Lambda

1
Lambda

• Or,
• Do functions really need to have a name?

2
Last time we said

• (define f
• (local ((define (f x) body )
• f))
• (define (f x) body )
• Note Inner function only used once

3
Concrete example

• (define f
• (local ((define (g x) ( 1 x))
• g))
• by renaming
• (define f
• (local ((define (f x) ( 1 x))
• f))
• by our new rule
• (define (f x) ( 1 x))

4
Observation I

• The inner name in the new rule doesn't really
  matter. We could have
• (define f
• (local ((define (lalala x) body )
• lalala))
• and still, it would equal
• (define (f x) body )

5
Observation II

• This pattern comes up quite a bit. It would be
  nice to have less to write
• (define f
• (lambda (x) body ))
• and still, it would equal
• (define (f x) body )

6
So, on one hand

• Lambda is a shorthand
• (lambda (x y z) body )
• by definition
• (local ((define (lalala x y z) body ))
• lalala)

7
On the other hand

• Our property is now much easier to express
• (define f
• (lambda (x) body ))
• by the property from last class
• (define (f x) body )

8
More on Hand Evaluation

• What are expressions like this supposed to do?
• ((lambda (x) exp_1 ) value_2 )
• ? exp_1 xvalue_2
• Example
• ((lambda (x) ( 1 x)) ( 3 4))
• ? ( 1 x) x( 3 4) ( 1 ( 3 4))
• ? ( 1 7) ? 8
• ((lambda (x) ( 1 x)) ( 3 4))
• ((lambda (x) ( 1 x)) 7)
• ( 1 x) x7 ( 1 7) ? 8

9
More on Hand Evaluation

• What are expressions like this supposed to do?
• ((lambda (x) exp_1 ) exp_2 )
• Example
• ((lambda (x) ( 1 x)) ( 3 4))
• ?
• ?

10
More on Hand Evaluation

• Can we write down a general rule?
• ((lambda (x) exp_1 ) exp_2 )
• ? ??? exp_1 ??? exp_2 ???
• Is it really correct?

11
Lambda calculus

• Very nice model of "programs"
• Alonzo Church, 1930's
• ltexpgt ltvargt
• (lambda (x) ltexpgt)
• (ltexpgt ltexpgt)
• Originally (and still) used to formalize logic
• LISP, 1960's. Basis for Scheme
• Gordon Plotkin, 1975. Basis for ML, Haskell