I have a few clarifying questions about the rules that I am trying to understand.
First Question: Are you allowed to tell Omega an integer that you can’t actually calculate but that Omega can, if you know that integer exists and can uniquely define it even if you can’t calculate it?
Example: You say “The integer calculated as a result of the maximal strategy that you, Omega can construct without falling into a loop.” Does that represent an extremely high integer to Omega, even though you couldn’t actually calculate it yourself because you have less room for computations than Omega does, or is it in an invalid response?
Second question: If you and Mrs. X both give the exact same response (for instance, if the above answer is valid and you both say it), then what happens?
Edit: I read your linked post and you said:
It should be noted that God, or any being capable of hypercomputation, has real problems in these situations: they actually have infinite options (not a finite options of choosing their future policy), and so don’t have any solution available.
Which means that that particular example is probably bad, since there is no integer calculated as a result of the maximal strategy that Omega can construct without falling into a loop.
However, the question “Can you ask Omega to do you calculations that you yourself can’t do?” is still relevant for answers like “The number I am about to say, raised to the power of the number I am about to say, the number I am about to say times. (Long pause as you calculate your Maximum integer.) My Maximum integer.”
Although, I suppose if you can run calculations on a hypercomputer, and you live indefinitely long while talking to it, You could potentially end up getting caught in a infinite loop of saying “The number I am about to say, raised to the power of the number I am about to say, the number I am about to say times… And that number can be calculated by taking the number I am about to say, raised to the power of the number I am about to say, the number I am about to say times, and that number can be calculated...”
So if you could have the hypercomputer help you with calculations, and wanted the highest finite number you and the hypercomputer together could generate, you would also have to ask the hyper computer to increment your number in such a way that you didn’t get caught in an infinite loop while attempting to increment it.
Yes, I think that you and I are talking about the same thing.
Attempting to rephrase, In essence, my question is how specific do I have to make my number, function, terminating algorithm, or noncomputable algorithm.
Clearly 99999999 is valid as a number,
And presumably 3^^^^3 as a function,
But is a program “Hyper G” that generates a number using a terminating algorithm involving Graham’s number being Knuth up arrowed to Graham’s number, having the result stored in a variable, and then having the variable Knuth up arrowed to itself iteratively until the process has repeated Graham’s number times valid as a terminating algorithm?
Is “The result of the Busy Beaver Function of a Turing Machine with Hyper G States and Hyper G symbols” valid? You might be able to say that names a large integer, but since it isn’t even a computable function anymore. I don’t know if Omega would accept that as an answer.
I’d say your Knuth up arrow is in, but the Busy Beaver number is out—you can’t use Omega’s (or anyone else’s) hypercomputation to do the job for you, and you can’t compute the Busy Beaver without hypercomputation.
Okay. As another question, to what extent should quantum effects be considered in the area?
1: If there are essentially no Quantum effects, then I don’t have access to a source of true randomness, just pseudorandom numbers. This would influence my coding, because true randomness can be used to lengthen a program in ways that fake randomness cannot, so I would have to adjust my code to take that into account.
[My understanding may be off here, but I think that given a pseudorandom algorithm, there are events which can be defined as to never take place. Ergo, a bad pseudorandom algorithms might never generate “0.01” 4 times in a row. But given quantum randomness, any defined event will eventually happen with probabilities approaching 1 as runtime increases]
2: On the other hand, if there are quantum effects, I can attempt to make programs like the following:
X=0;
DountilHalt;
X=X+1;
Write “S” in Memory Register X;
If the Character in Memory Register X is “0” then halt.
Else goto DountilHalt;
Which would keep running until there is a Quantum bitflip of “S” into “0″ at just the right moment (or some other bitflip in the program that also caused a halt.)
3: Alternatively, I could view it as “Your program can call Quantum Mechanical randomness, if you want it to, but neither your program, nor it’s output, will be changed by Quantum Mechanical effects unless you program that in.”
Which means that the Program in 2 would never halt because I did not call a Quantum function anywhere inside the program.
It seems sort of like the implicit scenario is 3, but I may be incorrect (or I may have cast 1,2 or 3 incorrectly.)
If I were Omega (feels good to think about the possibility), I would demand a program written in a specified high-level computer language which prints a string in the form SSSS...S0 (or something equivalent). This would exclude all sophistries from “the number my opponent chose plus one” to “the largest number you, Omega, can calculate [under specific conditions]”.
Are you allowed to tell Omega an integer that you can’t actually calculate but that Omega can, if you know that integer exists and can uniquely define it even if you can’t calculate it?
It doesn’t really matter much. It changes the methods you use somewhat, but the basic result is the same. For that matter, you could even allow infinite numbers without doing a whole lot.
If you and Mrs. X both give the exact same response (for instance, if the above answer is valid and you both say it), then what happens?
I have a few clarifying questions about the rules that I am trying to understand.
First Question: Are you allowed to tell Omega an integer that you can’t actually calculate but that Omega can, if you know that integer exists and can uniquely define it even if you can’t calculate it?
Example: You say “The integer calculated as a result of the maximal strategy that you, Omega can construct without falling into a loop.” Does that represent an extremely high integer to Omega, even though you couldn’t actually calculate it yourself because you have less room for computations than Omega does, or is it in an invalid response?
Second question: If you and Mrs. X both give the exact same response (for instance, if the above answer is valid and you both say it), then what happens?
Edit: I read your linked post and you said:
Which means that that particular example is probably bad, since there is no integer calculated as a result of the maximal strategy that Omega can construct without falling into a loop.
However, the question “Can you ask Omega to do you calculations that you yourself can’t do?” is still relevant for answers like “The number I am about to say, raised to the power of the number I am about to say, the number I am about to say times. (Long pause as you calculate your Maximum integer.) My Maximum integer.”
Although, I suppose if you can run calculations on a hypercomputer, and you live indefinitely long while talking to it, You could potentially end up getting caught in a infinite loop of saying “The number I am about to say, raised to the power of the number I am about to say, the number I am about to say times… And that number can be calculated by taking the number I am about to say, raised to the power of the number I am about to say, the number I am about to say times, and that number can be calculated...”
So if you could have the hypercomputer help you with calculations, and wanted the highest finite number you and the hypercomputer together could generate, you would also have to ask the hyper computer to increment your number in such a way that you didn’t get caught in an infinite loop while attempting to increment it.
By calculating it you mean writing the decimal expansion? Or is it enough to write a terminating algorithm that does so? Or something else?
Yes, I think that you and I are talking about the same thing.
Attempting to rephrase, In essence, my question is how specific do I have to make my number, function, terminating algorithm, or noncomputable algorithm.
Clearly 99999999 is valid as a number,
And presumably 3^^^^3 as a function,
But is a program “Hyper G” that generates a number using a terminating algorithm involving Graham’s number being Knuth up arrowed to Graham’s number, having the result stored in a variable, and then having the variable Knuth up arrowed to itself iteratively until the process has repeated Graham’s number times valid as a terminating algorithm?
Is “The result of the Busy Beaver Function of a Turing Machine with Hyper G States and Hyper G symbols” valid? You might be able to say that names a large integer, but since it isn’t even a computable function anymore. I don’t know if Omega would accept that as an answer.
I’d say your Knuth up arrow is in, but the Busy Beaver number is out—you can’t use Omega’s (or anyone else’s) hypercomputation to do the job for you, and you can’t compute the Busy Beaver without hypercomputation.
Okay. As another question, to what extent should quantum effects be considered in the area?
1: If there are essentially no Quantum effects, then I don’t have access to a source of true randomness, just pseudorandom numbers. This would influence my coding, because true randomness can be used to lengthen a program in ways that fake randomness cannot, so I would have to adjust my code to take that into account.
[My understanding may be off here, but I think that given a pseudorandom algorithm, there are events which can be defined as to never take place. Ergo, a bad pseudorandom algorithms might never generate “0.01” 4 times in a row. But given quantum randomness, any defined event will eventually happen with probabilities approaching 1 as runtime increases]
2: On the other hand, if there are quantum effects, I can attempt to make programs like the following:
X=0;
DountilHalt;
X=X+1;
Write “S” in Memory Register X;
If the Character in Memory Register X is “0” then halt.
Else goto DountilHalt;
Which would keep running until there is a Quantum bitflip of “S” into “0″ at just the right moment (or some other bitflip in the program that also caused a halt.)
3: Alternatively, I could view it as “Your program can call Quantum Mechanical randomness, if you want it to, but neither your program, nor it’s output, will be changed by Quantum Mechanical effects unless you program that in.”
Which means that the Program in 2 would never halt because I did not call a Quantum function anywhere inside the program.
It seems sort of like the implicit scenario is 3, but I may be incorrect (or I may have cast 1,2 or 3 incorrectly.)
If I were Omega (feels good to think about the possibility), I would demand a program written in a specified high-level computer language which prints a string in the form SSSS...S0 (or something equivalent). This would exclude all sophistries from “the number my opponent chose plus one” to “the largest number you, Omega, can calculate [under specific conditions]”.
It doesn’t really matter much. It changes the methods you use somewhat, but the basic result is the same. For that matter, you could even allow infinite numbers without doing a whole lot.
I’d guess that you both get half a utility.