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Lecture 11
Turing Machines II
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Turing Machines (II)

• Turing machines are not only interesting as
  language accepters, they provide us with a simple
  abstract model for digital computers in general.
• The input for a computation is all the non-blank
  symbols on the tape at the initial time. The
  output is whatever on the tape after the
  computation.

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Computability of a function defined by a TM

• A function f with domain D is computable if there
  exists some TM
• such that
• We will see shortly, all the common
  mathematical functions, no matter how
  complicated, are computable.

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Example 1

• Given two positive integers x and y, design a TM
  that
• computes xy.
• For simplicity, we use unary notation in which
  any positive integer x is represented by w(x) in
  1, such that w(x) x. We assume that w(x)
  and w(y) are on the tape in unary notation,
  separated by a single 0, with the head on the
  leftmost symbol of w(x). After the computation,
  w(xy) will be on the tape followed by a single
  0, with the head at the left end of the result.

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Example 1 (Conti.)

• We therefore want to design a TM for performing
  the computation
• All we need to do is to move the separating 0
  to the right end of w(y), so that the addition
  amounts to nothing more than the coalescing of
  the two strings. To achieve this, we construct
  with

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Other Basic Operations

• Unary notation is very convenient for programming
  TMs. The resulting programs are much simpler than
  binary or decimal representations.
• Other basic operations are copying strings and
  simple comparisons. These can also be done easily
  on a TM.

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Example 2

• Design a TM that copies strings of 1s.
• More precisely, we want a TM to perform
• To solve the problem, we implement the following
  intuitive process
• 1. Replace every 1 by x.
• 2. Find the rightmost x and replace it with 1.
• 3. Travel to the right end and create a 1 there.
• 4. Repeat steps 2 and 3 until there are no more
  xs.

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Example 2 (Conti.)

• A Turing machine implementing the process is
• where is the only final state. You should
  try to trace the program with some input strings.

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Example 3

• Design a TM as a comparer. More precisely, we
  want a TM to perform
• if
• if
• The idea is to match each 1 on the left of
  dividing 0 with the 1 on the right. At the end of
  the matching, we will have on the tape, one of
  the following

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Example 3 (Conti.)

• Depending on whether or In the
  1st case, we encounter the blank at the right of
  the working place when we attempt to match
  another 1. In the 2nd case, we still find a 1 on
  the right when all 1s on the left have been
  replaced. We use these signs to enter the two
  different states. The complete program is left as
  exercise.

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Combining TMs for Complicated Tasks

• Study how to combine together the TMs which
  perform basic operations into a single TM for
  complicated tasks.
• We start with a high-level description (block
  diagram or pseudo-code), then refine it
  successively until the program is in the actual
  language with which we are working.

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Example 4

• Design a TM that compute the function
• where x and y are positive integers in unary
  representation.

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Example 4 (Conti.)

• The computation of f(x,y) can be described by the
  diagram below.

xy
Adder
Comparer
x,y
f(x,y)
Eraser
0
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Example 5

• Design a TM that multiplies two integers in unary
  notation.
• Assume the initial and final tape contents are
  as follows 00y0x and 0xy0y0x.
• The multiplication can be seen as a repeated
  copying of the multiplicand y for each 1 in the
  multiplier x. The following pseudo-code shows the
  main steps of the process.

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Example 5 (Conti.)

• 1. Repeat the following steps until x contains no
  more 1s
• Find a 1 in x and replace it with another symbol
  a.
• Replace the leftmost 0 by 0y.
• 2. Replace all as with 1s.

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Turings Thesis

• The Turings thesis states that any computation
  that can be carried out by mechanical means
  (digital computer) can be performed by some
  Turing machine.
• No one has been able to suggest a problem,
  solvable by an algorithm, for which a TM program
  cannot be written.

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Homework

• Construct TMs to compute the following functions,
  where x and y are positive integers represented
  in unary.
• 1.
• 2.
• 3.

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Homework (Conti.)

• Write the pseudo-code for the TM program that
  compute the following functions (you may use
  adder, copier or multiplier in your program.)
• 1.
• 2.