Lambda Calculus

1. Lambda Calculus History and Syntax

2. History • The lambda calculus is a formal system designed to investigate function definition, function application and recursion. • It was introduced by Alonzo Church and Stephen Cole Kleene in the 1930s • Lambda calculus has greatly influenced functional programming languages, especially Lisp. • Church used the lambda calculus in 1936 to give a negative answer to the Entscheidungsproblem

3. ⋋-Calculus - 1930 LISP - 1950 Alonzo Church, Princeton John McCarthy, AI Lab, Stanford C - 1970 Dennis Ritchie, Bell Labs

4. General Proof Machine • Gottfried Leibniz asked if we can make a machine to validate all mathematical statemanets • He realized that the first step would have to be a clean formal language

5. Entscheidungsproblem • German: decision problem • In 1900, at a mathematical conference in Paris, David Hilbert presented his famous 23 unsolved problems of Mathematics. • Question that whether it was possible to determine the truth of any mathematical statement

6. Entscheidungsproblem • In 1931 Kurt Gödel, in his incompleteness Theorem, showed that there exist mathematical statements that are true but cannot be proved as true within the framework of a formal system. • The incompleteness theorem showed that mathematics was incomplete

7. Entscheidungsproblem • all that remained to be done was to find a way that would decide whether it was possible to decide whether a given function could be solved mechanically or not

8. Entscheidungsproblem • The negative answer to the Entscheidungsproblem was then given by Alonzo Church in 1936 and independently shortly thereafter by Alan Turing • Turing reduced the Entscheidungsproblem to the halting problem for Turing machines

9. λ – Calculus Origins • Originally, Church had tried to construct a complete formal system for the foundations of mathematics • System turned out to be susceptible to the analog of Russell's paradox • He separated out the lambda calculus and used it to study computability

10. Lambda Calculus, Definition • Provides a mathematical model for computation using recursive functions. • Shows that recursive functions have the same computing power as Turing machines. • Serves as the basis for functional programming languages like Haskell and ML

11. Some Informal Definitions • Consider definition in mathematic notations : ƒ(x) = x * x + 2 • In λ–Calculus we denote This in form of λx.(x*x + 2) • And if we have ƒ(5) return number is 27 • In λ–Calculus λx.(x*x + 2)(5) we replace x by 5 : λx.(x*x + 2)(5) = (5*5 + 2) = 27

12. Some informal definitions • introduce variables ranging over values • define functions by (λ-) abstracting over variables • apply functions to values x + 1 x. x + 1 (x. x + 1) 2

13. -ExpressionsA lambda expression is any of the following: 1. Variable: x, y, z, … A variable denotes an abstraction to which it is bound, or denotes itself if it is not bound. 2. Function application: if x and y are  -expressions, then (x y) is a  -expression called function application. It denotes the result of applying the function x to the argument y.

14. -Expressions 3. Abstraction or function: If x is a variable and y is a  -expression, then ( x . y) is an abstraction or function with dummy variable x and body y. ( x . y) denotes the function which when applied to argument a returns as result the abstraction (( x . y) a) = y [ a | x ]

15. Examples of -expressions • ( x . x) is the identity function. • (( x . x) y)  y is a function application returning a variable y. • ( f . ( x . ( f x))) is a function returning a function as value, since its body which is to be returned, ( x . ( f x)), is a function. • ( ( f . ( x . ( f x))) w)  ( x . (w x)) is a function application returning a function.

16. Formal Definition • Formal Definition of λ expression is : • <expr> ::= <identifier> • <expr> ::= (λ <identifier> . <expr>) • <expr> ::= (<expr> <expr>) • Where identifier is a member of countable infinite set like {a, b, c, ..., x, y, z, x1, x2, ...}

17. Notational Conventions 1. Function application is left-associative. w x means (w x) w x y means ((w x) y) w x y z means (((w x) y) z) 2. The scope of the dummy variable extends as far to the right as possible.  w . x y z means ( w . ((x y) z))

18. Notational Conventions 3. ‘.’ is right associative.  w .  x .  y . z means ( w . ( x . ( y . z))) 4.  distributes up to the ‘.’.  w x y . z means  w .  x .  y . z which in turn means ( w . ( x . ( y . z)))

19. Example Notational Shortcuts 1.  x y . f y x means ( x . ( y . ((f y) x))), a function. 2. ( x . y x)( y . x y) means ( ( x . (y x))( y . (x y)) ), a function application.

20. Bound Variables Def. A variable x occurs bound in a  -expression if there is an enclosing abstraction of which x is the dummy variable. ( x . ( f … x … ))   dummy bound

21. Free Variables Def. An occurrence of a variable y is free if it is not bound -- that is, in the scope of the dummy variable x, it is not the variable x. ( x . (y …. x))   free bound

22. ExampleBound and free variables •  x y . u x y = ( x . ( y . ((u x) y) )) u is free x and y are bound

23. -Conversion Def. The variable x in ( x . … x … ) can be renamed in  x and in all of its bound occurrences to a new variable u, giving ( u . … u … ). To denote this renaming, we write: ( x .y)  ( u . (y [ u | x ] ) where y [ u | x ] means all occurrences of x bound by  x are replaced by u. The variable u must be fresh, and must not get bound when substituted in y, except by  u.

24. -Conversion, Examples 1.  x .w x u .w u 2.  x .w x  w .w w is not true since the original w becomes bound. 3.  x . ( u .u) x   u . ( u .u) u is okay   y . ( u .u) y is better. 4.  x . ( u .w u x )   u . ( u .w u u ) is not true, since u becomes bound.   y . ( u .w u y ) is better.

25. -Conversion, Examples 5.( w . x .w x ) x( w . u .w u ) x since the outer x is not in scope of  x. 6.  w .w x   w .w u is not correct since x is not the dummy variable.

26. Evaluating -expressions: -reduction -reduction describes how to evaluate an application of a function to its actual argument.

27. Function Application Def. Function application is carried out as follows: ( x . B )  B [ A | x ] where B [ A | x ] means all occurrences of x not bound by other  inside the body B are replaced by A. Further, the free variables occurring in A can not become bound when substituted in A.

28. -Redex Rem. A -expressionconsisting of a function applied to an argument is called a -reducible expression or is a -redex.

29. -reduction, Examples 1. ( x . x) a a 2.( x .  y . x) a b ( y . a) b  a 3. ( x . x a) ( x . x)  ( x . x) a  a 4.( x .  y . x y) y  ( x .  z . x z) y  ( z . y z)

30. -Functions Def. A -function is a name given to a  -expression. Whenever the  -expression is needed, we can use the name instead. It is a shortcut or macro for the  -expression. Example: Some -functions are true, false, not, and, or, 0, 1, 2, .., n, iszero, suc, and add.

31. -Functions 1. true =  x y . x 2. false =  x y . y 3. (x?y:z) =x y z 4.not =  x . x false true 5.and = x y . x y false 6. or = x y . x true y

32. -Functions 7. 0 =  f x . x 8. 1 =  f x . f x 9. 2 =  f x . f (f x) 10.n =  f x . f n x 11.iszero = n . n ( x . false) true 12. succ = n f x . n f (f x) 13. add =  m n f x . m f (nf x)

33. -Reduction Any abstraction of the form  x . (E x) in which x has no free occurrences in E can be reduced to E. For example  x . (succ x)  succ An abstraction which is -reducible is called an -redex.

34. Rationale for -Reduction If x has no free occurrences in E, then (( x . (E x)) A) b(E x) [ A | x ] = (E A) Thus the effect of ( x . (E x)) on A is the same as the effect of E on A.

35. Normal Forms Def. A -expression is in normal form if it contains no -redexes or -redexes. A -expression has a normal form if there exists a finite sequence of reduction steps that results in a nromal form.

36. Examples of Normal Forms 1. x is in normal form 2. ( x . x x) ( y . y) is not in normal form but has a normal form, since ( x . x x) ( y . y)  ( y . y) ( y . y)  ( y . y)

37. Examples of Normal Forms 3. ( x . x x) ( y . y y) is not in normal form and has no normal form, since the reduction sequence does not terminate: ( x . x x) ( y . y y)  ( y . y y) ( y . y y)  . . .

38. Reduction Order A  -expression can contain several redexes. For example, ( x . ( y . y) ( z . z)) ( u . u) 1 5 An outer redex starts at character 1. An inner redex starts at character 5. In this case, more than one reduction sequence is possible.

39. Reduction Sequences Inner redex: ( x . ( y . y) ( z . z)) ( u . u)  ( x . ( z . z)) ( u . u)  ( z . z) Outer redex: ( x . ( y . y) ( z . z)) ( u . u)  ( y . y) ( z . z)  ( z . z) The normal form is the same for the two.

40. Reduction Sequences, Example 2 Inner redex: ( y . z . z) (( x . x x) ( x . x x))  ( y . z . z) (( x . x x) ( x . x x))  . . . Outer redex: ( y . z . z) (( x . x x) ( x . x x))  ( z . z) 1st sequence is infinite. 2nd has normal form.

41. Reduction Strategies 1. Normal order strategy - repeatedly reduce the leftmost-outermost redex. 2. Applicative order strategy - repeatedly reduce the leftmost-innermost redex.

42. Normalization Theorem If a -expression E has a normal form, then the normal order strategy will terminate in a normal form. (Curry & Feys, 1958)

43. Church-Rosser Corollary The normal form of a -expression, if it exists, is unique.

44. Church’s Thesis The class of effectively computable functions is the same as the class of functions that can be defined in -calculus (1936)

45. Church’s Theorem The class of Turing-computable functions is the same as the class of -definable functions.

46. New Models • Australian Mathematician, published a paper16 in late 2001 that stated that “incomputable” problems can be computed by exploiting the laws of Quantum Mechanics • A Quantum computer would be able to perform an infinite search in a finite amount of time and hence would solve the halting problem and would show the computability of incomputable functions.

47. Question