Does it matter which Turing Machine Number is used when you run it with the Universal Turing Machine?

More specifically, you are asking about the infinitely many Turing Machine Numbers, all of which translate back to the same Turing Machine. Does it matter which one of them is used with the Universal Turing Machine?

Theoretically, it shouldn't matter; but in a computer simulation of a Turing Machine, it may.

First of all, a Turing Machine works with an infinitely long tape. Also, there is no restriction on the number of states it can have, or how long any particular "instruction" can be. However, any computer implementation of Turing Machine has such restrictions, including the Excel file that can be downloaded from the Home page of this website.

Secondly, a perfectly good Universal Turing Machine may not have been designed to work with Turing Machine Numbers that were constructed by inserting extra leading zeros in places it did not expect. The same is true of all Turing Machines, in the sense that a Turing Machine can only work with the data for which it was designed, and otherwise is unpredictable. For example, you cannot expected a Turing Machine designed to work with Unary data to give equivalently correct answers if the data is in Expanded Binary notation.

Perhaps it would be instructive to look at three specific examples:

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Example 1:

Run the Universal Turing Machine, using the built-in example from the book UN+1 and also using the built-in sample data.

As you are setting up everything for this run, you may notice that the Turing Machine Number that is brought in is the eighteen-digit binary number 101011010111101010.

You may also notice that the sample data (also from the book The Emperor's New Mind) that is automatically brought in is 00000111100000.

Notice that the Universal Turing Machine will take 5,148 steps and stop, and it does give the correct answer.

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Example 2:

Run the Universal Turing Machine again, but this time, when you're asked What Turing Machine shall the Universal Turing Machine imitate? then answer with I will provide my own Turing Machine.

So, what Turing Machine Number should you provide? Well, if you already saw the page with the answer to the FAQ How do I figure out the Turing Machine Number?, then you saw that if we skipped step 4 of the 6-step process of converting the Turing Machine UN+1 to its Turing Machine Number, we would end up with the nineteen-digit number 1010110010111101010.

The difference between this number and the one given in Example 1 is the 0 (in red) that was inserted in the middle of the number. It's really a leading zero of an "instruction" that would have been removed in Step 4 of the process of converting a Turing Machine to its Turing Machine Number.

So now, when it asks you How will you provide the Binary Number of the Turing Machine that the Universal Turing Machine will emulate?, just answer I'd just like to type it in. And then, just type in this nineteen-digit number.

Finally, when it asks you How will you provide the data for your Turing Machine?, just answer I'd just like to type it in. And then, type in 00000111100000, the same number that was used in Example 1.

Notice that this time the Universal Turing Machine will take 5,338 steps and stop, and again it does give the same correct answer.

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Example 3:

Run the Universal Turing Machine again, and just like in Example 2, when you're asked What Turing Machine shall the Universal Turing Machine imitate? then answer with I will provide my own Turing Machine.

So, what Turing Machine Number shall we provide this time? What we can do this time is take the eighteen-digit number from Example 1, and insert one leading 0 before the last instruction. This will again give us a nineteen-digit binary number 1010110101111001010. Notice that this time, the 0 (in red) was inserted into a different place than in Example 2.

So now, when it asks you How will you provide the Binary Number of the Turing Machine that the Universal Turing Machine will emulate?, just answer I'd just like to type it in. And then, just type in this nineteen-digit number.

Finally, when it asks you How will you provide the data for your Turing Machine?, just answer I'd just like to type it in. And then, type in 00000111100000, the same number that was used in Example 1.

This time, the Universal Turing Machine will run for 180,816 steps and then crash with the Error Message that there aren't enough columns in this Excel file for this Turing Machine to run to completion.