Foundations of Software DesignFall 2002Marti Hearst: Foundations of Software Design Fall 2002 Marti Hearst Lecture 29: Computability, Turing Machines, Can Computers Think?
Computability: Computability Is there anything a computer cannot compute?
Linked to the notion of what is an algorithm.
Alan Turing: Alan Turing An amazing scientist
Helped solved the Enigma Machine (WWII)
Advances in Probability Theory
Invented the theory behind computers
Turing Machine
Turing Test
Turing Machines: Turing Machines Anything that can be computed by a finite set of rules can be computed by a Turing machine.
Turing Equivalence
We saw that finite automata are less powerful than TMs
Can’t compute
Thus not Turing-equivalent
Modern computers and programming languages are all Turing-equivalent
Slide5: Regular Languages Context-Free Languages Languages accepted by
Turing Machines
A Turing Machine: A Turing Machine ...... ...... Tape Read-Write head Control Unit
The Tape: The Tape ...... ...... Read-Write head No boundaries -- infinite length The head moves Left or Right
Slide8: ...... ...... Read-Write head The head at each time step:
1. Reads a symbol
2. Writes a symbol
3. Moves Left or Right
Slide9: ...... ...... Example: Time 0 ...... ...... Time 1 1. Reads 2. Writes 3. Moves Left
The Input String: The Input String ...... ...... Blank symbol head Head starts at the leftmost position
of the input string Input string
States & Transitions: States & Transitions Read Write Move Left Move Right
Slide12: Example: ...... ...... Time 1 current state
Slide13: ...... ...... Time 1 ...... ...... Time 2
Determinism: Determinism Allowed Not Allowed Turing Machines are deterministic
Halting: Halting The machine halts if there are
no possible transitions to follow
Slide16: Example: ...... ...... No possible transition HALT!!!
Final States: Final States Allowed Not Allowed Final states have no outgoing transitions
In a final state the machine halts
Acceptance: Acceptance Accept Input If machine halts
in a final state Reject Input If machine halts
in a non-final state
or
If machine enters
an infinite loop
Infinite Loop Example: Infinite Loop Example A Turing machine for language
Slide20: Time 0
Slide21: Time 1
Slide22: Time 2
Slide23: Time 2 Time 3 Time 4 Time 5 ... Infinite Loop
Church-Turing Thesis: Church-Turing Thesis The Church-Turing thesis says that Turing machine algorithms are the same as our intuitive notion of algorithms.
Most people think the Church Turing thesis is correct.
It means, among other things that
You can simulate any computer program that runs on any hardware in any language with any other program on any other hardware using any other language.
Tasks that humans agree is an algorithm can always be executed on a computer.
The Halting Problem: The Halting Problem An example of something that is not computable.
Created by Turing in 1936 to define a problem which no algorithmic procedure can solve.
Can we write a program that will take in a user's program and inputs and decide whether
it will eventually stop, or
it will run infinitely in some infinite loop ?
Proof (by contradiction): Proof (by contradiction) Assume that it is possible to write a program to solve the Halting Problem.
Denote this program by HaltAnswerer(prog,inputs).
HaltAnswerer(prog,inputs) will
return yes if prog will halt on inputs and
no otherwise.
A program is just a string of characters
E.g. your Java program is just a long string of characters
An input can also be considered as just a string of characters
So HaltAnswerer is effectively just working on two strings
Proof (cont.): Proof (cont.) We can now write another program Loopy(prog) that uses HaltAnswerer
The program Loopy(prog) does the following:
[1] If HaltAnswerer(prog,prog) returns yes,
Loopy will go into an infinite loop.
[2] If HaltAnswerer(prog,prog) returns no,
Loopy will halt.
Proof (cont.): Proof (cont.) [1] If HaltAnswerer(prog,prog) returns yes,
Loopy will go into an infinite loop.
[2] If HaltAnswerer(prog,prog) returns no,
Loopy will halt.
Consider what happens when we run Loopy(Loopy).
If Loopy loops infinitely,
HaltAnswerer(Loopy,Loopy) return no which by [2] above means Loopy will halt.
If Loopy halts,
HaltAnswerer(Loopy,Loopy) will return yes which by [1] above means Loopy will loop infinitely.
Conclusion: Our assumption that it is possible to write a program to solve the Halting Problem has resulted in a contradiction.
Diagonalization: Diagonalization Not Diagon Alley
A proof by contradiction technique
Uses the notion of Infinity
The name comes from a proof that shows you can’t ever list all numbers.
Assume you list all possible binary numbers
Diagonalization shows you can always construct a new number that is not yet in the list
Have to assume numbers can have infinite length
Construct the new number by choosing the opposite of the number on the diagonal.
Diagonalization: Diagonalization B1 0 0 0 0 0 …
B2 0 0 1 0 1 …
B3 1 0 1 1 0 …
B4 0 1 1 1 1 …
B5 1 1 0 0 0 …
…
D 1 1 0 0 1 …
The Universal Turing Machine: The Universal Turing Machine There are an infinite number of Turing Machines
There are an infinite number of calculations that can be done with a finite set of rules.
However, we can define a Universal Turing Machine which can simulate all possible TMs
Comes from the definition of TMs
You convert the description of the TM and its input into two tapes, and use these as the input to the UTM
The Halting Problem: The Halting Problem Halting Problem:
There is no procedure for telling whether an arbitrary TM will halt on a given input.
Use Diagonalization to show this.
Again, proof by contradiction
Assume there is a rule for deciding if a TM will halt.
Construct a table as follows:
List all Turing machines down the side
List the possible inputs across the top
In position (j,i) put the result of executing Turing machine j on input i
If it halts, output H
If it doesn’t halt, output ?
Diagonalization: Diagonalization 1 2 3 4 5 …
T1 H H ? H H …
T2 ? ? H H H …
T3 H H H H H …
T4 H H H ? H …
T5 H H H H H …
…
D ? H ? H ? …
Diagonalization on the Halting Problem: Diagonalization on the Halting Problem Now define a new TM called D that will halt for all inputs. It outputs
H if TMi(i) does not halt
? if TMi(i) does halt
We already said that the assumption is that we can always decide if a TM halts.
Also, we said this table lists all possible TMs.
So D must be in the table.
But this means that we are saying that D outputs halt if it doesn’t halt!
To see this, give D as input to itself.
This is a contradiction. Hence the premise does not hold: We cannot determine if an arbitrary program will halt.
“Going Meta”: “Going Meta” This proof based in part on Gödel’s Theorem
If you are interested in these kinds of questions (and other things related to “going meta” like compiler compilers), see
Gödel, Escher, Bach: An Eternal Golden Braid, by Douglas Hofstadter.
The Turing Test: The Turing Test An observer
Interacts with a keyboard and monitor
Has to distinguish which of two respondents is a computer and which is human.
There is a contest with a $100,000 prize!
For the first computer whose responses are indistinguishable from a human's.
The Loebner Prize http://www.loebner.net/Prizef/loebner-prize.html
It’s actually pretty easy to fool people over the short term.
Chat room ’bots work quite well.
What is Intelligence?: What is Intelligence? Do androids dream … ?
What would it take for a computer’s thoughts to be indistinguishable from a human’s?
THIS is the deepest question of CS.