Computability

1
Computability

• Universal Turing Machine. Countability. Halting
  Problem.
• Homework Show that the integers have the same
  cardinality (size) as the natural numbers.
  Postings.

2
Enumerator definition?

• Need to describe parts
• Transition function
• Output

3
Universal Turing Machine

• is a Turing Machine U in which the input comes
  in two parts
• Description/encoding of a Turing Machine M and
• Input string w
• U (ltM,wgt) is the same as M(ltwgt)
• Accepts if M accepts w
• Rejects if M rejects w
• Loops (fails to halt) if M fails to halt

4
Claim

• We can build such a U.
• Use 3 tapes.
• For initial information (definition of M and w)
• Hold status info for which state of M and
  position in w
• Working tape.
• First step is to set up tape 2 with initial state
  and starting position and tape 3 with w.
• Operation states of U use status to simulate M
  operating on tape 3.

5
Stored program

• The notion of a universal Turing machine MAY
  have helped in development of stored programs.
• Note calculators do not have stored programs.
• The notion of a program being treated as data,
  such as done when compiling or interpreting code,
  was critical in the development of computers.
• Topic research idea.

6
Last class

• Atm ltM,wgtM is a TM and w is a string and M
  accepts w is
• Turing recognizable because U recognizes it.
• May not halt because U only simulates M and M
  may not halt.
• But maybe another technique could be better than
  M...
• This is the halting problem

7
Digression infinite sets

• How do we compare infinite sets?
• Certainly,
• the counting numbers (1, 2, 3, ) are contained
  in
• the integers( 0, 1, 2, , -1, -2, ) are
  contained in
• the rationals (all numbers of the form p/q, where
  p and q are integers) are contained in
• the reals (numbers with decimals, possibly
  infinite)

8
New concept cardinality

• Note Sipser uses size.
• Georg Cantor (1873) noticed that two finite sets
  are the same size if the elements in each can be
  paired.
• Definition Two sets A B have the same
  cardinality (size?) if there exists a function
• f A ? B, that is 1 to 1 and onto1 to 1 means
  if f(a) f(b) then ab.
• onto means if b is in B, then there is an a
  such that f(a) b

9
Example

• The natural numbers N are the same cardinality as
  the even natural numbers!
• Let f(n) 2 n. This is 1 to 1 and onto!

10
Countable

• A set is countable if it is finite OR if it has
  the same cardinality as the natural numbers.
• My words a set of countable if you can put all
  the elements in a list (describe the list in the
  case of an infinite set).

11
Rationals are countable!

• Construct a table. Redundant numbers will be
  removed. Red line represents the list.
• 1/1 ½ 1/3 ¼ 1/5
• 2/1 2/2 2/3 2/4 2/5 .
• 3/1 3/2 3/3 ¾ 3/5

12
Are the reals countable?

• Proof by contradiction.
• Suppose there is a list
• 1.0000000000
• 3.14159
• .111222333
• .
• Let x by a number such with whole number 1 and
  number at ith position from decimal point is NOT
  equal to the ith position of the ith element on
  the list. Avoid 0 or 9. Then x is NOT on the
  list!

13
Diagonalization

• General technique, possible only if there is a
  list.

14
Halting Problem

• Atm ltM,wgtM is a TM and w is a string and M
  accepts w is undecidable.
• Proof assume that H is a decider for Atm. Define
  a new Turing Machine D as follows
• D(ltMgt) run H on (M, ltMgt). Output reject if H
  accepts and accept if H rejects. (Like the
  diagonalization). Claim if H exists, then we can
  build D that calls it as a subroutine.

15
Proof, continued

• Then, what is D(ltDgt)? Run H(D, ltDgt). By
  definition, this is D running on ltDgt. If H
  returns accept (D running on ltDgt accepts) then H
  rejects. If it rejects, then accept.
• Contradiction!
• Contradiction arises because H could not be built
  to be used by D.

16
Practical implications

• Before there were computers, mainly stored
  program computers, there is a result that there
  can't be full-proof debuggers/checkers.

17
Homework

• Produce proof that the integers are countable.
• Posting on practical implications of Halting
  problem.
• Posting on history / origin of stored program
  concepts.