Introduction to lambda expression & lambda calculus for junior computer science student.

1. Requirements
• Common Knowledge of Programming
• Maybe Python/Ruby
• Recursion

2. LAMBDA INTRO
KIM LEO

3. λ

4. lambda expression
def square(x):
return x*x
=> square = lambda x: x*x
e.g.:
(lambda x: x*x)(8) =>
8*8 =>
64
But how could make use of LAMBDA?

5. lambda expression
alist: [8, 5, 2, 4, 62, 4, 6]
anOp: blablabla
apply the operation to the list.
like:
for i = 0 to len(alist):
anOp( alist[i] )
or:
foreach i in alist:
anOp( i )

6. Implement
apply_square_for_each(alist):
foreach i in alist: square( i );
apply_plusplus7_for_each(alist):
foreach i in alist: plusplus7( i );
apply_for_each(alist, op):
case
when op == "square":
foreach i in alist: square( i )
.....

7. lambda expression
apply_for_each(alist, alambda):
foreach i in alist:
alambda( i )
apply_for_each([1,2,3], square)
apply_for_each([ ... ],
anything_you_want_applied_to)
apply_for_each => map/each

8. General LISt Processing
Map ：
[x,y,z] -> map(op,[x,y,z])
=> [op(x),op(y),op(z)]
Reduce ：
[x,y,z] -> reduce(op,[x,y,z],v)
=> op(op(op(v,x),y),z)
Filter ：
[x,y,z] -> filter(op,[x,y,z])
=> a subset of the list for each
op(i) => true

9. more..
each, find, every/all, any/some, invoke,
where, sort, group/groupBy, countBy
....
(underscore.js/Ruby Enumerable Module)
var all_of_us
def lonelyboy
def lonelygirl
zip( filter( lonelyboy, all_of_us),
filter( lonelygirl, all_of_us) )

10. Lambda Calculus
lambda x: x
lambda x,y: y(x)
(lambda x:x)(1)
1
xx.λ
xyyx ..λλ
1).( xxλ
xfxf ..λλ

11. Recursion
Y := λg.(λx.g (x x)) (λx.g (x x))
Y f
=> λg.(λx.g (x x)) (λx.g (x x)) f
=> (λx.f (x x)) (λx.f (x x))
=> f ((λx.f (x x)) (λx.f (x x)))
=> f (Y f)

12. SKI Combinator
I := λx.x
K := λx.λy.x
S := λx.λy.λz.x z (y z)
OR:
Ix = x
Kxy = x
Sxyz = xz(yz)

13. self-application
SIIα = Iα(Iα) = αα
SII(SII)
= I(SII)(I(SII))
= I(SII)(SII)
= SII(SII)

14. reverse
S(K(SI))K αβ →
K(SI)α(Kα) β →
SI(Kα) β →
Iβ(Kαβ) →
Iβα
βα

15. Y in SKI
Y = S (K (S I I)) (S (S (K S) K) (K (S I
I)))
Y' = S S K (S (K (S S (S (S S K)))) K)