CSE-321 Programming Languages -Calculus (II) POSTECH March 26, 2007 박성우

2 Values and Reductions redex = reducible expression :  -reduction

3 Call-by-name Call-by-value

4 Outline Abstract syntax of the -calculus V Operational semantics of the -calculus V Substitutions Programming in the -calculus

5 [e' / x] e Informally "substitute e' for every occurrence of x in e." Examples

6 Easy Cases First

7 Two Remaining Cases

8 First (stupid) attempt Second attempt But wait:

9 Names of bound variables do not matter. Hence –for a fresh variable y, Bound Variables

10 One Remaining Case

11 A Naive Attempt An anomaly: something for y

12 Free Variables Variables that are bound nowhere FV(e) = set of free variables in e

13 Free Variables Remain Free From the point of view of an outside observer, a free variable remains free until it is explicitly replaced. outside observer ? variable capture

14 Capture-Avoiding Substitution What happens if –the free variable y is captured and becomes a bound variable. –To an outside observer, it suddenly disappears!

15 Substitution Completed

16 We have to rename bound variables as necessary. Capture-Avoiding Substitution in Action

17  Conversion Renaming bound variables when necessary Okay because the names of bound variables do not matter. Examples

18 Formalization of  -Conversion See the course notes! –It's more interesting than you might think.

19 Outline Abstract syntax of the -calculus V Operational semantics of the -calculus V Substitutions V Programming in the -calculus

20 A boolean value –"Give me two options and I will choose one for you!" Syntactic sugar Booleans

21 Examples Under the call-by-name strategy,

22 Logical Operators

23 Natural Numbers A natural number n –has the capability to repeat a given process n times. –"Give me a function f and I will return f n = f o f... f o f "

24 Church Numerals

25 Addition Key observation:

26 Multiplication Key observation: Alternatively

27 Recursive Functions in -Calculus Plan of attack 1.Assume a recursive function construct fun f x. e 2.Rewrite it as an expression in the -calculus i.e., show that it is syntactic sugar.

28 Plan of attack 1.Assume a recursive function construct 2.Rewrite it as an expression in the -calculus i.e., show that it is syntactic sugar. Example: Factorial

29 fac to FAC

30 FAC produces a new function from f

31 Now we only need to find a fixed point of:

32 Fixed Point Fixed point of function f V such that V = f (V) Ex. f (x) = 2 - x fixed point of f = 1 1 = f (1)

33 Magic Revealed Fixed point combinator / Y combinator (call-by-value) fix F returns a fixed point of F.

34 Now we only need to find a fixed point of: Now we only need to compute:

35 Answer