Computer Science 101 A Survey of Computer Science

1
Computer Science 101A Survey of Computer Science

• Church-Turing Thesis

2
Where we are

• We have looked at the definition of Turing
  machines simple models of computing agents.
• We have seen a series of examples showing that we
  can do numerical algorithms, rearranging, etc.
• We have seen how we can piece together machines
  for simple problems to solve more complex
  problems.
• With more time, we could do much more impressive
  things.
• So far, we have used a new machine for each
  problem. Now we look at the notion of a
  programmable Turing machine.

3
Encoding a Turing Machine

• To encode symbols of the alphabet Use 1 for
  b Use 11 for second symbol Use 111 for third
  symbol, etc.
• To encode states Use 1 for first state Use
  11 for second state, etc.
• To encode directions Use 1 for L Use 11 for R

4
Encoding a Turing Machine (cont.)

• To encode instructions Encode the components of
  the instructions and separate by 0s.
• Example Suppose alphabet is b,0,1 so b is
  encoded as 1, 0 as 11, 1 as 111.Then the
  instruction (3,1,0,2,L) would be encoded as
  111011101101101.
• To encode a Turing machine Encode the
  instructions and separate by 0s.

5
Encoded TM

• Here is the code (serial number) for a Turing
  Machine with alphabet b,1 1011011010101010110
  1101or Turing Machine number 5985645

6
Universal Turing Machine

• A Universal Turing Machine, U
• U is a Turing Machine
• U takes as input
• The encoding of another TM, M, (this is the
  program) and
• Some string S of input symbols for M
• U produces as output
• Whatever output M would have produced if it had
  been given input S

7
Universal Turing Machine (cont.)

• So a universal Turing machine is programmable in
  that you can program it with the encoding of
  another machine and it will simulate that machine
  (program).
• The details of how U works are beyond the scope,
  but basically
• It keeps sections of its tape
• For the current state of M
• A copy of what Ms tape would look like
• It scans to see what state M would be in, what it
  would be reading. Then it finds an instruction
  that matches. It then carries out the actions
  that M would do.

8
Church Turing Thesis

• Church-Turing Thesis If there is an algorithm to
  do a symbol manipulation task, then there is a
  Turing machine to do that task.
• Or, contrapositively If such a task can not be
  done with a Turing machine, then there is no
  algorithm to do the task.

9
Church Turing Thesis (cont.)

• The Church-Turing Thesis is not a theorem to be
  proven, but an accepted theoretical definition of
  a computation or algorithm.
• Why is Church-Turing Thesis accepted?
• Every algorithm presented was shown to be
  solvable by TM
• Other very different attempts (by very smart
  people) to define computability were shown to be
  equivalent to TMs

10
Proof by Contradiction

• Proof by contradiction is an important proof
  technique used by mathematicians and logicians.
• To prove statement P true
• Make the assumption that P is false
• Show that this leads to a logical contradiction
• Conclude that the assumption must be false i.e.,
  that P must be true.

11
The Halting Problem

• The problem Find TM (algorithm), H, that takes
  as input
• Encoding, M of a TM (program), M, and
• Input for M (input for the program), S
• and outputs
• 1 if M will eventually halt on input S
• 0 if M will not halt on input S
• Note such an algorithm would let us know whether
  or not it is safe to execute the program on the
  given input.

12
Halting Problem (cont.)

• Suppose we have TM, H, as proposed

M b S
H
Halts with 1 on tape if M halts with input S
Halts with 0 on tape if Mnever halts with input S
13
Halting Problem (cont.)

• Modify H by adding instructions to cause it to
  loop forever once it reaches the halt with 1
  configuration.

14
Halting Problem (cont.)

• Modify K by having it first make copy of its
  input and then simulate K

15
Halting Problem (cont.)

• Now suppose we give Q its own encoding for input

16
Halting Problem (cont.)

• So
• Q never halts on input Q if Q halts on input Q
• Q halts on input Q if Q never halts on input Q
• This is a contradiction

• Suppose we have TM, H, as proposed
• Conclusion No such H exists Halting problem is
  unsolvable by algorithm

17
Unsolvable Problems

• Problems for which no algorithm can exist (1000s
  known)
• Decide whether Java program will terminate on
  given input.
• Decide whether two Java programs will always give
  the same output.
• Decide whether a given polynomial has roots that
  are whole numbers.
• Decide whether a given TM will halt if given a
  blank tape