.m)n 1β [N/x]M P 1β Q: Replace an occurrence of (λx.m)n in P by [N/x]M. Example Whiteboard. β-reduction (P β Q) P is changed to Q by a finite (perhaps empty) series of β-contractions and α-conversions. Example (λx.(λy.yx)z)v β zv. Lambda Calculus 10/49

11 Beta-Reduction β-normal form A term which contains no β-redex. β-nf: The set of all β-normal forms. Example Whiteboard. Lambda Calculus 11/49

12 Beta-Reduction Theorem (The Church-Rosser theorem for β (the diamond property)) P P β M P β N T.M β T N β T M T N Corollary If P has a β-normal form, it is unique modulo α ; that is, if P has β-normal forms M and N, then M α N. Proof Whiteboard. Lambda Calculus 12/49

13 Beta-Equality β-equality or β-convertibility (P = β Q) Exist P 0,..., P n such that P 0 P P n Q ( i n 1)(P i 1β P i+1 P i+1 1β P i P i α P i+1 ) Theorem (Church-Rosser theorem for = β ) Proof Whiteboard. P = β Q T.P β T Q β T Lambda Calculus 13/49

14 Beta-Equality Corollary If P, Q β-nf and P = β Q, then P α Q. Corollary The relation = β is non-trivial (not all terms are β-convertible to each other). Proof Whiteboard. Lambda Calculus 14/49

15 What is the Combinatory Logic? Invented by Moses Schönfinkel (1920) and Haskell Curry (1927). Intended for clarify the role of quantified variables. Idea: To do logic and mathematics without use bound variables. Combinators: Operators which manipulate expressions by cancellation, duplication, bracketing and permutation. Combinatory Logic 15/49

16 Introduction Example (Informally) The commutative law for addition xy.x + y = y + x, can be written as A = CA, where Axy represents x + y and C is a combinator with the property Cfxy = fyx. Combinatory Logic 16/49

17 Introduction Example (Combinators) B f g x = f (g x) B f g x = g (f x) I x = x K x y = x S f g x = f x (g x) W f x = f x x A composition operator A reversed composition operator The identity operator A projection operator A stronger composition operator A doubling operator Combinatory Logic 17/49

18 Introduction Definition (CL-terms) The set of CL-terms is inductively defined by v V v CL-terms c C c CL-terms X, Y CL-terms (XY ) CL-terms where V : Set of variables C = {I, K, S,... } : Set of atomic constants Combinatory Logic 18/49

19 Introduction F V (X): The set of variables occurring in X. Atoms, basic combinators and combinator A atom is a variable or atomic constant. The basic combinators are I, K and S. A combinator is a CL-term whose only atoms are basic combinators. Substitution ([U/x]Y ) The result of substituting U for every occurrence of x in Y : [U/x]x U [U/x]a a [U/x](V W ) ([U/x]V [U/x]W ) for all atoms a x Combinatory Logic 19/49

20 Weak Reduction Weak redex The CL-terms I X, K X Y and S X Y Z. Weak contraction (U 1w V ) Replace an occurrence of a weak redex in U using: Weak reduction (U w V ) I X by X, K X Y by X, S X Y Z by X Z (Y Z). U is changed to V by a finite (perhaps empty) series of weak contractions. Weak normal form A CL-term which contains no weak redex. Combinatory Logic 20/49

21 Weak Reduction Example (whiteboard) B S(KS)K. Then BXY Z w X(Y Z). W SS(KI). Then WXY w XY Y. WWW w WWW w Theorem (Church-Rosser theorem for w ) P w M P w N T.M w T N w T Corollary (Uniqueness of nf) A CL-term can have at most one weak normal form. Combinatory Logic 21/49

22 Weak Equality Weak equality or weak convertibility (X = w Y ) Exist X 0,..., X n such that X 0 X X n Y ( i n 1)(X i 1w X i+1 X i+1 1w X i ) Theorem (Church-Rosser theorem for = w ) Corollary X = w Y T.X w T Y w T If X and Y are distinct weak normal forms, them X w Y ; in particular S w K. Hence = w is non-trivial in the sense that not all terms are weakly equal. Combinatory Logic 22/49

23 Fixed-Point Combinators Idea For every term F there is a term X such F X = β X. The term X is called a fixed-point of F. Theorem F X.F X = β X. Proof. Let W λx.f (xx), and let X W W. Then X (λx.f (xx))w = β F (W W ) F X Combinatory Logic 23/49

24 Fixed-Point Combinators Fixed-point combinator A fixed-point combinator is any combinator Y such YF = β F (YF ), for all terms F. Theorem (Turing) The term Y UU, where U λux.x(uux) is a fixed-point combinator. Proof Whiteboard. Theorem (Curry and Rosenbloom) The term Y λf.v V, where V λx.f(xx) is a fixed-point combinator. Proof Whiteboard. Combinatory Logic 24/49

25 Fixed-Point Combinators Corollary For every term Z and n 0, the equation xy 1... y n = Z can be solved for x. That is, there is a term X such that Xy 1... y n = β [X/x]Z. Proof X Y(λxy 1... y n.z) (whiteboard). Combinatory Logic 25/49

26 Reduction Strategies Idea Proving that a given term has no normal form. Contraction (X R Y ) X R Y : R is an redex in X and Y is the result of contracting R in X. Example (λx.(λy.yx)z)v (λy.yx)z (λx.zx)v. Combinatory Logic 26/49

27 Reduction Strategies Reduction A reduction ρ is a finite or infinite sequence of contractions separated by α-conversions X 1 R1 Y 1 α X 2 R2... Question Given an initial term X, there is some way of choosing a reduction that will terminate if X has a normal form? Combinatory Logic 27/49

28 Reduction Strategies Outermost (maximal) redex A redex is called outermost iff it is not contained in any other redex. Normal-order evaluation (left-most reduction) (call-by-name) In every contraction, the contracted redex is the leftmost outermost. Eager evaluation (call-by-value) A redex is reduced only when its right hand side has reduced to a canonical form (variable, constant or lambda abstraction). Example (Whiteboard). 1. (λy.a)ω, where Ω (λx.xx)(λx.xx) 2. (λx.xx)((λy.yy)(λz.zz)) Combinatory Logic 28/49

29 Reduction Strategies Theorem (Standardization theorem (left-most reduction theorem)) If a term X has a normal form X, then the normal-order evaluation of X is finite and ends at X. Combinatory Logic 29/49

30 Lambda Calculus and Inconsistencies Curry paradox (λ-calculus + logic) Rosser (1984, p. 340) descriptions of the paradox: Suppose we have two familiar logical principles: Φ Φ Applying (10) to the second Φ gives P P (8) (P (P Q)) (P Q) (9) together with modus ponens (if P and P Q, then Q). We undertake to prove an arbitrary proposition A. We construct a Φ such that Φ = Φ A (10) To do this, we take F = λx(x A) in the fixed-point theorem. By (8), we get Φ (Φ A) By (9) and modus ponens, we get Φ A By (10) reversed, we get By modus ponens and the last two formulas, we get A Lambda Calculus and Inconsistencies 30/49 Φ

31 Lambda Calculus and Inconsistencies Rusell paradox (λ-calculus + set theory) See (Paulson 2000, 4.6). Lambda Calculus and Inconsistencies 31/49

32 Encoding Data in the Lambda Calculus (From (Paulson 2000, Ch. 3)) Booleans true λxy.x false λxy.y if λpxy.pxy where if true M N = β M if false M N = β N Encoding Data in the Lambda Calculus 32/49

33 Encoding Data in the Lambda Calculus Ordered pairs pair λxyf.fxy fst λp.p true snd = λp.p false where fst (pair M N) = β M snd (pair M N) = β N Encoding Data in the Lambda Calculus 33/49

34 Encoding Data in the Lambda Calculus Natural numbers Notation: X n Y X(X(... (X Y )... )) if n 1, }{{} n X s X 0 Y Y. The Church numerals: n λfx.f n x Encoding Data in the Lambda Calculus 34/49

35 Encoding Data in the Lambda Calculus Some operations: add λmnfx.mf(nfx) mult λmnfx.m(nf)x iszero λn.n(λx.false) true where add m n = β m + n mult m n = β m n iszero 0 = β true iszero n + 1 = β false Encoding Data in the Lambda Calculus 35/49

36 Recursion Using Fixed-Points Example (Informally (Peyton Jones 1987)) fac λn.if n = 0 then 1 else n fac (n 1) fac λn.(... fac... ) fac (λfn.(... f... )) fac h λfn.(... f... ) - - not recursive! fac h fac - - fac is a fixed-point of h! fac Yh Encoding Data in the Lambda Calculus 36/49

37 Recursion Using Fixed-Points Example (cont.) fac 1 Yh 1 = β h(yh) 1 (λfn.(... f... ))(Yh) 1 β if 1 = 0 then 1 else 1 (Yh 0) β 1 (Yh 0) = β 1 (h(yh) 0) 1 ((λfn.(... f... ))(Yh)0) β 1 (if 0 = 0 then 1 else 1 (Yh ( 1))) β 1 1 β 1 Encoding Data in the Lambda Calculus 37/49

38 Representing the Computable Functions Representability Let ϕ a partial function ϕ : N n N. A term X represents ϕ iff Example ϕ(m 1,..., m n ) = p Xm 1... m n = β p, ϕ(m 1,..., m n ) does not exits Xm 1... m n has no nf. The successor function succ(n) = n + 1 is represented by succ λnfx.f(nfx) Theorem (Representation of Turing-computable functions) In λ-calculus every Turing-computable function can be represented by a combinator. Encoding Data in the Lambda Calculus 38/49

39 Undecidability Gödel number #M #x i = 2 i #(λx i.m) = 3 i 5 #M #(MN) = 7 #M 11 #N Notation: M = #M Theorem (Double fixed-point theorem) F X.F X = β X Proof Whiteboard. Encoding Data in the Lambda Calculus 39/49

40 Undecidability Theorem (Rice s theorem for the λ-calculus) Let A λ-terms such as A is non-trivial (i.e. A, A λ-terms). Suppose that A is closed under = β (i.e. M A, M = β N N A). Then A is no recursive, that is #A = {#M M A} is not recursive. Proof Whiteboard (see (H. Barendregt 1990)). Theorem The set NF = {M M has a normal form} is not recursive. Proof. The set NF is not trivial and it is closed under = β. Encoding Data in the Lambda Calculus 40/49

41 ISWIM: Lambda Calculus as a Programming Language ISWIM: If you See What I Mean Landin (1966) ISWIM 41/49

42 ISWIM Features (From (Paulson 2000, Ch. 3)) Simple declaration let x = M in N (λx.n)m Example let n = 0 in succ n let m = 0 in (let n = 1 in add m n) Function declaration let fx 1... x k = M in N (λf.n)(λx 1... x k.m) Example let succ n = λfx.f(nfx) in succ 0 ISWIM 42/49

43 ISWIM Features Recursive declaration letrec fx 1... x k = M in N (λf.n)(y(λfx 1... x k.m)) Example letrec fac n = if (n == 0) 1 (n fac(n 1)) in fac 0 Pairs (M, N) : pair constructor fst, snd : projections let λ(x, y).e λz.(λxy.e)(fst z)(snd z) Example let (x, y) = (2, 3) in add x y ISWIM 43/49

44 The Formal Theory λβ of β-equality Formulas M = N, where M, N λ-terms. Axiom-schemes (α) λx.m = λy.[y/x]m if y F V (M), (β) (λx.m)n = [N/x]M, (ρ) M = M. Rules of inference M = M (µ) NM = NM M = M λx.m = λx.m (ξ) M = N (σ) N = M M = M (ν) MN = M N M = N N = P (τ) M = P Formal Theory 44/49

45 The Formal Theory λβ of β-equality Deductions If there is a deduction of B from the assumptions A 1,..., A n in λβ) is denoted by λβ, A 1,..., A n B. Theorems If the formula B is probable in λβ) is denoted by Remark λβ B. λβ is a equational theory and it is a logic-free theory (there are not logical connectives or quantifiers in its formulae). Formal Theory 45/49

46 The Formal Theory λβ of β-equality Example Let M and N two closed terms (λx.(λy.x))m = [M/x]λy.x λy.m (ν) (λx.(λy.x))mn = (λy.m)n (λy.m)n = [N/y]M M (τ) (λx.(λy.x))mn = M That is to say, λβ (λxy.x)mn = M. Theorem M = β N λβ M = N. Formal Theory 46/49

47 The Formal Theory λβ of β-reduction Similar to the formal theory of β-equality, but: 1. Formulas: M β N. 2. To change = by β. 3. Remove the rule (σ). Theorem M β N λβ M β N. Remark Formal theories for combinatory logic. Remark λβ is not a first-order theory. Formal Theory 47/49

48 References Barendregt, H. P. (2004). The Lambda Calculus. Its Syntax and Semantics. Revised edition, 6th impression. Vol Studies in Logic and the Foundations of Mathematics. Elsevier (cit. on p. 2). Barendregt, Henk (1990). Functional Programming and Lambda Calculus. In: Handbook of Theoretical Computer Science. Ed. by van Leeuwen, J. Vol. B. Formal Models and Semantics. MIT Press. Chap. 7. doi: /B (cit. on p. 40). Barendregt, Henk and Barendsen, Erik (2000). Introduction to Lambda Calculus. Revisited edition, Mar (cit. on p. 2). Hindley, J. R. and Seldin, J. (2008). Lambda-Calculus and Combinators. An Introduction. Cambridge University Press (cit. on p. 2). Landin, P. J. (1966). The Next 700 Programming Languages. In: Communications of the ACM 9.3, pp (cit. on p. 41). Paulson, Lawrence C. (2000). Foundations of Functional Programming. Lecture notes. url: (visited on 09/12/2016) (cit. on pp. 2, 31, 32, 42).

49 References Peyton Jones, Simon L. (1987). The Implementation of Functional Programming Languages. Prentice-Hall International (cit. on p. 36). Rosser, J. Barkley (1984). Highlights of the History of Lambda-Calculus. In: Annals of the History of Computing 6.4, pp doi: /MAHC (cit. on p. 30).