SLIDE 1
Church Numerals
Encoding natural numbers as lambda-terms zero
= f :x :x- ne
Encoding natural numbers datatype nat = Z | S of nat val zero = Z - - PowerPoint PPT Presentation
Encoding natural numbers datatype nat = Z | S of nat val zero = Z - - PowerPoint PPT Presentation
Encoding natural numbers datatype nat = Z | S of nat val zero = Z val one = S Z val two = S (S Z) val three = S (S (S Z)) Define fold s.t. fold f x replaces S 7! f and Z 7! x fun fold f x Z = x | fold f x (S n) = f (fold f x n) Example:
SLIDE 2
SLIDE 3
Church Numerals in
- zero
= \f.\x.x; succ = \n.\f.\x.f (n f x); plus = \n.\m.n succ m; times = \n.\m.n (plus m) zero; ...
- > four;
\f.\x.f (f (f (f x)))
- > three;
\f.\x.f (f (f x))
- > times four three;
\f.\x.f (f (f (f (f (f (f (f (f (f (f (f x)))))))))))
SLIDE 4
Reduction rules
Central rule based on substitution
(x :M )N- ! M
(BETA) Structural rules: Beta-reduce anywhere, any time N
- ! N′
MN
- ! MN′
M
- ! M′
MN
- ! M′N
M
- ! M′
- !
SLIDE 5
Free variables
x is free in x x is free in M
_ x is free in Nx is free in MN x is free in M x
6= x′x is free in
x′ :M SLIDE 6
Exercise: Free Variables
What are the free variables in: \x.\y. y z \x.x (\y.x) \x.\y.\x.x y \x.\y.x (\z.y w) y (\x.z) (\x.\y.x y) y
SLIDE 7
Exercise: Free Variables
What are the free variables in: \x.\y. y z
- z
\x.x (\y.x)
- nothing
\x.\y.\x.x y
- nothing
\x.\y.x (\z.y w)
- w
y (\x.z)
- y z
(\x.\y.x y) y
- y
SLIDE 8
Capture-avoiding substitution
x [x
7! M ℄ =M y
[x 7! M ℄ =y
(YZ )[x 7! M ℄ = (Y [x 7! M ℄)(Z [x 7! M ℄) (x :Y )[x 7! M ℄ = x :Y (y :Z )[x 7! M ℄ = y :Z [x 7! M ℄if x not free in Z or y not free in M
(y :Z )[x 7! M ℄ = w :(Z [y 7! w ℄)[x 7! M ℄where w not free in Z or M Last transformation is renaming of bound variables
SLIDE 9
Renaming of bound variables
So important it has its own Greek letter: w not free in Z
y :Z- !