Turing Machine Design

Question

I recently encountered the following problem:

Give a Turing machine diagram for a Turing machine that on input a string x ∈ {0, 1}∗ halts (accepts) with its head on the left end of the tape containing the string x′ ∈ {0, 1}∗ at the left end (and blank otherwise) where x′ is the successor string of x in lexicographic order; ie the next string in the sequence ε, 0, 1, 00, 01, 10, 11, 000, . . . ε, 0, 1, 00, 01, 10, 11, 000, . . . in which the strings are listed in order of increasing length with ties broken by their corresponding integer value. (Briefly document your TM.)

I am confused as to how to start designing an appropriate solution for it. Could I get a couple of pointers for firstly designing this machine, and then Turing machines in general?

Answer 1

Turing Machines

First off, you need to understand what a Turing machine is, and by Turing machine, I'm assuming you're talking about a Universal Turing Machine . This is a conceptual machine created by the Godfather of Computer Science, Alan Turing.

The machine is built up of some components. First, an infinite tape that contains input. Something like..

1-0-1-1-1-1-0-1-0-1-0

Then a set of rules..

if 1 then 0
if 0 then 1

So when the machine hits the 1 , the output is 0 , as per the rule. We define the machine hitting a value when the read head is set to it. The read head is like the current position in the turing machine. So it will go..

1-0-1-1
^------------Current head.

Then the next iteration:

1-0-1-1
  ^----------Current Head

This machine is actually simulating bitwise NOT functionality. You can also set a state in a turing machine, so for example:

if 1 then enter state 1
if 0 then enter state 0

Big deal right? Example suddenly, now you can do stuff like:

1. if 1 and in state 1 output 1 and enter state 0
2. if 1 and in state 0 output 0 and enter state 1
3. if 0 and in state 0 output 1 and enter state 1
4. if 0 and in state 1 output 0 and enter state 0

And we define a state as our default state. In this example, let's call it state 0 . So when the machine starts, it sees a 1. Well I'm in state 0 and I've just got a 1 so I'm going to do rule number 2 . Output a 0 and enter state 1 . The next number is a 0 . Well I'm in state 1 , so I call rule number 4 . See where this is going? By adding states, you really open up what you can do.

Now, a Universal Turing Machine is what's known as Turing Complete . This means that it can compute any computable sequence. Including, the specification for your assignment!

Your Assignment

So let's look at your assignment in the context of a turing machine.

The aim of the machine is to print out..

0 1 00 01 11 000 001 011 111

So we know that we need to maintain a state. We also know that the state needs to get deeper and deeper. So if you user types in 000 , we need to be able to know that we have three characters to output.

And in terms of homework help, that's I'm afraid all I should be responsibly giving you. A good understanding of the turing machine, plus an understanding of what you need this to do, should result in a start in your research to a solution.