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Introduction to the λ-calculus Part II CS209 - Functional Programming Dr. Greg Lavender Department of Computer Science Stanford University iszero Predicate Test for zero iszero = λn.n(λx.f)t iszero 0 =

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Introduction to the λ-calculus Part II CS209 - Functional Programming Dr. Greg Lavender Department of Computer Science Stanford University iszero Predicate Test for zero iszero = λn.n(λx.f)t iszero 0 = (λn.n(λx.f)t) λsz.z = (λsz.z) (λx.f)t = T iszero S0 = (λn.n(λx.f)t) λsz.s(z) = (λsz.s(z)) (λx.f)t = (λx.f)t = F convince yourself that iszero Sn always returns F 2 1 Ordered Pairs Lambda expressions for ordered pairs fst = λp.p T = λp.p (λtf.t) snd = λp.p F = λp.p (λtf.f) (E 1, E 2 ) = λf.f E 1 E 2 fst (E1, E2) = (λp.pt) (E 1, E 2 ) = (E 1, E 2 ) T = (λf.f E 1 E 2 ) T = T E 1 E 2 = (λtf.t) E 1 E 2 = E 1 Similarly for snd (E 1, E 2 ) 3 n-tuples Tuples are defined in terms of pairs (E 1,E 2,, E n ) = (E 1, (E 2, ( (E n-1, E n ) ))) E 1 = fst E E 2 = fst(snd E) E i = fst(snd(snd( (snd E) ))) if i n E n = snd(snd( (snd E) ))) n-1 applications of snd (E 1,E 2,,E n ) 2 = (E 1,(E 2,( ))) 2 = fst (snd(e 1, (E 2, ( )))) = fst (E 2,( )) = E 2 Prove that (E 1,E 2,,E n ) i = E i for 1 = i = n 4 2 Predecessor function Predecessor is subtraction by 1 BUT, pred 0 = 0 pred = λn f x. snd(n (prefn f) (T, x)) where prefn is defined as: prefn = λf p. (F, (fst p - snd p (f (snd p)))) where (E - E 1 E 2 ) is syntactic sugar for (test E E 1 E 2 ) Show that: pred (succ n) = n pred 0 = 0 5 Fixed points A fixed point is a value x in the domain of a function that is the same in the range f(x). Every value in the domain of the identity function is a fixed point λx.x = x can you think of others? factorial(1) = 1 fibonacci(0) = 0, fibonacci(1) = 1 square(0) = 0, square(1) = 1 abs(x) = x, if x = 0 sin(0) = 0 Functionals may also have fixed points D x (e x ) = e x 6 3 Some Computability Theory Godel numbering every program is represented by a finite string of symbols: P = int main() { printf ( Hello world\n ); } a general algorithm can be defined that converts any program into a unique natural number e, called the godel number of the program godel(p) = e for any Turing machine M representing a program, we can assign it a Godel number e and denote the Turing machine that computes that program by M e Let U be a Turing machine that using inputs e and x computes M e (x): call e the program and x the input to program e U = if (e is a program) then M e (x) else output 0 U is thus a universal Turing machine this is the key idea that led John von Neumann to invent the stored program concept used in modern computers 7 Godel Numbering Function How do we write a Godel numbering function? a program is just a sequence of characters from some alphabet. For example, the ASCII alphabet has 128 characters: map chr [0..127]= \NUL\SOH\STX\ETX\EOT\ENQ\ACK\a\b\t\n\v\f\r\SO\SI\DLE\DC1\DC2\DC3\DC cdefghijklmnopqrstuvwxyz{ }~\DEL define a n-degree polynomial where each coefficient is the integer value of the corresponding ascii character and x is the size of the alphabet: P(x) = a n x n + a n-1 x n a 1 x + a 0 for any given program text convert it to an array of ascii values and use those values as the coefficients a n a 0 then evaluate the polynomial with x = 128 using Horner s rule 8 4 Godel Numbering Function Here is an example in Haskell ascii :: [Char] - [Integer] ascii s = map fromintegral (map ord s) -- horner s rule for evaluating a polynomial horner :: (Num a) = a - [a] - a horner x (y:ys) = foldl (\h a - a + (x * h)) y ys godel :: String- Integer godel p = horner 128 (ascii p) godel int main() {printf(\ hello, world\n\ ); } = Kleene s Fixed Point Theorem Also known as Kleene s recursion theorem let φ e = λx.u(e,x) For every computable function f there is a number n such that φ n = φ f(n) Corollary There is a Godel number n such that φ n is the constant function with output n Hence, n is the Godel number of a self-reproducing program. i.e., a Turing machine whose program, denoted by godel number n, does nothing on any input x except print its own code, i.e., the string = ungodel(n) This is the idea behind a quine the name quine was first used by Hofstadter in his book Godel, Escher, Bach in honor of the logician W.V.O. Quine Omega = (λx.xx)(λx.xx) is an example of a self-reproducing program in the lambda calculus compare this to the following in Scheme and C: ((lambda (x) `(,x ',x)) '(lambda (x) `(,x ',x))) main(a){printf(a= main(a){printf(a=%c%s%c,34,a,34);} ,34,a,34);} godel main(a){printf(a=\ main(a){printf(a=%c%s%c,34,a,34);}\ ,34,a,34);} = Recursive Definition in the λ-calculus How do we define recursive expressions? will this work? mult λm n. (iszero m - 0 add n (mult (pred m) n)) No! There is a problem with this definition we cannot define mult in terms of itself without some way to handle the self-referential naming -- recursive self-reference introduces a challenge most programming languages do this automagically, but in the pure λ calculus, we have to come up with a syntactic way to allow recursive definitions recall that in Scheme, letrec is a special syntactic form for writing recursive definitions (ML has val rec ) we have to come up with some way in the lambda calculus to express recursion syntactically without defining a function directly in terms of itself We need a special mathematical device called a fixed point operator that works for any function defined in the lambda calculus 11 Defining Recursive Functions Fixed Point Theorem for λ calculus For all F, there exists an X such that F(X) = X Proof: let W = λx.f(xx) and let X = WW then X = WW = λx.f(xx)w = F(WW) = F(X) X = WW is called a fixed point of F Fact: we can generate a fixed point for any function F Let F = λy.y W = λx.f(xx) = λx.(λy.y)(xx) Proof that X = WW is a fixed point for λy.y X = WW = (λx.(λy.y)(xx))(λx.(λy.y)(xx)) = (λy.y)((λx.(λy.y)(xx))(λx.(λy.y)(xx))) = F(X) note that (λy.y)((λx.(λy.y)(xx))(λx.(λy.y)(xx)) = (λx.xx)(λx.xx) so (λx.xx)(λx.xx) is a fixed point for λy.y, i.e., λx.x Homework: Generate a fixed point for F = λxy.xy 12 6 Fixed Point Operators Consider a fixed point operator Fix Fix F = F (Fix F) Fix applied to any function F gives F and repeats F by applying Fix to F one more time there are many such fixed point operators The fixed point operator commonly defined in the λ- calculus is called the (lazy) Y combinator Y = (λf. (λx.f(xx)) (λx.f(xx))) Y E = (λf. (λx.f(xx)) (λx.f(xx))) E = (λx.e(xx)) (λx.e(xx)) = E ((λx.e(xx)) (λx.e(xx))) = E (Y E) since Y E = E (Y E) = E (E (Y E)) = E (E (E (Y E))..) i.e., Y is a fixed point operator that when applied to any expression E applies E to a copy of itself repeatedly (i.e., recursively) the recursion only terminates if E has a terminating condition Homework: evaluate Y λx.x 13 Using the Y combinator Any expression of the form f x 1 x n = E is called recursive if f occurs free in E if you want: f x 1 x n = ~~~~ f ~~~~ then define F = Y (λf x 1 x n. ~~~~ f ~~~~) for example: mult = λm n. (iszero m - 0 add n (mult (pred m) n)) Becomes multfn = λf m n. (iszero m - 0 add n (f pred m) n)) mult = Y multfn mult x y = (Y multfn) x y = multfn (Y multfn) x y = multfn mult x y = (λf m n. (iszero m - 0 add n (f pred m) n))) mult x y = (iszero x - 0 add y (mult (pred x) y)) = (iszero x - 0 add y ((Y multfn) (pred x) y)) = 14 7 Fix in Haskell This works because of lazy evaluation. if Haskell had eager evaluation, f (fix f) would never terminate: f (fix f) = f f (fix f) = f f f (fix f) = fix f = f (fix f) fact :: Integer- Integer fact = fix (\f n - if n==0 then 1 else n*f(n-1)) fact 3 = (\f n - if n==0 then 1 else n*f(n-1)) (fix f) 3 = (if 3==0 then 1 else 3*(fix f)(3-1)) = (3*((\f n - if n==0 then 1 else n*f(n-1))(fix f)(3-1)) = = (3*(2*(1*1))) = 6 15 What about Eagerly Evaluated Scheme? We can t use the (lazy) Y combinator in Scheme because it would not terminate under applicative order (eager) evaluation: (define Y (lambda (f) ((lambda (x) (f (x x))) (lambda (x) (f (x x)))))) We need a fixed point combinator that works with applicative order evaluation. (define T (lambda (f) ((lambda (x) (f (lambda (y) ((x x) y)))) (lambda (x) (f (lambda (y) ((x x) y))))))) (define fact (T (lambda (g) (lambda (n) (if (zero? n) 1 (* n (g (- n 1)))))))) 16 8

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