Models of computation

Transcription

1 Lambda-Calculus (I) 2nd Asian-Pacific Summer School on Formal ethods Tsinghua University, August 23, 2010 Plan computation models lambda-notation bound variables odels of computation Computation models [machines] automata theory -- Turing machines [character strings] formal grammars, Thue systems, Post conversion rules [numbers] Kleene recursive functions theory reductions [terms] Church lambda-calculus, term rewriting systems normal forms numeral systems lambda-definability Applications to logic [cut elimination] 2nd order arithmetic -- Howard, Girard Barendregt, Henk, The Lambda Calculus. Its Syntax and Semantics, Elsevier, 2nd edition, Barendregt, Henk; Dezani, ariangiola, Lambda calculi with Types, [higher order dependent types] HOL, Isabelle, Coq -- Coquand, Huet

2 Computing with terms Computing with terms (2 + 3) ((2 + 3) + 4) (4 + 5)... (λf.λx.f (f x))(λx.x + 2)3... Computing with terms Computation model (λx. x + 1) (λx.2 x + 2) (λf.f 3)(λx. x + 2) (λx. x + 2) (λf.λx.f (f x))(λx.x + 2)... 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?

3 Abbreviations λ-calculus 1 2 n for ( ((1 )2 ) n ) (λx1 x2 xn. ) for (λx1.(λx2. (λxn. ) )) 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), (λxy.x), (λxy.y ), (λxy.y )() The lambda-calculus (λx.x) Lambda terms,, P ::= x, y, z,... (variables) ( λx. ) ( as function of x) ( ) ( applied to ) c, d,... (constants ) Calculations reductions ((λx.)) {x := } Examples (λf.f )(λx.x) (λx.x) (λx.xx)(λx.x) (λx.x)(λx.x) (λx.xx)(λx.xx) (λx.xx)(λx.xx) Yf = (λx.f (xx))(λx.f (xx)) f (Yf ) f (f (Yf )) (λx.x) f ((λx.f (xx))(λx.f (xx))) = f (Yf ) f n (Yf )

4 Recapitulation Abstract syntax calculus is more complex than expected Example: (λx.(λy.λx.y x)(λz.z x))x y looping expressions!! is recursion operator seems definable when termination? y consistency?!x x computing power?!y!z!x Abstract syntax Bound variables The syntax of lambda-terms can be abstracted as:,, P ::= x, y, z,... (variables) x (λx.(λy.λx.y x)(λz.z x))x y (rightmost x, y are free) ( λx. ) ( as function of x)!x Exercice 2 Show binders of bound variables in (λf.(λx.f (xx))(λx.f (xx)))(λx.λy.x) ( ) ( applied to ) (λf.(λx.f (xx))(λx.f (xx)))(λf xy.x(f y)) (λf.f ((λx.x)3))(λx.λy.x) c, d,... (constants ) c

5 Bound variables Bound variables (λy.λx.y)x λx.x incorrect (dynamic binding: Lisp) (λy.λx.y)x λx.x correct (lexical binding: Scheme) 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: is (λx.(λy.λx.y x)(λz.z x))x y (λ.(λ.λ.1 0)(λ.0 1))x y!x x y!y!z!x Exercice 2bis Why Lisp is consistent? Bound variables Substitution (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 x{y := P} = x y{y := P} = P c{y := P} = c (){y := P} = {y := P} {y := P} (λy.){y := P} = λy. (λx.){y := P} = λx.{x := x }{y := P} where x = x if y not free in or x not free in P, otherwise x is the first variable not free in and P. (we suppose that the set of variables is infinite and enumerable) π/2 0 cos(x)dx = π/2 0 cos(x )dx λx.x +2 = α λy.y +2 9 i=1 a i = 9 j=1 a j λxy.x + y = α λyx.y + x Free variables var(x) ={x} var(c) = var() = var() var() var(λx.) = var() {x}

6 Conversion rules Lambda theories λx. α λx.{x := x } (x var()) (λx.) β {x := } λx.x η (x var()) = β when and are related by a zigzag of reductions and are said interconvertible left-hand-side of conversion rule is a redex (reductible expression) "-redex, #-redex, $-redex,... we forget indices when clear from context, often # Reduction step let R be a redex in. Then one can contrat redex R in and get : R Also = α, = η, = β,η,... Interconvertibility is symmetric, reflexive, transivite closure of reduction relation or with notations of mathematical logic: α =, β =, η =, β + η =,... the syntactic equality = will often stand for = α. Reductions Exercice 3 when = n = (n 0) same with explicit contracted redexes = 0 R 1 1 R 2 2 and with named reductions ρ : = 0 R 1 1 R 2 2 R n R n 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 Find terms such that: = n = ( i all distinct) = β x = β λx. = β = β 1 2 n for all 1, 2,... n Find term Y such that, for any : Y = β (Y) Find Y such that, for any : Y (Y ) (difficult) Show there is only one redex R such that R R

7 ormal forms An expression without redexes is in normal form If reduces to a normal form, then has a normal form δ-rules, in normal form Exercice 4 which of following terms are in #-normal form? in #$-normal form? λx.x λx.x(λxy.x)(λx.x) λxy.x λxy.x(λxy.x)(λx.yx) λxy.xy λxy.x((λx.xx)(λx.xx))y λxy.x((λx.y (xx))(λx.y (xx))) Exercice 5 Adding!-rules: PCF Show that if is in normal form and, then = Terms of PCF Show that: 1- λx. 2- or,, P implies = λx. and P implies λx., 3- x1 2 n, and P = and {x := } implies 1 1, 2 P 2,... n n and x1 2 n = 4- {x := } or λy.p implies ::= x, y, z,... (variables) λx. ( function of x) ( applied to ) n ifz P then else (conditionnal) (integer constant) Conversion rules (λx.) λy. and {x := } x1 2 n and 1 {x := } n {x := } P λy.p (arithmetic operation, +, *, -, / ) m n {x := } m n ifz 0 then else ifz n+1 then else

8 Examples (bis) (2 + 3) ((2 + 3) + 4) (4 + 5)... Examples (bis) (λf.λx.f (f x))(λx.x + 2)3... Examples (bis) Examples (λx. x + 1) (λx.2 x + 2) (λf.f 3)(λx. x + 2) (λx. x + 2) (λf.λx.f (f x))(λx.x + 2)... 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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12 Computing without!-rules Booleans λ-definability True = λx.λy.x = K False = λx.λy.y True False π1, π2, Pairs and Projections, = λx.x π1 = λx.x True π2 = λx.x False on-negative integers... 0 = True, True n + 1 = False, n iszero = π1 Computing without!-rules Computing without!-rules... integers umbers will be in unary-code Succ = λx. False, x =0 S( ) Pred = λx. iszero x 0 π2 with following implementation: 0 = True,? 1 = False, 0 = False, True,? 2 = False, 1 = False, False, True,?... n = False, n 1 = False, False, True,? n iszero 0 True iszero(n + 1) False

13 Other numeral system also named Church s numerals Other numeral system... successor and predecessor or n = λf.λx.f (f ( f (x) )) n Succ = λn.λf.λx.nf (f x) Pred = λn. π3 3 (n φ 1, 1, 1) φ = λt.(λx.λy.λz.succ x, x, y)(π1 3 t)(π3 2 t)(π3 3 t) n = λf. f f f n where π 3 1, π3 2, π3 3 are the 3 projections on triples was n+1 in Church s original monograph φ shift register! FIFO Other numeral system Church numeral system Lambda-I calculus λx. ( depends upon x) K = λx.λy.x Church numerals n = λf.λx.f n (x) ni I n 1 I = λx.x no Pairs and projections, = λx.x π 1 = λp.p(λx.λy. yix) π 2 = λp.p(λx.λy. xiy π 1 m, n π 2 m, n m n Alonzo Church Stephen Kleene

14 Programming languages

15 Towards programming languages any!-rules Adding types never following terms : λx.x 4(5) ext class =??? 20(λx.x) λf.(λx.f (xx))(λx.f (xx)) ifz λx.x then 1 else 3 λx.xx Adding store and mutable values Functional programming ext class Scheme, SL, Ocaml, Haskell are functional programming languages they manipulate functions and try to reduce the number of memory states confluency F# Haskell SL Ocaml L Jocaml Ocaml Scheme Camllight Lisp Caml consistency