Lambda-Calculus (I) 2nd Asian-Pacific Summer School on Formal Methods Tsinghua University, August 23, 2010

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1 Lambda-Calculus (I) 2nd Asian-Pacific Summer School on Formal Methods Tsinghua University, August 23, 2010

2 Plan computation models lambda-notation bound variables conversion rules reductions normal forms numeral systems lambda-definability Barendregt, Henk, The Lambda Calculus. Its Syntax and Semantics, Elsevier, 2nd edition, Barendregt, Henk; Dezani, Mariangiola, Lambda calculi with Types, 2010.

3 Models of computation

4 Computation models [machines] automata theory -- Turing machines [character strings] formal grammars, Thue systems, Post [numbers] Kleene recursive functions theory [terms] Church lambda-calculus, term rewriting systems Applications to logic [cut elimination] 2nd order arithmetic -- Howard, Girard [higher order dependent types] HOL, Isabelle, Coq -- Coquand, Huet

5 Computing with terms (2 + 3) ((2 + 3) + 4) (4 + 5)...

6 Computing with terms (λx. x + 1) (λx.2 x + 2) (λf.f 3)(λx. x + 2) (λx. x + 2) (λf.λx.f (f x))(λx.x + 2)...

7 Computing with terms (λf.λx.f (f x))(λx.x + 2)3...

8 Computation model define a minimum set no instructions, no states, only expressions no arithmetic just a calculus of functions functions applied to functions functions as results interesting?

9 λ-calculus

10 The lambda-calculus Lambda terms M, N, P ::= x, y, z,... (variables) ( λx.m ) (M as function of x) ( M N ) (M applied to N) c, d,... (constants ) Calculations reductions ((λx.m)n) M{x := N}

11 Abbreviations MM 1 M 2 M n for ( ((MM 1 )M 2 ) M n ) (λx 1 x 2 x n. M) for (λx 1.(λx 2. (λx n. M) )) external parentheses and parentheses after a dot may be forgotten Exercice 1 Write following terms in long notation: λx.x, λx.λy.x, λxy.x, λxyz.y, λxyz.zxy, λxyz.z(xy), (λx.λy.x)mn, (λxy.x)mn, (λxy.y)mn, (λxy.y)(mn)

12 Examples (λx.x)n N (λf.f N)(λx.x) (λx.x)n N (λx.xx)(λx.xn) (λx.xn)(λx.xn) (λx.xn)n NN (λx.xx)(λx.xx) (λx.xx)(λx.xx) Y f =(λx.f (xx))(λx.f (xx)) f ((λx.f (xx))(λx.f (xx))) = f (Y f ) f (Y f ) f (f (Y f )) f n (Y f )

13 Recapitulation calculus is more complex than expected looping expressions!! recursion operator seems definable when termination? consistency? computing power?

14 Abstract syntax The syntax of lambda-terms can be abstracted as: M, N, P ::= x, y, z,... (variables) x λx ( λx.m ) (M as function of x) M ( M N ) (M applied to N) M N c, d,... (constants ) c

15 Abstract syntax Example: (λx.(λy.λx.y x)(λz.z x))x y is y λx x λy λz λx

16 Bound variables (λx.(λy.λx.y x)(λz.z x))x y (rightmost x, y are free) Exercice 2 Show binders of bound variables in (λf.(λx.f (xx))(λx.f (xx)))(λx.λy.x) (λf.(λx.f (xx))(λx.f (xx)))(λf xy.x(f y)) (λf.f ((λx.x)3))(λx.λy.x)

17 Bound variables (λy.λx.y)x λx.x incorrect (dynamic binding: Lisp) (λy.λx.y)x λx.x correct (lexical binding: Scheme) Exercice 2bis Why Lisp is consistent?

18 Bound variables (λy.λx.y)x λx.x (λy.λx.y)x = α (λy.λx.y)x λx.x renaming of bound variables names of bound variables are not important standard in many other calculi π/2 0 cos(x)dx = π/2 0 cos(x )dx 9 i=1 a i = 9 j=1 a j λx.x +2 = α λy.y +2 λxy.x + y = α λyx.y + x

19 Bound variables de Bruijn indices is a systematic computer representation of bound variables for each occurence of a bound variable, one counts the number of binders to traverse to reach its binder. Example: (λx.(λy.λx.y x)(λz.z x))x y y is (λ.(λ.λ.1 0)(λ.0 1))x y λx x λy λz λx

20 Substitution x{y := P} = x c{y := P} = c y{y := P} = P (MN){y := P} = M{y := P} N{y := P} (λy.m){y := P} = λy.m (λx.m){y := P} = λx.m{x := x }{y := P} where x = x if y not free in M or x not free in P, otherwise x is the first variable not free in M and P. (we suppose that the set of variables is infinite and enumerable) Free variables var(x) ={x} var(c) = var(mn) = var(m) var(n) var(λx.m) = var(m) {x}

21 Conversion rules λx.m α λx.m{x := x } (x var(m)) (λx.m)n β M{x := N} λx.mx η M (x var(m)) left-hand-side of conversion rule is a redex (reductible expression) α-redex, β-redex, η-redex,... we forget indices when clear from context, often β Reduction step let R be a redex in M. Then one can contrat redex R in M and get N: M R N

22 Reductions M N when M = M 0 M 1 M 2 M n = N (n 0) same with explicit contracted redexes M = M 0 R 1 M 1 R 2 M 2 R n M n = N and with named reductions ρ : M = M 0 R 1 M 1 R 2 M 2 R n M n = N we speak of redex occurences when specifying reduction steps, but it is convenient to confuse redexes and redex occurences when clear from context

23 Lambda theories M = β N when M and N are related by a zigzag of reductions M and N are said interconvertible M N Also M = α N, M = η N, M = β,η N,... Interconvertibility is symmetric, reflexive, transivite closure of reduction relation or with notations of mathematical logic: α M = N, β M = N, η M = N, β + η M = N,... the syntactic equality M = N will often stand for M = α N.

24 Exercice 3 Find terms M such that: M M M = M 0 M 1 M 2 M n = M (M i all distinct) M = β xm M = β λx.m M = β MM M = β MN 1 N 2 N n for all N 1, N 2,... N n Find term Y such that, for any M: YM = β M(YM) Find Y such that, for any M: Y M M(Y M) (difficult) Show there is only one redex R such that R R

25 Normal forms An expression M without redexes is in normal form M If M reduces to a normal form, then M has a normal form M N, N in normal form Exercice 4 which of following terms are in β-normal form? in βη-normal form? λx.x λxy.x λxy.xy λxy.x((λx.y(xx))(λx.y(xx))) λx.x(λxy.x)(λx.x) λxy.x(λxy.x)(λx.yx) λxy.x((λx.xx)(λx.xx))y

26 Exercice 5 Show that if M is in normal form and M N, then M = N Show that: 1- λx.m N implies N = λx.n and M N 2- MN P implies M M, N N and P = M N or M λx.m, N N and M {x := N } P 3- xm 1 M 2 M n N implies M 1 N 1, M 2 N 2,... M n N n and xn 1 N 2 N n = N 4- M{x := N} λy.p implies M λy.m and M {x := N} P or M xm 1 M 2 M n and NM 1 {x := N} M n {x := N} λy.p

27 δ-rules

28 Adding δ-rules: PCF Terms of PCF M, N, P ::= x, y, z,... (variables) λx.m (M function of x) M N (M applied to N) n (integer constant) M N (arithmetic operation, +, *, -, / ) ifz P then M else N (conditionnal) Conversion rules (λx.m)n M{x := N} m n ifz 0 then M else N ifz n+1 then M else N m n M N

29 Examples (bis) (2 + 3) ((2 + 3) + 4) (4 + 5)...

30 Examples (bis) (λx. x + 1) (λx.2 x + 2) (λf.f 3)(λx. x + 2) (λx. x + 2) (λf.λx.f (f x))(λx.x + 2)...

31 Examples (bis) (λf.λx.f (f x))(λx.x + 2)3...

32 Examples Fact(3) Fact = Y (λf.λx. ifz x then 1 else x f (x 1)) Y = λf.(λx.f (xx))(λx.f (xx)) can be written as a single term in: (λ Fact. Fact(3)) ((λy.y (λf.λx. ifz x then 1 else x f (x 1))) (λf.(λx.f (xx))(λx.f (xx))) )

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45 λ-definability

46 Computing without δ-rules Numbers will be in unary-code =0 S ( ) with following implementation: 0=True,? 1 = False,0 = False, True,? 2 = False,1 = False, False, True,?... n = False, n 1 = False, False, True,? n

47 Computing without δ-rules Booleans True = λx.λy.x = K False = λx.λy.y Pairs and Projections M, N = λx.xmn π 1 = λx.x True π 2 = λx.x False True MN False MN π 1 M, N π 2 M, N M N M N Non-negative integers... 0=True, True n +1=False, n iszero = π 1 iszero 0 True iszero(n + 1) False

48 Computing without δ-rules... integers Succ = λx.false, x Pred = λx. iszero x 0 π 2

49 Other numeral system also named Church s numerals or n = λf.λx.f (f ( f (x) )) n n = λf. f f f n was n+1 in Church s original monograph

50 Other numeral system Lambda-I calculus λx.m (M depends upon x) Church numerals no K = λx.λy.x n = λf.λx.f n (x) n 1 Pairs and projections M, N = λx.xmn π 1 = λp.p(λx.λy. yix) π 2 = λp.p(λx.λy. xiy ni I I = λx.x π 1 m, n π 2 m, n m n

51 Other numeral system... successor and predecessor Succ = λn.λf.λx.nf (f x) Pred = λn. π 3 3 (n φ 1, 1, 1) φ = λt.(λx.λy.λz.succ x, x, y)(π 3 1 t)(π3 2 t)(π3 3 t) where π 3 1, π3 2, π3 3 are the 3 projections on triples φ shift register! FIFO

52 Church numeral system Alonzo Church Stephen Kleene

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56 Programming languages

57 Towards programming languages Many δ-rules Adding types never following terms : + =??? 3+λx.x 4(5) 20(λx.x) ifz λx.x then 1 else 3 λf.(λx.f (xx))(λx.f (xx)) λx.xx Adding store and mutable values

58 Functional programming Scheme, SML, Ocaml, Haskell are functional programming languages they manipulate functions and try to reduce the number of memory states F# Haskell SML Ocaml Jocaml ML Ocaml Scheme Lisp Camllight Caml

59 Next class

60 Next class confluency consistency