Transcript Lecture 6: Lambda Calculus God created the integers – all else is the result of man. Leopold Kronecker God created application – all else.

Lecture 6:
Lambda Calculus
God created the integers – all else is the result of man.
Leopold Kronecker
God created application – all else is the result of man.
Alonzo Church (not really)
CS655: Programming Languages
David Evans
University of Virginia
http://www.cs.virginia.edu/~evans
Computer Science
Language Complexity
(Number of Morphemes)
Language Complexity and Brain Cell Decay
English
(OED: 500,000
words)
C++
(680 pages)
Scheme
(50 pages)
??? (.5 page)
0
6 Feb 2001
5
10
15
Years Spent in School
CS 655: Lecture 6
20
25
2
Complexity in Scheme
• Special Forms
– if, cond, define, etc.
• Primitives
If we have lazy evaluation,
(and don’t care about
abstraction) we don’t need these.
– Numbers (infinitely many)
– Booleans: #t, #f
Hard to get rid of?
– Functions (+, -, and, or, etc.)
• Evaluation Complexity
– Environments (more than ½ of our interpreter)
Can we get rid of all this and still have a useful language?
6 Feb 2001
CS 655: Lecture 6
3
-calculus
Alonzo Church, 1940
(LISP was developed from -calculus,
not the other way round.)
term = variable
| term term
| (term)
|  variable . term
6 Feb 2001
CS 655: Lecture 6
4
Mystery Function (Teaser)
p  xy.  pca.pca (x.x xy.x) x) y
(p ((x.x xy.y) x) (x. z.z (xy.y) y)
m  xy.  pca.pca (x.x xy.x) x) x.x
(p y (m ((x.x xy.y) x) y))
f  x.
 pca.pca
((x.xxxy.x)
if
= 0 x)
(z.z (xy.y)
1 (x.x))
(m x (f x((x.x
* f (x xy.y)
– 1) x)))
6 Feb 2001
CS 655: Lecture 6
5
Evaluation Rules
-reduction
(renaming)
y. M  v. (M [y  v])
where v does not occur in M.
-reduction
(substitution)
(x. M)N   M [ x  N ]
6 Feb 2001
CS 655: Lecture 6
6
Identity ()
xy
if x and y are same name
M1 M2  N1 N2 if M1  N1  M2  N2
(M)  (N)
if M  N
x. M  y. N if x  y  M  N
6 Feb 2001
CS 655: Lecture 6
7
Defining Substitution (  )
x [x  N]  N
y [x  N]  y where x  y
M1 M2 [x  N]  M1 [x  N] M2 [x  N]
(x. M) [x  N]  x. M
6 Feb 2001
CS 655: Lecture 6
8
Defining Substitution (  )
(y. M) [x  N]  y. (M [x N])
where x  y and y does not appear free in N
or x does not appear free in M.
(y. M) [x  N]  z. (M [y  z]) [x  N]
where x  y, z  x and z  y and z does not
appear in M or N, x does occur free in M and
y does occur free in N.
6 Feb 2001
CS 655: Lecture 6
9
Reduction (Uninteresting Rules)
y. M  v. (M [y  v])
where v does not occur in M.
MM
M  N  PM  PN
M  N  MP  NP
M  N  x. M  x. N
M  N and N  P  M  P
6 Feb 2001
CS 655: Lecture 6
10
-Reduction (the source of all
computation)
(x. M)N  M [ x  N ]
6 Feb 2001
CS 655: Lecture 6
11
Recall Apply in Scheme
“To apply a procedure to a list of
arguments, evaluate the procedure in a
new environment that binds the formal
parameters of the procedure to the
arguments it is applied to.”
• We’ve replaced environments with
substitution.
• We’ve replaced eval with reduction.
6 Feb 2001
CS 655: Lecture 6
12
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
6 Feb 2001
CS 655: Lecture 6
13
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.
6 Feb 2001
CS 655: Lecture 6
14
Example
 f. (( x.f (xx)) ( x. f (xx)))
6 Feb 2001
CS 655: Lecture 6
15
Alyssa P. Hacker’s 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))
 ...
6 Feb 2001
CS 655: Lecture 6
16
Ben Bitdiddle’s 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)
 ...
6 Feb 2001
CS 655: Lecture 6
17
Be Very Afraid!
• Some -calculus terms can be reduced forever!
• The order in which you choose to do the
reductions might change the result!
6 Feb 2001
CS 655: Lecture 6
18
Take on Faith (for now)
• 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 is there is
one.
– This is normal order reduction – corresponds to
normal order (lazy) evaluation
6 Feb 2001
CS 655: Lecture 6
19
Who needs primitives?
T  xy. x
F  xy. y
if  pca . pca
6 Feb 2001
CS 655: Lecture 6
20
Evaluation
T  xy. x
F  xy. y
if  pca . 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
6 Feb 2001
CS 655: Lecture 6
21
Who needs primitives?
and  xy. if x y F
or  xy. if x T y
6 Feb 2001
CS 655: Lecture 6
22
Coupling
[M, N]  z.z M N
first  p.p T
second  p.p F
first [M, N]
= p.p T (z.z M N)   (z.z M N) T
= (z.z M N) xy. x   (xy. x) M N   M
6 Feb 2001
CS 655: Lecture 6
23
Tupling
n-tuple:
[M] = M
[M0,..., Mn-1, Mn] = [M0, [M1 ,..., [Mn-1, Mn ]... ]
n-tuple direct:
[M0,..., Mn-1, Mn] = z.z M0,..., Mn-1, Mn
Pi,n = x.x Ui,n
Ui,n = x0... xn. xi
What is P1,2?
6 Feb 2001
CS 655: Lecture 6
24
Primitives
What about all those pesky
numbers?
6 Feb 2001
CS 655: Lecture 6
25
What are numbers?
• We need three (?) functions:
succ: n  n + 1
pred: n  n – 1
zero?: n  (n = 0)
• Is that enough to define add?
6 Feb 2001
CS 655: Lecture 6
26
Adding for Post-Docs
add  xy.if (zero? x) y
(add (pred x) (succ y)
6 Feb 2001
CS 655: Lecture 6
27
Counting
0I
1  [F, I]
2  [F, [F, I]]
3  [F, [F [F, I]]
...
n + 1  [F, n]
6 Feb 2001
CS 655: Lecture 6
28
Arithmetic
Zero?  x.x T
Zero? 0 = (x.x T) I = T
Zero? 1 = (x.x T) [F, I] = F
succ  x.[F, x]
pred  x.x F
pred 1 = (x.x F) [F, I] = [F, I]F = I = 0
pred 0 = (x.x F) I = IF = F
6 Feb 2001
CS 655: Lecture 6
29
Factorial
mult 
xy. if (zero? x) 0
(add y (mult (pred x) y))
fact 
x. if (zero? x) 1
(mult x (fact (pred x)))
Recursive definitions should make you uncomfortable.
Need for definitions should also bother you.
6 Feb 2001
CS 655: Lecture 6
30
Summary
• All you need is application and abstraction
and you can compute anything
• This is just one way of representing
numbers, booleans, etc. – many others
are possible
• Integers, booleans, if, while, +, *, =, <,
subtyping, multiple inheritance, etc. are for
wimps! Real programmers only use .
6 Feb 2001
CS 655: Lecture 6
31
Charge
• Problem Set 2 out today
– Make a quantum scheme interpreter
– Longer and harder than PS1
– New teams assigned
• Read Prakash’s Notes
• Be uncomfortable about recursive
definitions
6 Feb 2001
CS 655: Lecture 6
32