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COMP SCI 731 - Lecture 4 - Tuesday, Nov 7, 2000
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Today: o The TM model
       o P 


The TM model
------------

To make things precise we will need a computational model. Most popular model
is a TM.  Popular because of it's simplicity.

Draw a picture.

Definition:

    A TM is a tuple  M = (Q, Sigma, Gamma, delta)

* Q is a finite set , "states".  It contains three special states:
   q_0,  q_Y, q_N.
These are all different from each other.

* Sigma is the "input alphabet".  It is a finite non-empty set of characters.

* Gamma is the "tape alphabet".  It is a superset of Sigma.  It contains one
special character, b, called the blank symbol.  This symbol is not
an element of Sigma.

* delta: Q-{q_Y,q_n} x Gamma -> Q x Gamma x {-1,+1}
is the "transition function".  It tells the machine where to move.

                          Finite 
                          Control
                            |
                            |
                           \|/    read/write head
   -------------------------------------------
      |   |   |   |   |   |   |   |   |   |          Tape
   -------------------------------------------
        -4  -3  -2 -1   0   1   2   3   4

Describe the operation of a TM.

Describe the conventions for how the input is presented to the TM.

Example:  * Make a TM to decide if its input is even.

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Problem 2. Specify a TM that, when started on a blank tape, visits 
           every single tape cell.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++



P 
-


P = { L: L is a language and there exists a TM M s.t.
          * x in L     ==> M(x) says YES
          * x not in L ==> M(x) says NO
          * For some polynomial poly, for all x, #steps_M(x) <= poly(|x|)


Discuss the robustness of P to changes in the model of computation.

Discuss the sense in which P has been a good approximation, for practice,
to what is computable by "reasonable" algorithms.