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Chapter 9 Turing Machines

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Chapter 9 Turing Machines

What would happen if we change the stack in Pushdown Automata into some other storage device?

Truing Machines, which maintains that any computational process, such as those carried out by present-day computers.

9.1The standard Turing Machine

Control unit

Read-write head

Tape

State p

State q

Example 9.1 The following figure shows the situation before and after the move caused by the transition .

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Halt state: A Turing machine will halt whenever it enters a final state or reaches a configuration for which is not defined.

Control unit

Processing unit with a finite memory.

Read-write head

Tape

Example 9.2 Consider the Turing machine defined by

A secondary storage of unlimited capacity.

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We can think of a Turing machine as a computer.

Example 9.3 Consider the Turing machine defined by

It is clear that the machine, whatever the initial information on its tape, will run forever, with the read-write head moving alternately right then left, but making no modifications to the tape. This is an instance of a Turing machine that does not halt. We say that the machine is in an infinite loop.

3. Input (all or some of them)

1. The tape is unbounded in both directions, allowing any number of left and right moves.

3. Output (all or some of them)

What is a Standard Turing machine ?

2. The Turing machine is deterministic.

We denote the following configuration of a Turing machine

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- Solution
- Starting at the left end of the input and check if the first one is a 0.
- If we reach a blank without encountering anything but 0, we terminate and accept the string. If the input contains a 1 anywhere, the string is not in L(00*), and we halt in a nonfinal state.

Solution

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Solution In the tape, represent integers x and y by a number x of 1’s and a number y of 1’s, and put a 0 between x and y.

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Solution: To solve the problem, we implement the following intuitive process:

- Replace every 1 by x.
- Find the rightmost x and replace it with 1.
- Travel to the right end of the current nonblank region and create a 1 there.
- Repeat Steps 2 and 3 until there are no more x’s.

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Chapter 12 Limits of Algorithmic Computation

Definition 12.1 An algorithm is a finite sequence of precise instructions for performing a computation or for solving a problem.

Computability and Decidablity

Definition 12.2

When the results of a computation for a problem is a simple “yes” or “no”, the problem is called adecision problem.

If a decision problem is computable, the problem is called decidable, otherwise, it is undecidable.

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Turing Machine Halting Problem

Theorem 12.1 There does not exist any Turing machine H that behaves as required by Definition 12.3. There halting problem is therefore undecidable.