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David Evans http://www.cs.virginia.edu/evans

Lecture 39: Lambda Calculus. David Evans http://www.cs.virginia.edu/evans. CS150: Computer Science University of Virginia Computer Science. Equivalent Computers. . z. z. z. z. z. z. z. term = variable | term term | ( term )

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David Evans http://www.cs.virginia.edu/evans

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  1. Lecture 39: Lambda Calculus David Evans http://www.cs.virginia.edu/evans CS150: Computer Science University of Virginia Computer Science

  2. Equivalent Computers ... z z z z z z z • term = variable • | term term • | (term) • |  variable .term ), X, L  ), #, R (, #, L 2: look for ( 1 Start (, X, R • y. M v. (M [yv]) • where v does not occur in M. • (x. M)N   M [ xN ] HALT #, 0, - #, 1, - Finite State Machine Lambda Calculus Turing Machine

  3. What is Calculus? • In High School: d/dxxn = nxn-1 [Power Rule] d/dx (f + g) = d/dx f + d/dx g [Sum Rule] Calculus is a branch of mathematics that deals with limits and the differentiation and integration of functions of one or more variables...

  4. Real Definition • A calculus is just a bunch of rules for manipulating symbols. • People can give meaning to those symbols, but that’s not part of the calculus. • Differential calculus is a bunch of rules for manipulating symbols. There is an interpretation of those symbols corresponds with physics, slopes, etc.

  5. Lambda Calculus • Rules for manipulating strings of symbols in the language: term = variable | term term | (term) |  variable .term • Humans can give meaning to those symbols in a way that corresponds to computations.

  6. Why? • Once we have precise and formal rules for manipulating symbols, we can use it to reason with. • Since we can interpret the symbols as representing computations, we can use it to reason about programs.

  7. Evaluation Rules -reduction (renaming) y. M v. (M [yv]) where v does not occur in M. -reduction (substitution) (x. M)N   M [ xN ] Note the syntax is different from Scheme: (x.M)N ((lambda (x) M) N)

  8. -Reduction (the source of all computation) (x. M)N  M [ xN ] Replace all x’s in M witn N

  9. Evaluating Lambda Expressions • redex: Term of the form (x. M)N Something that can be -reduced • An expression is in normal form if it contains no redexes (redices). • To evaluate a lambda expression, keep doing reductions until you get to normal form.

  10. Some Simple Functions I x.x Cxy.yx Abbreviation for x.(y. yx) CII = (x.(y. yx)) (x.x) (x.x)  (y. y (x.x))(x.x) x.x (x.x) x.x = I

  11. Example  f. (( x.f (xx))( x. f (xx)))

  12. Possible Answer ( f. (( x.f (xx))( x. f (xx)))) (z.z) (x.(z.z)(xx))( x. (z.z)(xx))  (z.z) ( x.(z.z)(xx)) ( x.(z.z)(xx))  (x.(z.z)(xx)) ( x.(z.z)(xx))  (z.z) ( x.(z.z)(xx)) ( x.(z.z)(xx))  (x.(z.z)(xx)) ( x.(z.z)(xx)) ...

  13. Alternate Answer ( f. (( x.f (xx))( x. f (xx)))) (z.z)  (x.(z.z)(xx))( x. (z.z)(xx))  (x.xx)(x.(z.z)(xx))  (x.xx)(x.xx)  (x.xx)(x.xx) ...

  14. Be Very Afraid! • Some -calculus terms can be -reduced forever! • The order in which you choose to do the reductions might change the result!

  15. Take on Faith (until CS655) • All ways of choosing reductions that reduce a lambda expression to normal form will produce the same normal form (but some might never produce a normal form). • If we always apply the outermost lambda first, we will find the normal form if there is one. • This is normal order reduction – corresponds to normal order (lazy) evaluation

  16. Universal Language • Is Lambda Calculus a universal language? • Can we compute any computable algorithm using Lambda Calculus? • To prove it isn’t: • Find some Turing Machine that cannot be simulated with Lambda Calculus • To prove it is: • Show you can simulate every Turing Machine using Lambda Calculus

  17. Simulating Every Turing Machine • A Universal Turing Machine can simulate every Turing Machine • So, to show Lambda Calculus can simulate every Turing Machine, all we need to do is show it can simulate a Universal Turing Machine

  18. Simulating Computation z z z z z z z z z z z z z z z z z z z z • Lambda expression corresponds to a computation: input on the tape is transformed into a lambda expression • Normal form is that value of that computation: output is the normal form • How do we simulate the FSM? ), X, L ), #, R (, #, L 2: look for ( 1 Start (, X, R HALT #, 0, - #, 1, - Finite State Machine

  19. Simulating Computation z z z z z z z z z z z z z z z z z z z z Read/Write Infinite Tape Mutable Lists Finite State Machine Numbers Processing Way to make decisions (if) Way to keep going ), X, L ), #, R (, #, L 2: look for ( 1 Start (, X, R HALT #, 0, - #, 1, - Finite State Machine

  20. Making “Primitives”from Only Glue ()

  21. In search of thetruth? • What does true mean? • True is something that when used as the first operand of if, makes the value of the if the value of its second operand: ifTMN M

  22. Don’t search for T, search for if T  x (y. x)  xy. x F  x ( y. y)) if pca . pca

  23. Finding the Truth T  x . (y. x) F  x . (y. y) if p . (c . (a . pca))) if T M N ((pca . pca) (xy. x)) M N  (ca . (x.(y. x)) ca)) M N   (x.(y. x))M N  (y. M ))N  M Is the if necessary?

  24. and and or? and x (y. ifxyF)) or x (y. ifxTy))

  25. Lambda Calculus is a Universal Computer? z z z z z z z z z z z z z z z z z z z z • Read/Write Infinite Tape Mutable Lists • Finite State Machine Numbers • Processing Way to make decisions (if) Way to keep going ), X, L ), #, R (, #, L 2: look for ( 1 Start (, X, R HALT #, 0, - #, 1, - Finite State Machine ...to be continued Wednesday...

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