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Computability
Universal Turing Machine. Countability. Halting
Problem.
Homework: Show that the integers have the
same cardinality (size) as the natural numbers.
Postings.
Enumerator definition?
• Need to describe parts…
• Transition function
• Output
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 (<M,w>) is the same as M(<w>)
– Accepts if M accepts w
– Rejects if M rejects w
– Loops (fails to halt) if M fails to halt
Claim
We can build such a U.
• Use 3 tapes.
1. For initial information (definition of M and w)
2. Hold status info for which state of M and
position in w
3. 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.
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.
Last class
• Atm = {<M,w>|M 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
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)
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 onto
1 to 1 means: if f(a) = f(b) then a=b.
onto means: if b is in B, then there is an a such
that f(a) = b
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!
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).
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 …
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!
Diagonalization
• General technique, possible only if there is a
list….
Halting Problem
Atm = {<M,w>|M 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(<M>) : run H on (M, <M>). 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.
Proof, continued
Then, what is D(<D>)? Run H(D, <D>). By
definition, this is D running on <D>. If H
returns accept (D running on <D> accepts)
then H rejects. If it rejects, then accept.
Contradiction!
Contradiction arises because H could not be
built to be used by D.
Practical implications
• Before there were computers, mainly stored
program computers, there is a result that
there can't be full-proof debuggers/checkers.
Homework
• Produce proof that the integers are countable.
• Posting on practical implications of Halting
problem.
• Posting on history / origin of stored program
concepts.