1. David Evans http://www.cs.virginia.edu/~evans
Class 36:
The Meaning of Truth
CS200: Computer Science
University of Virginia
Computer Science
David Evans

2. Menu Lambda Calculus Review
How to Prove Something is a Universal Computer
Making “Primitives”
19 April 2002
CS 200 Spring 2002

3. Lambda Calculus -reduction (renaming) y. M  v. (M [y v])
term ::= variable |term term | (term)|  variable . term
-reduction (renaming)
y. M  v. (M [y v])
where v does not occur in M.
-reduction (substitution)
(x. M)N   M [ x N ]
19 April 2002
CS 200 Spring 2002

4. 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
19 April 2002
CS 200 Spring 2002

5. 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.
19 April 2002
CS 200 Spring 2002

6. Example ( f. (( x.f (xx)) ( x. f (xx)))) (z.z) 19 April 2002
CS 200 Spring 2002

7. 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))
 ...
19 April 2002
CS 200 Spring 2002

8. 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)
 ...
19 April 2002
CS 200 Spring 2002

9. Be Very Afraid! Some -calculus terms can be -reduced forever!
The order in which you choose to do the reductions might change the result!
19 April 2002
CS 200 Spring 2002

10. Take on Faith (until CS655)
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
19 April 2002
CS 200 Spring 2002

11. Church-Turing Thesis
Any mechanical computation can be performed by Lambda Calculus reduction rules
There is a Lambda Calculus term equivalent to every Turing Machine
19 April 2002
CS 200 Spring 2002

12. Proof Show you can implement a Turing Machine using Lambda Calculus
19 April 2002
CS 200 Spring 2002

13. What do we need? Read/Write Infinite Tape Mutable Listsz z z z z z z z z z z z z z z z z z z z
), X, L
), #, R
(, #, L 1
2: look for (
Read/Write Infinite Tape
Mutable Lists
Finite State Machine
Numbers to keep track of state
Processing
Way of making decisions (if)
Way to keep going
Start
(, X, R
HALT
#, 1, -
#, 0, -
Finite State Machine
19 April 2002
CS 200 Spring 2002

14. Lambda Calculus Summary so Far
Rules for substitution
Rules for reduction (only -reduction does real work: substitution)
Normal form (no more reductions possible)
On faith: if you do outermost -reduction first, you find the normal form if there is one
19 April 2002
CS 200 Spring 2002

15. Lambda Calculus Intuition
Lambda expression corresponds to a computation
Normal form is that value of that computation
But...can we do useful computations without primitives?
19 April 2002
CS 200 Spring 2002

16. Making “Primitives”
19 April 2002
CS 200 Spring 2002

17. In search of the truth? What does true mean?
True is something that when used as the first operand of if, makes the value of the if the value of its second operand:
if T M N  M
19 April 2002
CS 200 Spring 2002

18. Don’t search for T, search for if
T  x (y. x)
 xy. x
F  xy. y
if  pca . pca
19 April 2002
CS 200 Spring 2002

19. Finding the Truth if T M N T  x . (y. x) F  x . (y. y)
if  p . (c . (a . 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
Is the if necessary?
19 April 2002
CS 200 Spring 2002

20. Charge Schedule Progress Meetings Now! For your meeting be prepared
To demonstrate and explain what you have done so far
To explain your plans for finishing
19 April 2002
CS 200 Spring 2002