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.

9.1 The standard Turing Machine Control unit Read-write head Tape abc State p dbc State q Example 9.1 The following figure shows the situation before and after the move caused by the transition. Halt state: A Turing machine will halt whenever it enters a final state or reaches a configuration for which is not defined.

Control unit Read-write head Tape We can think of a Turing machine as a computer. Processing unit with a finite memory. A secondary storage of unlimited capacity. Example 9.2 Consider the Turing machine defined by aababb bb

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.

What is a Standard Turing machine ? 1. The tape is unbounded in both directions, allowing any number of left and right moves. 3. Input (all or some of them) 3. Output (all or some of them) 2. The Turing machine is deterministic.

We denote the following configuration of a Turing machine

*

Solution (1)Starting at the left end of the input and check if the first one is a 0. (2) 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

*

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. * * *

* 1.Replace every 1 by x. 2.Find the rightmost x and replace it with 1. 3.Travel to the right end of the current nonblank region and create a 1 there. 4.Repeat Steps 2 and 3 until there are no more x’s. Solution: To solve the problem, we implement the following intuitive process: **** *** ****** * ****** ** **** ***

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. * Definition 12.2 When the results of a computation for a problem is a simple “yes” or “no”, the problem is called a decision problem. If a decision problem is computable, the problem is called decidable, otherwise, it is undecidable. Computability and Decidablity

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