Turing Machines and Undecidability

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Turing Machinesand Undecidability
Eric Roberts Symbolic Systems 100 May 19, 2009
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Can Machines Think?

• In his essay in Mind from 1950, Alan Turing began
  his inquiry into this question with the following
  paragraph

Computing Machinery and Intelligence A. M. Turing
1 The Imitation Game I propose to consider the
question, Can machines think? This should
begin with definitions of the meaning of the
terms machine and think. The definitions
might be framed so as to reflect so far as
possible the normal use of the words, but this
attitude is dangerous. If the meaning of the
words machine and think are to be found by
examining how they are commonly used it is
difficult to escape the conclusion that the
meaning and the answer to the question, Can
machines think? is to be sought in a statistical
survey such as a Gallup poll. But this is absurd.

• In my first lecture back in April, I offered some
  brief historical insights into what philosophers
  mean by think today, my goal is to explore
  what computer scientists mean by machine.

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Mathematics Enters the 20th Century
If we would obtain an idea of the probable
development of mathematical knowledge in the
immediate future, we must let the unsettled
questions pass before our minds and look over the
problems which the science of today sets and
whose solution we expect from the future.

• In 1900, the eminent German mathematician David
  Hilbert set out a series of challenging problems
  for mathematicians in the twentieth century.
• Many of those problems turned out to be
  relatively easy, but several remain unsolved even
  today.
• Several of Hilberts 23 original problems, along
  with others he devised later, were resolved in a
  way that shook the foundations of the
  mathematical community.

David Hilbert (1862-1943)
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Hilberts Entscheidungsproblem

• Of the problems that have significance to
  computer science, the most important is the
  Entscheidungsproblem, which was posed in 1928.
  In informal terms, the Entscheidungsproblem can
  be expressed as follows

Is it possible to find a mechanical procedure
that can determine, given a specific proposition
in a formal system of symbolic logic, whether
that proposition is provable within that system?

• Hilbert and the mathematicians of his day assumed
  that such a mechanical procedure was indeed
  possible, but a series of mathematical results in
  the 1930sby Kurt Gödel, Alonzo Church, Alan
  Turing, and Emil Postshowed that such a
  mechanical procedure is a logical impossibility.

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Alan Turings Contribution

• Alan Mathison Turing, a young British
  mathematician just out of Cambridge, helped
  settle the Entscheidungsproblem by developing a
  model for computation by a mechanical procedure.
• Turings modelwhich is now known as a Turing
  machineis a central concept in theoretical
  computer science.
• Turing is widely recognized as one of the most
  important figures in the history of computer
  science. The fields most prestigious prize is
  the Turing Award, which is given in his honor.

Alan Turing (1912-1954)
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Optional Movie
Tonight, May 19, 900 P.M. Cubberley 334
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Designing the Turing Machine

• In his groundbreaking 1936 paper, On computable
  numbers, with an application to the
  Entscheidungsproblem, Turing described the
  process of computation in informal terms

• These operations and the notion of a state of
  mind form the basis for the Turing machine.

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Turing Machine Components
Computation requires Scratch paper An
unbounded amount of space At least two
symbols A read/write mechanism Some form of
program control
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A Sample Turing Machine
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1R2
1L2
1L1
1L0
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Representing Numbers

• Even though the standard Turing machine alphabet
  consists of the digits 0 and 1, it is not
  practical to represent numbers in binary.

• Even though the standard Turing machine alphabet
  consists of the digits 0 and 1, it is not
  practical to represent numbers in binary. Why?

• Instead, numbers will be written in unary in
  which each number is written as a sequence of 1s.
  The 0 symbol is used to indicate the start and
  end of a number.

• An input configuration for the Turing machine is
  well-formed if it consists of a single number in
  which the tape head appears over the first 1
  digit.

• A Turing machine program is a function if it
  starts with one well-formed number and ends with
  a well-formed number.

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The Add3 Function (M3)
1L2
1L1
1
1L3
2
1R3
1L0
3
Try it with an input value of 2
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
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The Doubler Function (M2x)
0R0
0R2
1
0R3
1R2
2
1R4
1R3
3
1L5
4
0L6
1L5
5
0R1
1L6
6
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Composing Machines (M2x3)
Suppose you wanted to compute the function 2x
3, given that you have the machines M2x and M3.
0R0
0R2
0R7
1
0R3
1R2
2
1R4
1R3
3

• Start with the two machines.

M2x
1L5
4

• Renumber the states in M3.

0L6
5
1L5

• Combine the machines.

0R1
1L6
6

• Change halt transitions in M2x to jump to M3.

M3
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The Busy Beaver Problem

• Although it is possible to introduce the notion
  of undecidable problems using Turings original
  argument involving a universal Turing machine,
  it is much easier to do so in the context of a
  more recent problem posed by Tibor Radó in the
  early 1960s

What is the largest finite number of 1s that can
be produced on blank tape using a Turing machine
with n states?
Tibor Radó (1895-1965)

• This problem is called the Busy Beaver Problem.

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The Function BB(n)

• The Busy Beaver problem has a natural expression
  as a mathematical function. If n represents the
  number of states, let BB(n) represent the largest
  finite number of 1s that can be written on blank
  tape by a machine of that size.
• For very small values on n, it is fairly easy to
  determine the value of the BB function

BB(1) 1
BB(2) 4
BB(3) 6

• From there, the situation gets much harder.
  Proving that BB(4) 14 was a Ph.D. thesis. No
  one is yet sure of the values for any higher
  number, although conjectures exist for BB(5) and
  BB(6).

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Known Bounds for BB(n)
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Computing BB(n)

• Given that BB(n) is a mathematical function, it
  makes sense to ask whether that function can be
  computed by a Turing machine. In other words, is
  there a machine MBB that takes a number of 1s
  representing n as input and writes out a number
  of 1s representing BB(n) as output?
• It turns out that the answer is no. There is no
  Turing machine MBB that computes the BB function.
  Whats more, well be able to prove that such a
  function cannot be computed at all.
• The BB function is an example of an uncomputable
  function, which is the profound new idea that
  Alan Turing and several of his contemporaries
  introduced to the mathematical world.

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Observations about BB(n)

• In order to resolve the question about whether
  BB(n) can be computed by a Turing machine, it
  helps to make two observations about the BB
  function

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Proof by Contradiction

• In seeking to prove that BB(n) is not computable
  by a Turing machine, the simplest approach is to
  employ a strategy called proof by contradiction.
  In proof by contradiction, you start by assuming
  the opposite of what you wish to prove, and then
  showtypically by constructing a specific
  examplethat leads to an absurd conclusion or
  that violates one of the assumptions. If the
  steps in your construction are correct, the only
  questionable part of the process is the original
  assumption.
• Thus, to prove that BB(n) is not computable by a
  Turing machine, we start by assuming that it is.
  That means that we can assume the existence of a
  machine MBB with ? states that takes n as input
  and writes out BB(n) 1s as output.
• The essence of the contradiction is to construct
  a machine with k states that writes out more than
  BB(k) 1s.

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Steps in the Proof
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But Wait . . . Why Cant You . . .

• Despite the proof by contradiction, the idea that
  BB(n) is uncomputable seems wrong. After all, we
  can simulate a Turing machine. Why isnt it
  possible to solve this problem using the
  following approach

• There is a problem here. Some of the machines go
  on forever, so there is no way to terminate the
  computation in step 2.
• If it were possible to tell whether a Turing
  machine would halt, it would be possible to
  compute the BB(n) function.

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The Halting Problem in Java
/ File HaltCheck.java --------------------
This program purports to determine whether
some other program, specified by entering the
name of the file containing the code, goes
into an infinite loop when executed. / public
class HaltCheck extends ConsoleProgram
public void run() print("Enter name of
source file ") String filename
readLine() if (doesProgramHalt(filename))
println("This program halts.")
else println("This program goes into
an infinite loop.") public
boolean doesProgramHalt(String filename)
. . . code to test the test whether the program
halts . . .
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The Halting Problem in Java
/ File Paradox.java ------------------
This program uses doesProgramHalt to establish a
contradiction. / public class Paradox extends
ConsoleProgram public void run() if
(doesProgramHalt("Paradox.java"))
while (true) / loop forever /
else println("This
program halts.") public boolean
doesProgramHalt(String filename) . . .
code to test the test whether the program halts .
. .
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The Church-Turing Thesis

• The question of what is computable by a Turing
  machine is important in a search for what is
  generally computable mostly because no one has
  ever found a more powerful model.
• Most computer scientists believe what has come to
  be known as the Church-Turing thesis

No method of computation carried out by a
mechanical process can be more powerful than a
Turing machine.
Alonzo Church (1903-1995)

• This claim remains a conjecture, and it is not
  clear there is any way to prove it. At the same
  time, it has so far resisted all efforts to
  disprove it.

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The Plan for Thursday

• I will look more closely at the Church-Turing
  thesis and how one can prove that two
  computational models are equivalent.
• Discuss the arguments in the section of Turings
  paper entitled Contrary views on the main
  question.

• The theological objection
• The head in the sands objection
• The mathematical objection
• The argument from consciousness
• Arguments from various disabilities
• Lady Lovelaces objection
• Arguments from continuity in the nervous system
• The argument from informality of behaviour
• The argument from extra-sensory perception

You should come to class prepared to discuss any
of these viewpoints and be prepared to defend at
least one.
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The End