real count

May 2013
94
10
here are some sample machines that my program has generated so far.
{'H': [0, 'H', 1, 'H'], 'A': [1, 'L', 'I', 1, 'R', 'I'], 'B': [0, 'R', 'E', 1, 'R', 'K'], 'C': [0, 'R', 'I', 1, 'R', 'H'], 'D': [1, 'L', 'J', 1, 'R', 'L'], 'E': [1, 'L', 'A', 1, 'L', 'E'], 'F': [0, 'L', 'H', 1, 'R', 'B'], 'G': [1, 'R', 'L', 0, 'R', 'L'], 'I': [1, 'R', 'L', 0, 'R', 'C'], 'J': [0, 'L', 'N', 0, 'R', 'A'], 'K': [0, 'L', 'C', 0, 'L', 'E'], 'L': [1, 'L', 'M', 0, 'R', 'O'], 'M': [1, 'R', 'G', 0, 'R', 'D'], 'N': [1, 'L', 'L', 1, 'R', 'B'], 'O': [1, 'R', 'P', 1, 'R', 'H'], 'P': [0, 'L', 'P', 1, 'L', 'F'], 'Q': [0, 'L', 'D', 1, 'L', 'J']}
{'H': [0, 'H', 1, 'H'], 'A': [0, 'R', 'I', 0, 'R', 'A'], 'B': [1, 'R', 'B', 1, 'R', 'E'], 'C': [1, 'L', 'H', 0, 'L', 'H'], 'D': [0, 'R', 'C', 0, 'R', 'B'], 'E': [0, 'R', 'J', 0, 'L', 'I'], 'F': [0, 'L', 'A', 0, 'R', 'J'], 'G': [0, 'L', 'A', 1, 'L', 'D'], 'I': [0, 'L', 'I', 1, 'R', 'J'], 'J': [1, 'R', 'G', 1, 'R', 'G'], 'K': [0, 'R', 'B', 1, 'R', 'A']}
{'H': [0, 'H', 1, 'H'], 'A': [1, 'L', 'C', 1, 'R', 'B'], 'B': [1, 'R', 'C', 0, 'L', 'B'], 'C': [1, 'R', 'H', 0, 'R', 'B'], 'D': [1, 'L', 'B', 0, 'L', 'C'], 'E': [1, 'R', 'C', 0, 'L', 'H'], 'F': [0, 'R', 'B', 0, 'R', 'H']}
{'H': [0, 'H', 1, 'H'], 'A': [0, 'L', 'E', 0, 'R', 'I'], 'B': [1, 'R', 'G', 1, 'L', 'M'], 'C': [0, 'L', 'B', 1, 'R', 'N'], 'D': [0, 'R', 'A', 0, 'R', 'F'], 'E': [0, 'L', 'M', 0, 'L', 'E'], 'F': [0, 'L', 'A', 1, 'L', 'E'], 'G': [0, 'R', 'D', 1, 'R', 'L'], 'I': [1, 'R', 'A', 0, 'L', 'K'], 'J': [0, 'L', 'M', 1, 'R', 'F'], 'K': [1, 'L', 'D', 1, 'R', 'O'], 'L': [0, 'R', 'C', 1, 'L', 'I'], 'M': [0, 'L', 'H', 0, 'R', 'O'], 'N': [1, 'L', 'N', 0, 'R', 'G'], 'O': [1, 'R', 'J', 1, 'R', 'F'], 'P': [1, 'R', 'G', 0, 'L', 'A']}
{'H': [0, 'H', 1, 'H'], 'A': [1, 'R', 'X', 1, 'L', 'E'], 'B': [0, 'L', 'FB', 0, 'L', 'IB'], 'C': [0, 'L', 'Z', 0, 'L', 'N'], 'D': [0, 'L', 'R', 0, 'L', 'H'], 'E': [1, 'R', 'I', 0, 'R', 'H'], 'F': [1, 'R', 'P', 0, 'R', 'IB'], 'G': [1, 'R', 'IB', 1, 'R', 'V'], 'I': [1, 'R', 'I', 0, 'R', 'GB'], 'J': [0, 'R', 'C', 1, 'R', 'F'], 'K': [1, 'L', 'K', 1, 'R', 'CB'], 'L': [0, 'R', 'Z', 1, 'L', 'G'], 'M': [0, 'L', 'M', 0, 'R', 'X'], 'N': [1, 'R', 'U', 0, 'R', 'EB'], 'O': [0, 'L', 'C', 0, 'R', 'O'], 'P': [0, 'R', 'IB', 1, 'R', 'W'], 'Q': [1, 'L', 'U', 0, 'L', 'V'], 'R': [1, 'R', 'I', 0, 'R', 'K'], 'S': [0, 'L', 'EB', 1, 'R', 'H'], 'T': [1, 'R', 'O', 1, 'R', 'T'], 'U': [1, 'R', 'IB', 1, 'R', 'P'], 'V': [0, 'L', 'Z', 1, 'R', 'EB'], 'W': [1, 'L', 'D', 1, 'R', 'K'], 'X': [0, 'L', 'Y', 1, 'R', 'P'], 'Y': [1, 'L', 'M', 0, 'R', 'U'], 'Z': [1, 'R', 'J', 0, 'R', 'IB'], 'AB': [1, 'L', 'JB', 1, 'R', 'L'], 'BB': [1, 'R', 'X', 1, 'L', 'T'], 'CB': [0, 'R', 'B', 0, 'R', 'M'], 'DB': [1, 'R', 'H', 0, 'L', 'C'], 'EB': [1, 'L', 'DB', 0, 'L', 'S'], 'FB': [1, 'L', 'T', 1, 'L', 'H'], 'GB': [1, 'L', 'O', 0, 'R', 'V'], 'IB': [0, 'L', 'O', 0, 'R', 'Z'], 'JB': [1, 'L', 'I', 0, 'R', 'M'], 'KB': [1, 'R', 'Y', 0, 'R', 'S'], 'LB': [0, 'L', 'CB', 1, 'R', 'F']}
{'H': [0, 'H', 1, 'H'], 'A': [1, 'L', 'A', 0, 'R', 'A'], 'B': [0, 'L', 'H', 1, 'R', 'C'], 'C': [1, 'R', 'H', 1, 'R', 'C'], 'D': [0, 'R', 'D', 1, 'L', 'B']}
{'H': [0, 'H', 1, 'H'], 'A': [0, 'L', 'EB', 1, 'R', 'AB'], 'B': [0, 'R', 'X', 1, 'L', 'G'], 'C': [1, 'R', 'CB', 0, 'R', 'G'], 'D': [1, 'R', 'CB', 1, 'R', 'H'], 'E': [1, 'L', 'AB', 1, 'L', 'V'], 'F': [0, 'L', 'I', 0, 'L', 'W'], 'G': [0, 'L', 'I', 1, 'R', 'F'], 'I': [1, 'R', 'J', 1, 'L', 'R'], 'J': [1, 'R', 'EB', 0, 'R', 'CB'], 'K': [1, 'L', 'BB', 1, 'L', 'CB'], 'L': [1, 'R', 'H', 1, 'L', 'FB'], 'M': [1, 'L', 'E', 0, 'R', 'O'], 'N': [0, 'L', 'Q', 1, 'L', 'V'], 'O': [0, 'L', 'L', 0, 'L', 'Y'], 'P': [0, 'L', 'E', 0, 'R', 'D'], 'Q': [0, 'R', 'U', 1, 'R', 'FB'], 'R': [0, 'R', 'N', 1, 'R', 'CB'], 'S': [0, 'R', 'U', 1, 'L', 'E'], 'T': [1, 'R', 'Z', 0, 'R', 'X'], 'U': [1, 'R', 'C', 0, 'L', 'K'], 'V': [1, 'R', 'Z', 1, 'L', 'I'], 'W': [0, 'L', 'Y', 1, 'L', 'O'], 'X': [1, 'L', 'Q', 1, 'L', 'Q'], 'Y': [0, 'L', 'BB', 1, 'L', 'K'], 'Z': [1, 'L', 'Q', 1, 'L', 'I'], 'AB': [0, 'L', 'M', 1, 'L', 'R'], 'BB': [1, 'L', 'DB', 0, 'L', 'F'], 'CB': [1, 'L', 'W', 1, 'R', 'L'], 'DB': [1, 'R', 'N', 0, 'R', 'EB'], 'EB': [1, 'R', 'D', 1, 'L', 'X'], 'FB': [1, 'L', 'V', 1, 'L', 'W'], 'GB': [0, 'L', 'EB', 0, 'R', 'EB']}
{'H': [0, 'H', 1, 'H'], 'A': [0, 'L', 'A', 0, 'L', 'C'], 'B': [1, 'R', 'A', 0, 'L', 'D'], 'C': [1, 'R', 'D', 1, 'R', 'A'], 'D': [1, 'R', 'F', 1, 'R', 'J'], 'E': [1, 'R', 'C', 1, 'R', 'H'], 'F': [1, 'L', 'I', 0, 'L', 'I'], 'G': [0, 'L', 'G', 0, 'L', 'H'], 'I': [0, 'R', 'H', 0, 'L', 'I'], 'J': [1, 'R', 'B', 0, 'L', 'I'], 'K': [0, 'R', 'A', 1, 'R', 'C']}
{'H': [0, 'H', 1, 'H'], 'A': [0, 'L', 'H', 1, 'L', 'E'], 'B': [0, 'R', 'H', 1, 'L', 'B'], 'C': [0, 'L', 'H', 1, 'L', 'H'], 'D': [0, 'R', 'H', 1, 'R', 'D'], 'E': [0, 'R', 'H', 1, 'L', 'C'], 'F': [0, 'R', 'A', 1, 'L', 'H']}
{'H': [0, 'H', 1, 'H'], 'A': [1, 'L', 'C', 0, 'R', 'B'], 'B': [0, 'R', 'H', 1, 'L', 'D'], 'C': [0, 'R', 'D', 0, 'R', 'A'], 'D': [1, 'R', 'C', 0, 'R', 'D'], 'E': [1, 'L', 'B', 1, 'L', 'B'], 'F': [0, 'R', 'F', 0, 'L', 'D']}
{'H': [0, 'H', 1, 'H'], 'A': [1, 'L', 'D', 1, 'L', 'B'], 'B': [1, 'R', 'H', 0, 'L', 'C'], 'C': [0, 'R', 'D', 0, 'L', 'D'], 'D': [1, 'L', 'B', 1, 'L', 'K'], 'E': [1, 'R', 'I', 0, 'L', 'C'], 'F': [1, 'R', 'L', 0, 'R', 'A'], 'G': [1, 'L', 'H', 0, 'R', 'D'], 'I': [0, 'R', 'I', 1, 'R', 'B'], 'J': [0, 'L', 'I', 1, 'L', 'A'], 'K': [0, 'R', 'B', 1, 'R', 'L'], 'L': [0, 'L', 'A', 1, 'L', 'K'], 'M': [0, 'R', 'E', 0, 'L', 'H'], 'N': [0, 'L', 'G', 1, 'R', 'J']}
{'H': [0, 'H', 1, 'H'], 'A': [1, 'R', 'C', 0, 'R', 'A'], 'B': [0, 'R', 'C', 1, 'R', 'H'], 'C': [0, 'L', 'A', 0, 'R', 'B']}
{'H': [0, 'H', 1, 'H'], 'A': [0, 'R', 'B', 1, 'L', 'H'], 'B': [0, 'R', 'B', 1, 'L', 'C'], 'C': [0, 'R', 'B', 1, 'L', 'A']}
{'H': [0, 'H', 1, 'H'], 'A': [0, 'R', 'A', 0, 'L', 'H']}
{'H': [0, 'H', 1, 'H'], 'A': [0, 'L', 'CB', 0, 'R', 'C'], 'B': [0, 'R', 'G', 0, 'R', 'A'], 'C': [1, 'L', 'H', 0, 'R', 'G'], 'D': [0, 'R', 'I', 1, 'L', 'P'], 'E': [1, 'L', 'L', 0, 'R', 'P'], 'F': [0, 'R', 'X', 0, 'L', 'EB'], 'G': [0, 'L', 'EB', 1, 'L', 'L'], 'I': [0, 'L', 'G', 1, 'L', 'EB'], 'J': [1, 'L', 'CB', 1, 'L', 'X'], 'K': [0, 'R', 'P', 0, 'L', 'L'], 'L': [0, 'L', 'Y', 0, 'R', 'T'], 'M': [0, 'L', 'V', 1, 'R', 'EB'], 'N': [1, 'L', 'J', 1, 'R', 'W'], 'O': [1, 'R', 'B', 1, 'R', 'D'], 'P': [0, 'R', 'CB', 0, 'R', 'EB'], 'Q': [0, 'L', 'N', 0, 'L', 'H'], 'R': [1, 'R', 'G', 0, 'L', 'L'], 'S': [0, 'R', 'L', 1, 'L', 'V'], 'T': [1, 'R', 'Q', 1, 'R', 'N'], 'U': [0, 'L', 'H', 0, 'R', 'O'], 'V': [1, 'L', 'FB', 1, 'R', 'G'], 'W': [1, 'R', 'B', 1, 'L', 'F'], 'X': [0, 'L', 'EB', 1, 'L', 'G'], 'Y': [1, 'R', 'H', 0, 'L', 'B'], 'Z': [0, 'L', 'Z', 0, 'L', 'W'], 'AB': [1, 'L', 'S', 0, 'L', 'Y'], 'BB': [0, 'R', 'L', 1, 'L', 'M'], 'CB': [1, 'L', 'EB', 0, 'L', 'F'], 'DB': [1, 'L', 'G', 1, 'L', 'N'], 'EB': [0, 'L', 'K', 0, 'L', 'O'], 'FB': [0, 'R', 'T', 1, 'L', 'DB'], 'GB': [1, 'L', 'Z', 0, 'R', 'M']}
{'H': [0, 'H', 1, 'H'], 'A': [1, 'R', 'H', 0, 'L', 'L'], 'B': [0, 'L', 'D', 1, 'R', 'X'], 'C': [1, 'R', 'F', 1, 'L', 'R'], 'D': [0, 'R', 'F', 1, 'R', 'D'], 'E': [1, 'L', 'S', 1, 'L', 'D'], 'F': [0, 'L', 'I', 0, 'L', 'S'], 'G': [1, 'R', 'C', 1, 'R', 'D'], 'I': [1, 'R', 'I', 0, 'R', 'Q'], 'J': [1, 'R', 'S', 0, 'R', 'E'], 'K': [0, 'L', 'N', 0, 'L', 'I'], 'L': [1, 'L', 'N', 1, 'L', 'H'], 'M': [1, 'L', 'C', 0, 'L', 'H'], 'N': [0, 'R', 'V', 1, 'R', 'C'], 'O': [0, 'L', 'V', 1, 'R', 'V'], 'P': [1, 'L', 'V', 0, 'R', 'I'], 'Q': [1, 'R', 'P', 1, 'R', 'T'], 'R': [0, 'R', 'U', 0, 'R', 'B'], 'S': [0, 'R', 'U', 0, 'L', 'M'], 'T': [0, 'L', 'U', 1, 'L', 'A'], 'U': [0, 'L', 'K', 1, 'L', 'N'], 'V': [0, 'R', 'J', 0, 'R', 'L'], 'W': [1, 'L', 'B', 1, 'R', 'V'], 'X': [0, 'L', 'B', 0, 'R', 'P'], 'Y': [0, 'R', 'N', 1, 'R', 'N']}
{'H': [0, 'H', 1, 'H'], 'A': [0, 'L', 'C', 0, 'R', 'F'], 'B': [0, 'L', 'K', 1, 'R', 'J'], 'C': [1, 'L', 'B', 1, 'L', 'D'], 'D': [1, 'R', 'J', 1, 'L', 'G'], 'E': [1, 'R', 'P', 0, 'L', 'E'], 'F': [0, 'R', 'B', 0, 'L', 'P'], 'G': [0, 'L', 'C', 1, 'L', 'A'], 'I': [1, 'R', 'O', 0, 'L', 'M'], 'J': [0, 'R', 'G', 0, 'L', 'D'], 'K': [1, 'R', 'I', 0, 'R', 'O'], 'L': [1, 'R', 'A', 0, 'R', 'L'], 'M': [1, 'L', 'A', 1, 'R', 'D'], 'N': [0, 'R', 'L', 1, 'L', 'P'], 'O': [0, 'R', 'P', 1, 'L', 'J'], 'P': [1, 'L', 'B', 0, 'R', 'H'], 'Q': [0, 'R', 'G', 0, 'L', 'M'], 'R': [0, 'R', 'H', 0, 'L', 'K']}
{'H': [0, 'H', 1, 'H'], 'A': [1, 'R', 'H', 0, 'R', 'B'], 'B': [1, 'R', 'A', 0, 'R', 'H']}
{'H': [0, 'H', 1, 'H'], 'A': [1, 'L', 'B', 1, 'L', 'H'], 'B': [1, 'R', 'A', 0, 'R', 'B']}
{'H': [0, 'H', 1, 'H'], 'A': [1, 'R', 'B', 1, 'R', 'B'], 'B': [1, 'L', 'B', 1, 'L', 'A']}
{'H': [0, 'H', 1, 'H'], 'A': [0, 'L', 'C', 0, 'L', 'B'], 'B': [0, 'L', 'D', 1, 'L', 'E'], 'C': [1, 'L', 'C', 1, 'L', 'A'], 'D': [0, 'R', 'D', 1, 'R', 'H'], 'E': [1, 'L', 'D', 0, 'R', 'H']}
{'H': [0, 'H', 1, 'H'], 'A': [0, 'R', 'H', 0, 'L', 'B'], 'B': [0, 'L', 'A', 0, 'R', 'B'], 'C': [0, 'R', 'B', 0, 'L', 'H']}
 
Aug 2012
2,463
759
i've decided to prove it by running the program. several bugs were in my original program, which i took care of.
it should take about 2 months to fully run the program. i haven't looked at the current known bits of chiatins omega.
but will after my program is complete.
I'm at the limit of my knowledge of computer science and Chaitin's ideas. I think your idea is interesting but I think you're wrong that your program will "prove it," whatever "it" is. Do you mean you intend to falsify Turings proof of the unsolvability of the Halting problem? That's crank territory. Perhaps you could be more circumspect as to what exactly it is you're doing.

I see that you are attempting to randomly generate a large number of TMs, see how many of them halt within a given number of steps; and thereby approximate Chaitin's Omega, the probability that a randomly chosen TM will halt.

There are some issues I have with your idea, which I'll just enumerate. As I say I'm right at the limits of my knowledge of theoretical CS so if I'm wrong or fuzzy about something it's perfectly possible.

* First, a TM is defined as a particular program written in some predefined formal languag, as Turing defined it. You are initializing random finite state machines and letting them run. It's not at all clear to me that one of your state machines can be used to approximate or represent a TM. For one thing, as I say I don't know whether a FSM is a good proxy for a TM in this context. But secondly, I'm concerned that you are not writing TM programs but rather initializing state machines. I don't know if these are equivalent ideas or not. But this is not how TMs are defined.

* Now why is it important that we define a TM as a particular program written in fixed and pre-decided formal language? It's because a program is then a finite string of symbols, and we can enumerate them first by length, and then alphabetically among programs of the same length.

Having now put all the possible TMs into a well-defined order, we can THEN write a 1-bit for all those programs that halt;, and a 0-bit otherwise; and thereby define Chaitin's Omega.

Your idea doesn't do that. What is Chaitin's Omega for you? How is it defined? You haven't shown us how to enumerate your finite state machines (which aren't TMs anyway!) You see you have a conceptual issue here you need to address.

* Note by the way that I've just showed that there's not a single Omega. Rather, for each choice of formal language, you get a different set of programs and a different Omega. So Omega is actually a class of numbers, not a single number. And in YOUR conception, you haven't defined it at all; because you haven't defined your TMs as programs, as Turing does; and you haven't shown us how to enumerate your state machines.

* Also note how I defined Omega. For each TM in some enumeration of all possible TMs, we write 1-bits for those that halt and 0-bits for those that don't. Well, how do I do that?? No Turing machine can determine that for all TMs. That's Turing's beautiful result. So we can DEFINE Omega; but we can't COMPUTE Omega. In fact Chaitin's Omega is in the class of real numbers that are definable (which has a particular technical definition) but not computable.

* And finally, say you get some result that says .45459485934950305438503 of all your state machines halt within N steps for some huge N. You have generated a rational number! You haven't generated Omega. I am far from convinced that you can approximate Omega this way.

I just wanted to toss out some ideas for you to think about and subjects for you to research as you go forward. There is no point is disagreeing with me on anything since as I say for all I know I have some of the computer science wrong. I won't even defend anything I've written and I claim no authority in the subject. These are just things I happen to think based on the small amount of theoretical CS that I've run across.

To sum up, the biggest issue is that TMs are programs written in a fixed formal language; and your "not actually TM's" are randomly initialized finite state machines, which (absent an enumeration) do not even define an Omega.

So that's everything I know about it. I think your idea is interesting. I just don't think it proves what you think it does; and there are some gaps in your theoretical understanding that you should patch up as you go.
 
Last edited:
Aug 2012
2,463
759
ps -- The more I read this Wiki page the less I understand what Omega is. It seems somewhat more subtle than just lining up the bits yes/no depending on whether the n-th TM halts. I would like to understand how this works. If anyone knows ...


Here's an article that says that:

Every Chaitin constant is simultaneously computably enumerable (the limit of a computable, increasing, converging sequence of rationals), and algorithmically random (its binary expansion is an algorithmic random sequence), hence uncomputable

Now that's interesting ... the limit of a computable sequence of rationals. Very much like your idea.

I don't doubt that you have something interesting going on here. But I don't actually understand the official definition of Omega so I should work on that.

Here's a good article by Chaitin. It seems more clear than the Wiki page.

Omega and why maths has no TOEs
 
Last edited:
May 2013
94
10
I'm at the limit of my knowledge of computer science and Chaitin's ideas. I think your idea is interesting but I think you're wrong that your program will "prove it," whatever "it" is. Do you mean you intend to falsify Turings proof of the unsolvability of the Halting problem? That's crank territory. Perhaps you could be more circumspect as to what exactly it is you're doing.

I see that you are attempting to randomly generate a large number of TMs, see how many of them halt within a given number of steps; and thereby approximate Chaitin's Omega, the probability that a randomly chosen TM will halt.

There are some issues I have with your idea, which I'll just enumerate. As I say I'm right at the limits of my knowledge of theoretical CS so if I'm wrong or fuzzy about something it's perfectly possible.

* First, a TM is defined as a particular program written in some predefined formal languag, as Turing defined it. You are initializing random finite state machines and letting them run. It's not at all clear to me that one of your state machines can be used to approximate or represent a TM. For one thing, as I say I don't know whether a FSM is a good proxy for a TM in this context. But secondly, I'm concerned that you are not writing TM programs but rather initializing state machines. I don't know if these are equivalent ideas or not. But this is not how TMs are defined.

* Now why is it important that we define a TM as a particular program written in fixed and pre-decided formal language? It's because a program is then a finite string of symbols, and we can enumerate them first by length, and then alphabetically among programs of the same length.

Having now put all the possible TMs into a well-defined order, we can THEN write a 1-bit for all those programs that halt;, and a 0-bit otherwise; and thereby define Chaitin's Omega.

Your idea doesn't do that. What is Chaitin's Omega for you? How is it defined? You haven't shown us how to enumerate your finite state machines (which aren't TMs anyway!) You see you have a conceptual issue here you need to address.

* Note by the way that I've just showed that there's not a single Omega. Rather, for each choice of formal language, you get a different set of programs and a different Omega. So Omega is actually a class of numbers, not a single number. And in YOUR conception, you haven't defined it at all; because you haven't defined your TMs as programs, as Turing does; and you haven't shown us how to enumerate your state machines.

* Also note how I defined Omega. For each TM in some enumeration of all possible TMs, we write 1-bits for those that halt and 0-bits for those that don't. Well, how do I do that?? No Turing machine can determine that for all TMs. That's Turing's beautiful result. So we can DEFINE Omega; but we can't COMPUTE Omega. In fact Chaitin's Omega is in the class of real numbers that are definable (which has a particular technical definition) but not computable.

* And finally, say you get some result that says .45459485934950305438503 of all your state machines halt within N steps for some huge N. You have generated a rational number! You haven't generated Omega. I am far from convinced that you can approximate Omega this way.

I just wanted to toss out some ideas for you to think about and subjects for you to research as you go forward. There is no point is disagreeing with me on anything since as I say for all I know I have some of the computer science wrong. I won't even defend anything I've written and I claim no authority in the subject. These are just things I happen to think based on the small amount of theoretical CS that I've run across.

To sum up, the biggest issue is that TMs are programs written in a fixed formal language; and your "not actually TM's" are randomly initialized finite state machines, which (absent an enumeration) do not even define an Omega.

So that's everything I know about it. I think your idea is interesting. I just don't think it proves what you think it does; and there are some gaps in your theoretical understanding that you should patch up as you go.
 
May 2013
94
10
Do you mean you intend to falsify Turings proof of the unsolvability of the Halting problem?
of course not. only give an aproximation of halting.
First, a TM is defined as a particular program written in some predefined formal languag, as Turing defined it. You are initializing random finite state machines and letting them run.
the only differance between a finte state machine and a TM, according to the link you provided, is a TM has infinite memory. well no computer has infinite memory, and i would be interested if you could provide a turing machine that can't be converted to a FSM.
I'm concerned that you are not writing TM programs but rather initializing state machines.
as an example, imagine a two tape turing machine, one tape for the state machine, the other for the program itself.
You haven't shown us how to enumerate your finite state machines
this shouldn't be nessicary. omega is based on the probability of a randomly given TM halting, so enumeration is "useless"
but it you want to enforce enumeration...
one state FSM.
{"H" : [0,"H",1,"H"]} "HALT STATE"
two states FSM
{"A":[0,"L","A",0,"L", "A"], "H" : [0,"H",1,"H"]} IF YOU READ A ZERO OR A ONE, WRITE A ZERO, GO LEFT AND GO TO STATE "A"
1 {"A":[0,"L","A",0,"R", "A"], "H" : [0,"H",1,"H"]}
2 {"A":[0,"R","A",0,"L", "A"], "H" : [0,"H",1,"H"]}
3 {"A":[0,"R","A",0,"R", "A"], "H" : [0,"H",1,"H"]}
4 {"A":[0,"L","A",1,"L", "A"], "H" : [0,"H",1,"H"]}
5 {"A":[0,"L","A",1,"R", "A"], "H" : [0,"H",1,"H"]}
...
64‬ {"A":[1,"R","H",1,"R", "H"], "H" : [0,"H",1,"H"]}
three states FSM
{"A":[0,"L","A",0,"R", "A"],"B":[0,"L","A",0,"R", "A"], "H" : [0,"H",1,"H"]}
say you get some result that says .45459485934950305438503 of all your state machines halt within N steps for some huge N. You have generated a rational number! You haven't generated Omega.
well that's true with any irrational number. for example i could generate the first 10 trillion digits of square root 2, but that's not square root 2.