Lecture L: Computability
Unit L1: What is Computation?
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Slide 1 of No.
Physical Computation
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Slide 2 of No.
Mathematically defined computation
f(x,0)
f(x,y)
g(z,0)
g(z,w)
h(r,0)
h(r,s)
=
=
=
=
=
=
1
g(y,f(x,y-1))
0
h(w,g(z,w-1))
r
h(r,s-1)+1
public class PrintPrimes {
public static void
main(String[] args){
for(j = 2; j < 1000; j++)
if (isPrime(j))
System.out.println(j);
}
private static boolean
isPrime(int n) {
for(int j=2; j < n; j++)
if (n%j == 0)
return false;
return true;
}
}
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Slide 3 of No.
Turing Machines
a
b
0
1
a 0, R, b 1, L, g
b 1, L, b 0, L, a
e
z
z 1, R, x 0, L, a
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Slide 4 of No.
Turing Machine for adding 1
0
a
b
1, _, b
0, _, b
1
start
0, L, a
1, _, b
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halt
Slide 5 of No.
Universal Computers
• Theoretical version: there exists a single Universal Turing
Machine that can simulate all other Turing machines

The input will be a coding of another machine + an input to the
other machine
• Hardware version: stored program computer
• Software version: interpreter
/**
* Runs the first method in the Java class that is given in
* “program” on the parameter given in “input”. Returns
* the value returned by the method. The method must be
* static, accept a single String parameter, and return a
* String value.
*/
static String simulate(String program, String input){ … }
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Slide 6 of No.
The Church-Turing Hypothesis
Everything computable is computable by a Turing machine


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Physical interpretation
Conceptual interpretation
Definition of Computation
Everything computable is computable by a Java program
All “good enough” computers are equivalent
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Slide 7 of No.
CTH: The Modern Version
Everything efficiently computable is computable efficiently by
a randomized Turing machine
c
 “Efficiently”: in “polynomial time”. Running time is O ( n ) for


some constant c.
“Randomized”: allow to “flip coins” – use randomization
All “good enough” computers are equivalent -- even if you worry
about efficiency
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Slide 8 of No.
CTH and Chaos
• The physical world is analog
• When we simulate it on a digital device we use a given
precision
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Input and output are given in finite precision
How many bits of precision do we need?
How much can  error in the input affect the output?
Normally: a little
Chaotic systems: a lot
• When we simulate a digital device on an analog device we
can encode many bits in one analog signal

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Limits to precision of analog systems: atoms, quantum effects
Hard to think of a physical signal of even 64 bits of precision
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Slide 9 of No.
CTH and Quantum Physics
• Only known challenge to the “modern” version of CTH
• We do not know how to simulate quantum mechanical
systems efficiently
• The quantum “probability amplitudes” are different from
normal probabilities
• Maybe “quantum computers” can efficiently do more than
normal randomized computers

Efficient “quantum algorithm” for factoring is known
• Can quantum computers be built?

Existing physics? New physical laws? Technology?
• Can quantum computers be simulated efficiently on a
randomized Turing machines?
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Slide 10 of No.
CTH and the Brain
• Is the brain just a computer?
• Metaphysics: The mind-body problem

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Monism: everything is matter -- including humans
Dualism: there is more (“soul”, “mind”, “consciousness” …)
• Technical differences

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Parallel, complicated, analog, …
Still equivalent to a Turing Machine
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Slide 11 of No.
The Brain vs. a Pentium
• There are 10^9 -- 10^10 neurons in the brain
• Each neuron receives input from 10^3 -- 10^4 synapses
• Each neuron can fire about 1 -- 10^2 times/sec
• Total computing power is 10^9 -- 10^16 ops/sec
• Pentium III computing power = 10^8 ops/sec
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Slide 12 of No.
AI
• Why can’t we program computers to do what humans
easily do?

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Recognize faces
Understand human language
• Processing power?
• Software?

Scruffy vs. Neat debate
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Slide 13 of No.
The Turing Test
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Slide 14 of No.
Lecture L: Computability
Unit L2: The limits of computation and
mathematics
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Slide 15 of No.
The Halting problem
/**
* Return True if the program finishes
* running when given the input as its
* parameter. Return false if it gets into
* an infinite loop.
*/
static boolean halt(String program, String input){
// fill in – can you do it?
}
•
program must be a Java class. The method that is run is the first one
in the class and must be of the form
static boolean XXXX(String input)
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Slide 16 of No.
Diagonalization
static boolean autoHalt(String program){
return halt(program, program);
}
static boolean paradox(String program){
if autoHalt(program)
while(true)
; // do nothing
else
return true;
}
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Slide 17 of No.
The Halting problem cannot be solved
String paradoxString =
“class Test {\n” +
“ static boolean paradox(String program){\n” +
…
“ static boolean halt(String program, String input){\n” +
…
Boolean answer = Test.paradox(paradoxString);
• Will paradox halt when running on its own code?


Yes ==> autoHalt(paradoxString) returns true ==> No
No ==> autoHalt(paradoxString) returns false ==> Yes
• Contradiction. Hence the assumption that
halt(program, input) exists is wrong.
• Theorem: the halt method can not be written.
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Slide 18 of No.
Diophantine Equations
/**
* Produce a set of diophantine equations that have a
* solution if the given program halts on the given input.
*/
static String compileToEquations(String prog, String inp) {
…
}
•
Fact: It is possible to write
compileToEquations above
x y z  xwz  3  0
•
Theorem: There does not exist
any program that can solve
diophantine equations
Proof: If there was, we could
write the halt method.
x z  3w z  2 z  0
•
3
2
2 3
...
z w  zw  2  0
7
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2
6
Slide 19 of No.
Formal Proofs
Theorem:
Proof:
Axiom
Follows from
previous
lines
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dfiecnmOWD
dklmejd
inedejde
8hdl9jlS
DJWUIMDKOPP
dfiecnmOWD
QED
Slide 20 of No.
Axiomatic System
• A syntactic way to show semantic mathematical truths
• Axioms + Deduction rules (logic)
• Must be effective

trivial to determine if a proof is correct
• Must be sound

Anything you prove must be true
• Can it be complete?

Prove all true statements
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Slide 21 of No.
Zermelo-Fraenkel Set Theory
• The basis of all mathematics (one possibility)
ST [z ( z  S  z  T )  S  T ]
STz ( z  S  z  T )
...
• Effective
• Sound, we think
• Anything you prove in Math courses really uses ZFC
• The details are not important, yet we will prove:
• Theorem: ZFC is not complete
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Slide 22 of No.
Proof Checking
/**
* Checks whether the proof is a valid proof of the
* theorem using ZFC axioms.
*/
Static boolean isProof(String theorem, String proof){
… // check line by line
}
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Slide 23 of No.
Computation theory is part of mathematics
/**
* Returns a formal mathematical statement that says
* that the program halts on the given input
*/
static String formalHalt(String program, String input){
…
}
/**
* Returns a formal mathematical statement that says
* that the program does not halt on the given input
*/
static String formalNoHalt(String program, String input){
…
}
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Slide 24 of No.
Godel’s incompleteness theorem
static boolean halt(String program, String input){
String thmYes = formalHalt(program, input);
String thmNO = formalNoHalt(program, input);
StringEnumerator e = new AllStringsEnumeration();
while(true) {
String proof = e.getNext();
if isProof(thmYes, proof)
return true;
if isProof(thmNo, proof)
return false;
}
}
Theorem: ZFC is not complete
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Slide 25 of No.