Transcript Class 32: The Meaning of Truth CS200: Computer Science University of Virginia Computer Science David Evans http://www.cs.virginia.edu/evans.

Class 32:
The Meaning
of Truth
CS200: Computer Science
University of Virginia
Computer Science
David Evans
http://www.cs.virginia.edu/evans
Menu
• Lambda Calculus Review
• How to Prove Something is a Universal
Computer
• Making “Primitives”
7 April 2004
CS 200 Spring 2004
2
Lambda Calculus Review
term ::= variable |term term | (term)|  variable . term
Parens are just for grouping, not like in Scheme
-reduction
(renaming)
y. M  v. (M [y  v])
where v does not occur in M.
-reduction
(substitution)
(x. M)N   M [ x  N ]
( f. (( x.f (xx)) ( x. f (xx)))) (z.z)
7 April 2004
CS 200 Spring 2004
3
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))
 ...
7 April 2004
CS 200 Spring 2004
4
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)
 ...
7 April 2004
CS 200 Spring 2004
5
Be Very Afraid!
• Some -calculus terms can be -reduced
forever!
• The order in which you choose to do the
reductions might change the result!
7 April 2004
CS 200 Spring 2004
6
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 if there is
one.
– This is normal order reduction – corresponds to
normal order (lazy) evaluation
7 April 2004
CS 200 Spring 2004
7
Universal Language
• Is Lambda Calculus a universal language?
– Can we compute any computable algorithm
using Lambda Calculus?
• To prove it isn’t:
– Find some Turing Machine that cannot be
simulated with Lambda Calculus
• To prove it is:
– Show you can simulate every Turing Machine
using Lambda Calculus
7 April 2004
CS 200 Spring 2004
8
Simulating Every Turing
Machine
• A Universal Turing Machine can simulate
every Turing Machine
• So, to show Lambda Calculus can
simulate every Turing Machine, all we
need to do is show it can simulate a
Universal Turing Machine
7 April 2004
CS 200 Spring 2004
9
Simulating Computation
z z
z
z
z
z
z
z
), X, L
), #, R
(, #, L
2:
look
for (
1
Start
(, X, R
#, 1, -
HALT
#, 0, -
Finite State Machine
7 April 2004
z
z
z
z
z
z
z
z z
z
z
z
• Lambda expression
corresponds to a computation:
input on the tape is
transformed into a lambda
expression
• Normal form is that value of
that computation: output is the
normal form
• How do we simulate the FSM?
CS 200 Spring 2004
10
Simulating Computation
z z
z
z
z
z
z
z
), X, L
), #, R
(, #, L
2:
look
for (
1
Start
(, X, R
#, 1, -
HALT
#, 0, -
Finite State Machine
7 April 2004
z
z
z
z
z
z
z
z z
z
z
z
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
CS 200 Spring 2004
11
Making “Primitives”
from Only Glue ()
7 April 2004
CS 200 Spring 2004
12
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
7 April 2004
CS 200 Spring 2004
13
Charge
• Remember: if you have team requests for
PS8, I must receive them before midnight
tonight!
• Friday: finish discovering truth (and proving
lambda calculus is as powerful as a Turing
Machine)
• Monday: Review session for Exam 2
7 April 2004
CS 200 Spring 2004
14