As far as 1 goes, a big part of the reason I want to do this is because I want to generalize computability more, and to look at computers more abstractly
It’s a noble cause, and I suspect you did perceive interesting things that I would like to know more about. However, notice that you seem to overcharge operator « computing » to a non standard (from CTP frame) notion for this word. [pause from internet][reading your comment about computability!=simulatability] Yes, this may be exactly that!
So in other words, and conditioned on I guessed your thoughts well, I would keep 2 as is (« in a manner of speaking » helps!), then rephrase 1, 3&5 as:
Let’s define simulability as the analog of computability and see why simulability is machine and constraint dependent, and why picking different machines will get you different results in what you can simuate, and why picking different time and space requirements will affect the sets of things you can simulate.
Let’s assume some key results from complexity apply to simulability as well, then Cobham’s thesis that the practical problems we can solve correspond to P in complexity theory in our specific universe is unlikely to be true, because of the Time Hierarchy theorem as well as issues with the constant potentially being arbitrarily large.
The underappreciated amount of things a Universal/Complete Turing Machine can simulate, and that it can actually emulate a halting oracle, even if it couldn’t solve the halting problem on it’s own, and the implications of that observation.
And then you could conclude with « Equivalently, I’m trying to say that computability and simulatability are different concepts, and one can simulate something without you being able to solve the computational problem it’s doing. »
Sorry for this comment being so long, I needed to respond to all of the objections properly.
I loved it! 😉 However I’d love more meat on what problems are better seen through the lens of simulability. Could you elaborate or provide a specific example?
I’ll try to mention that the fact that computability is not equivalent to simulatability is the reason I wouldn’t edit my post like that. I will always make clear when I’m switching to a non-standard frame.
I loved it! 😉 However I’d love more meat on what problems are better seen through the lens of simulability. Could you elaborate or provide a specific example?
For a good example from a Turing Machine, obviously, the halting function would be the obvious example of simulability but not computability. Basically any function that is turing reducible to the halting problem would be a good example of simulability but not computability for Universal Turing Machines.
Yes, I get that « periodically switching between non computable tasks » is your paradigmatic case for simulability. The question is: what do you think could get accomplished through this definition? In other words, I could take a pen and write « This is a non standard number. », but that would still be just one piece of paper. In what sense is the notion of simulability better than that?
Essentially, I’m using the definition of the first paragraph of this wikipedia article, where you imitate a real world system. although in this case, the model of the real world and the real world are the same object, so the simulation connotations usually used in lower-scale regimes break down, and thus I will be treating simulation and the real world as the same object.
Ok then that’s standard language and there’s no need to redefine it, you’re right.
Let’s try another example if you indulge my curiosity. If I use some cryptographic method you won’t be able to crack it without hypercomputing, but you could simulate cracking it if you could guess the secret. Is that the kind of thing you have in mind?
It’s a noble cause, and I suspect you did perceive interesting things that I would like to know more about. However, notice that you seem to overcharge operator « computing » to a non standard (from CTP frame) notion for this word. [pause from internet][reading your comment about computability!=simulatability] Yes, this may be exactly that!
So in other words, and conditioned on I guessed your thoughts well, I would keep 2 as is (« in a manner of speaking » helps!), then rephrase 1, 3&5 as:
Let’s define simulability as the analog of computability and see why simulability is machine and constraint dependent, and why picking different machines will get you different results in what you can simuate, and why picking different time and space requirements will affect the sets of things you can simulate.
Let’s assume some key results from complexity apply to simulability as well, then Cobham’s thesis that the practical problems we can solve correspond to P in complexity theory in our specific universe is unlikely to be true, because of the Time Hierarchy theorem as well as issues with the constant potentially being arbitrarily large.
The underappreciated amount of things a Universal/Complete Turing Machine can simulate, and that it can actually emulate a halting oracle, even if it couldn’t solve the halting problem on it’s own, and the implications of that observation.
And then you could conclude with « Equivalently, I’m trying to say that computability and simulatability are different concepts, and one can simulate something without you being able to solve the computational problem it’s doing. »
I loved it! 😉 However I’d love more meat on what problems are better seen through the lens of simulability. Could you elaborate or provide a specific example?
I’ll try to mention that the fact that computability is not equivalent to simulatability is the reason I wouldn’t edit my post like that. I will always make clear when I’m switching to a non-standard frame.
For a good example from a Turing Machine, obviously, the halting function would be the obvious example of simulability but not computability. Basically any function that is turing reducible to the halting problem would be a good example of simulability but not computability for Universal Turing Machines.
Yes, I get that « periodically switching between non computable tasks » is your paradigmatic case for simulability. The question is: what do you think could get accomplished through this definition? In other words, I could take a pen and write « This is a non standard number. », but that would still be just one piece of paper. In what sense is the notion of simulability better than that?
Essentially, I’m using the definition of the first paragraph of this wikipedia article, where you imitate a real world system. although in this case, the model of the real world and the real world are the same object, so the simulation connotations usually used in lower-scale regimes break down, and thus I will be treating simulation and the real world as the same object.
Link is below:
https://en.wikipedia.org/wiki/Simulation
Ok then that’s standard language and there’s no need to redefine it, you’re right.
Let’s try another example if you indulge my curiosity. If I use some cryptographic method you won’t be able to crack it without hypercomputing, but you could simulate cracking it if you could guess the secret. Is that the kind of thing you have in mind?