3 ms·

but that's exactly my point - there are clear-cut halting, clear-cut non-halting, and "we don't know yet" programs. My interest is in delineating the properties

but that's exactly my point - there are clear-cut halting, clear-cut non-halting, and "we don't know yet" programs. My interest is in delineating the properties that separate the 3. The collatz conjecture is exactly the example I use when explaining the "we don't know yet" programs. If I could build a logic system that can positively prove halting, and one that can positively prove non-halting, then anything that remains is "dunno". The halting problem is basically showing that you can never prove the category of "dunno" to be empty. Which is pretty self-evident given there are unsolved bounding problems like the collatz conjecture in mathematics. For a minimal example, if I create a program that takes an AST and runs it through two truth functions, one that checks if the AST === "return;" and prints "it halts" and one that checks if the AST === "while(true) {}" and prints "it never halts", then we've already created such an application, just with a "dunno" space containing the majority of programs. Adding practical axioms to a logic system could grow the "halts" and "never halts" spaces to the point where only a subset of applications cannot be excluded from the "dunno" space, and my question is: what are the unique properties of those applications?

lisper 8y ago
> the properties that separate the 3 But that is exactly my point: what separates these three classes is the state of our current mathematical knowledge, and that obviously changes over time. > Which is pretty self-evident No, that is not at all self-evident. (If it were, someone would have proven it long before Turing, and he wouldn't be famous for it.) I don't know what you mean by a "bounding" problem, but we can prove a lot of things about programs with a structure similar to Collatz. For example, we can prove that a program that searches for a prime larger than its input will always halt for any input.
- beaconstudios 8y ago
  it's self-evident _at this point in time_, given you could feed such a machine the collatz conjecture as an algorithm and we just don't know if any arbitrary input will halt. I'm not claiming it is self-evident in perpetuity. By bounding problem I guess I mean the mathematical equivalent of knowing it can halt. I'm not a mathematician as I previously stated so I don't know the terminology. I don't think the space I'm talking about is equivalent to the state of all of our mathematical knowledge, as AFAIK not all problems relate directly to this space. But a subset of all mathematics, sure (perhaps computability, or logic in general although of a higher kind than first-order). My interest is in creating a static analyser based on the set of axioms or deductions that can be derived from that space.
  - lisper 8y ago
    > it's self-evident _at this point in time_ No, it's not. Just because you and I don't know the answer is not proof that no one knows the answer. Someone may have discovered the answer in the last five minutes. (In fact, for any mathematical discovery there is always necessarily a period time when only one person knows the answer.) > AFAIK not all problems relate directly to this space And I am telling you that you are wrong about that. If you think that there is anything about computation that does not relate to math then you have a deep misunderstanding of either computation or math (possibly both).
    - beaconstudios 8y ago
      alright, if you're going to resort to nitpicking and being wilfully insulting then I see there's not much point discussing this further with you.
      - lisper 8y ago
        Let me rephrase that to hopefully make it sound a little less insulting: if you are able to identify an aspect of computation that cannot be described mathematically then you will be remembered alongside Turing, Godel and von Neuman as one of the greatest thinkers humanity has ever produced. (But a priori, the odds of your being able to do that seem pretty low to me.)
        beaconstudios 8y ago
        well you've just gone and been inaccurate yourself - I said that I didn't believe all mathematical discoveries had an impact on computability, but you just suggested I was saying that not all computation is based in mathematics. Which is absurd and it's clear you think I'm stupid, where to me it's pretty clear you are misinterpreting what I'm saying, whether that's poor communication on my part or uncharitable interpretation on yours. I've been friends with academics before who had an unyielding desire to interpret lay speak as an attempt to be rigorous and then criticise it as such, as you have done with me saying "self-evident", but surely you realise I'm not trying to say "self-evident" in some kind of rigorous mathematical sense? I mean if you're just trying to be obtuse it's working well but it comes off like you're trying to hold my conversational and largely intuition-based ideas to a much higher standard than I am trying to meet. I don't understand the impulse at all. If I were trying to be totally rigorous with my thoughts rather than just conversationally trying to communicate an idea that is still taking shape, I would be learning the correct syntax to write a paper and sharing that instead.
- vignesh_m 8y ago
  I think op's problem is different than the one normally studied - he wants to design a halting program and find the class of programs for which it works. This is typically not very popular as these statements depend a lot on the program, and the representations of programs unlike the halting problem computability statement.
  - lisper 8y ago
    Yes, I understand that. And what I'm saying is: you can't do that. The halting problem is formally equivalent to proving arbitrary mathematical theorems. For every program, there is a corresponding theorem which is true IFF the program halts, and for every theorem there is a corresponding program that halts IFF the theorem is true. So the question: for what programs can the halting problem be decided? is formally equivalent to the question: what theorems can we prove? And we can't prove any useful theorems about that because if we could, we could solve the halting problem! [UPDATE] I want to revise my claim that if we could prove interesting theorems about which programs have decidable halting properties that we could solve the halting problem itself. That's not true, because the word "interesting" is too poorly defined. But what is true is that if we could do this then we could prove theorems that we could not previously prove (that has to be true for any reasonable definition of "interesting"). Furthermore, we could now prove these theorems in a purely mechanistic way with no need for human creativity. By any stretch of the imagination that would be a major breakthrough.
    - chriswarbo 8y ago
      >> he wants to design a halting program and find the class of programs for which it works. > Yes, I understand that. And what I'm saying is: you can't do that. Not sure if it's relevant, but it's certainly possible to define an approximate halting-problem-decider, then figure out what it does and doesn't work for. For example, a trivial implementation which checks if the given program is equal to `42`, outputs "halts" if it is, outputs "don't know" if it isn't. The space of programs it works for is { `42` }, the space of programs it doesn't work for is `remove(42, enumerateAllPrograms)`

lisper 8y ago

> the properties that separate the 3 But that is exactly my point: what separates these three classes is the state of our current mathematical knowledge, and that obviously changes over time. > Which is pretty self-evident No, that is not at all self-evident. (If it were, someone would have proven it long before Turing, and he wouldn't be famous for it.) I don't know what you mean by a "bounding" problem, but we can prove a lot of things about programs with a structure similar to Collatz. For example, we can prove that a program that searches for a prime larger than its input will always halt for any input.

beaconstudios 8y ago
it's self-evident _at this point in time_, given you could feed such a machine the collatz conjecture as an algorithm and we just don't know if any arbitrary input will halt. I'm not claiming it is self-evident in perpetuity. By bounding problem I guess I mean the mathematical equivalent of knowing it can halt. I'm not a mathematician as I previously stated so I don't know the terminology. I don't think the space I'm talking about is equivalent to the state of all of our mathematical knowledge, as AFAIK not all problems relate directly to this space. But a subset of all mathematics, sure (perhaps computability, or logic in general although of a higher kind than first-order). My interest is in creating a static analyser based on the set of axioms or deductions that can be derived from that space.
- lisper 8y ago
  > it's self-evident _at this point in time_ No, it's not. Just because you and I don't know the answer is not proof that no one knows the answer. Someone may have discovered the answer in the last five minutes. (In fact, for any mathematical discovery there is always necessarily a period time when only one person knows the answer.) > AFAIK not all problems relate directly to this space And I am telling you that you are wrong about that. If you think that there is anything about computation that does not relate to math then you have a deep misunderstanding of either computation or math (possibly both).
  - beaconstudios 8y ago
    alright, if you're going to resort to nitpicking and being wilfully insulting then I see there's not much point discussing this further with you.
    - lisper 8y ago
      Let me rephrase that to hopefully make it sound a little less insulting: if you are able to identify an aspect of computation that cannot be described mathematically then you will be remembered alongside Turing, Godel and von Neuman as one of the greatest thinkers humanity has ever produced. (But a priori, the odds of your being able to do that seem pretty low to me.)
      - beaconstudios 8y ago
        well you've just gone and been inaccurate yourself - I said that I didn't believe all mathematical discoveries had an impact on computability, but you just suggested I was saying that not all computation is based in mathematics. Which is absurd and it's clear you think I'm stupid, where to me it's pretty clear you are misinterpreting what I'm saying, whether that's poor communication on my part or uncharitable interpretation on yours. I've been friends with academics before who had an unyielding desire to interpret lay speak as an attempt to be rigorous and then criticise it as such, as you have done with me saying "self-evident", but surely you realise I'm not trying to say "self-evident" in some kind of rigorous mathematical sense? I mean if you're just trying to be obtuse it's working well but it comes off like you're trying to hold my conversational and largely intuition-based ideas to a much higher standard than I am trying to meet. I don't understand the impulse at all. If I were trying to be totally rigorous with my thoughts rather than just conversationally trying to communicate an idea that is still taking shape, I would be learning the correct syntax to write a paper and sharing that instead.
vignesh_m 8y ago
I think op's problem is different than the one normally studied - he wants to design a halting program and find the class of programs for which it works. This is typically not very popular as these statements depend a lot on the program, and the representations of programs unlike the halting problem computability statement.
- lisper 8y ago
  Yes, I understand that. And what I'm saying is: you can't do that. The halting problem is formally equivalent to proving arbitrary mathematical theorems. For every program, there is a corresponding theorem which is true IFF the program halts, and for every theorem there is a corresponding program that halts IFF the theorem is true. So the question: for what programs can the halting problem be decided? is formally equivalent to the question: what theorems can we prove? And we can't prove any useful theorems about that because if we could, we could solve the halting problem! [UPDATE] I want to revise my claim that if we could prove interesting theorems about which programs have decidable halting properties that we could solve the halting problem itself. That's not true, because the word "interesting" is too poorly defined. But what is true is that if we could do this then we could prove theorems that we could not previously prove (that has to be true for any reasonable definition of "interesting"). Furthermore, we could now prove these theorems in a purely mechanistic way with no need for human creativity. By any stretch of the imagination that would be a major breakthrough.
  - chriswarbo 8y ago
    >> he wants to design a halting program and find the class of programs for which it works. > Yes, I understand that. And what I'm saying is: you can't do that. Not sure if it's relevant, but it's certainly possible to define an approximate halting-problem-decider, then figure out what it does and doesn't work for. For example, a trivial implementation which checks if the given program is equal to `42`, outputs "halts" if it is, outputs "don't know" if it isn't. The space of programs it works for is { `42` }, the space of programs it doesn't work for is `remove(42, enumerateAllPrograms)`