>It is not entirely clear what your original point was. The possibility of there being a simpler solution does not justify your claim "there is ALWAYS an opportunity for a reduction to be discovered" (and you specifically capitalized ALWAYS so we wouldn't misunderstand you!)
Let me clarify. If simplicity of a solution cannot be verified then a probability of the existence of a simpler solution will ALAWYS be greater than zero. This is in line with what I wrote but hopefully more clear.
>Furthermore, you were originally objecting to the statement that "the solution can be as simple as the problem but no simpler", but this is a reasonable statement given that, as we both seem to agree,
No I don't agree. The statement is easily disproven with an example of a problem that is more complex than the solution:
>there is no formal metric that captures the sort of complexity we are talking about here
I disagree, fuzzy human intuition can always be formalized given enough thought. If the intuition is flawed and irrational, formalizations of such intuitions will make this evident and further our overall understanding.
But either way even with this fuzzy definition of complexity my example of Newtons laws of motion, seemingly the simplest solution to the laws of motion was not only not the most general or simple solution but ultimately "less correct" than relativity. It always seems as if it's the simplest solution but you actually never know. This is in line with my experience.
>In being pedantic about the lack of a formal definition of such terms, you are avoiding (or, at least, unnecessarily complicating) the sort of issues he was discussing (and we are here).
I'm not trying to be pedantic. I am just trying to say that there are tons of instances where something looks like it's the simplest solution but it is actually not and you can never really know either.
Formal proof that something cannot ever be verified to be in it's most simplest form is just the ultimate supporting pillar. We can talk in terms of opinions and vague human definitions but this kind of argument leads to conversations that are never ending. A formal proof moves the argument towards a final resolution. I actually wasn't expecting a formal proof to be introduced into this argument, but it was introduced from your wikipedia source.
Before you introduced that source, I always knew that verifying that a solution is the simplest possible solution is an impossible endeavor in terms of science. Your source introduced to me that it is also an impossible endeavor in terms of logic.
Of course the strategy to win an argument that has already been verified by formal proof is to move the argument out of the realm of formal proof which you are doing now. It's a valid strategy but I still disagree, even on those fuzzy informal grounds.
So to keep in line with the overall topic. Can more complexity always be destroyed for a given solution? Can complexity always be reduced for a given statement?
The formal answer is: We can absolutely never know.
The thing is, prior to Gödel, one might have said "a probability of the existence of a proof of the decidability of Peano arithmetic will ALAWYS be greater than zero", and, by your argument here, that would have been correct - but, of course, one would have been mistaken.
It is your persistent and insistent use of the word 'ALAWYS' that is complicating things, and things would have been simpler if you had written of possibilities rather than certainties in the first place. You have written a great deal arguing for something that is not in doubt.
>>Furthermore, you were originally objecting to the statement that "the solution can be as simple as the problem but no simpler", but this is a reasonable statement given that, as we both seem to agree,
>No I don't agree. The statement is easily disproven with an example of a problem that is more complex than the solution:
(By the way, it is rather amusing how you cut off my sentence in the middle in order to give it a meaning that the original does not have!)
Here we have an additional confusion. The solution to a travelling salesman problem is just an ordered list of vertices, but doing what you are doing here - just writing down the solution - completely misses the point.
> So to keep in line with the overall topic. Can more complexity always be destroyed for a given solution? Can complexity always be reduced for a given statement?
> The formal answer is: We can absolutely never know.
It is probably just as well that your "Solution: 0" example is beside the point, for if it were not, I would doubt your assertion that we can absolutely never know whether more complexity could always be destroyed for this given solution.
>The thing is, prior to Gödel, one might have said "a probability of the existence of a proof of the decidability of Peano arithmetic will ALAWYS be greater than zero", and, by your argument here, that would be correct - but, of course, one would have been mistaken.
There is a difference you're not seeing. The wikipedia article introduces the fact that it has been Proven that the complexity of a solution can Never be verified. This is equivalent to saying: "It has absolutely been proven that the probability of the existence of a simpler solution is greater than 0"
This is entirely different from what you're saying. It has never been proven that decidability of Peano arithmetic will ALAWYS be greater than zero. We can only make your statement given lack of knowledge; given more knowledge the statement may be untrue. My statement is saying that there is no amount of knowledge in the universe that will allow us to know whether a solution is in its' simplest state ever.
>It is your persistent and insistent use of the word 'ALAWYS' that is complicating things, and things would have been simpler if you had written of possibilities rather than certainties in the first place. You have written a great deal arguing for something that is not in doubt.
Then I clarified it when you specified that it wasn't clear. The reason why I used the term ALWAYS was to emphasize that it is PROVEN to be greater than zero. You didn't like it and said so and then I clarified what I meant.
>(By the way, it is rather amusing how you cut off my sentence in the middle in order to give it a meaning that the original does not have!)
It's not my intention to do this. If you feel it was done, just point it out and I will address it accordingly. If I detect any further malice in your replies I will end this conversation.
>Here we have an additional confusion. The solution to a travelling salesman problem is just an ordered list of vertices, but doing what you are doing here - just writing down the solution - completely misses the point.
That's a valid point. If we called "0" the "conclusion," than shouldn't the "solution" include the computations to arrive at the "conclusion"?
If so I would say that it the computations in the "solution" are still simpler than the problem overall:
problem = "1*2*3*4*5*6*7*2*38*3432*0*232*22*33453*221*334/2*3929330"
def solve(problem):
return 0 if '0' in problem else eval(problem)
If you look carefully at the computational steps of "solve." One branch should do less calculations then the total amount of numbers in the problem. Therefore even the computational steps are smaller/simpler than the total size of the problem.
This is equivalent to saying that there exists problems of size N where the Big Oh complexity of the solution is of O(M) where M < N, which is certainly true. I know I sort of slaughtered the meaning of Big-Oh here but you get it.
>It is probably just as well that your "Solution: 0" example is beside the point, for if it were not, I would doubt your assertion that we can absolutely never know whether more complexity could always be destroyed for this given solution.
The whole point of Kolmogorov complexity is precisely the concept of a minimal program for solving problems of a certain type. The uncomputability of Kolmogorov complexity is beside the point here, just as Turing's halting theorem does not prove that it is always possible that any given program will fail to halt.
> One branch should do less calculations then the total amount of numbers in the problem.
Judging the complexity of a solution by how it responds to a subset of problems is the same sort of thing as just stating the solution, as that subset is the only set of problems that we are considering. In this case, that set is "products of a set of numbers including at least one zero."
>The uncomputability of Kolmogorov complexity is beside the point here, just as Turing's halting theorem does not prove that it is always possible that any given program will fail to halt.
I never implied this. First off the halting problem only proves that you can never CALCULATE or PROVE whether a given program will fail to halt. It does not prove that it will always fail to halt or never fail to halt. I never said or implied otherwise.
The uncomputability of Kolmogorov means you can never CALCULATE or PROVE the simplest form of a program. It does not mean that such a form doesn't exist nor does it mean that we haven't randomly found such a form for a specific program. It just means that even if we have found something that we think is the the Kolmogorov complexity of a given program we can never verify it to be true.
This fact is sufficient to show the following can never proven be true:
"There exists a program where the Complexity can never be reduced or destroyed"
>Judging the complexity of a solution by how it responds to a subset of problems is the same sort of thing as just stating the solution, as that subset is the only set of problems that we are considering. In this case, that set is "products of a set of numbers including at least one zero."
But Kolmogorov complexity calculates exactly what you don't want to judge. See the definition:
>In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output.
From the definition you need a predefined input from the set of all inputs and a predefined language from the set of all languages.
In terms of algorithmic complexity theory, I can define a problem to be as narrow or as large as I want. The domain for the previous problem can the set of sets of numbers that contain one or more zeros.
Subsets from larger problems can be used to create other problems and those problems and all problems in general can be used to make statements about complexity all together. If a subset shows something is not True then you have shown that you cannot make that blanket statement about the subset or the parent set.
>I never implied this... I never said or implied otherwise.
This is a very odd way of responding. When someone makes a point, it does not necessarily carry the implication that you said otherwise.
> First off the halting problem only proves that you can never CALCULATE or PROVE whether a given program will fail to halt.
It does not prove that. There are programs that one can prove will halt, and also others that you can prove will not halt.
> This fact is sufficient to show the following can never proven be true: "There exists a program where the Complexity can never be reduced or destroyed"
This seems to be demonstrating the same sort of misunderstanding as over what the halting theorem proves. Even where we cannot compute the Kolmogorov complexity, it does not follow that there is not a minimal one. As I mentioned before, the whole idea of Kolmogorov complexity is predicated on there being programs of minimal complexity.
Going back to that Wikipedia article, we see that "The theorem says that, among algorithms that decode strings from their descriptions (codes), there exists an optimal one."
>This is a very odd way of responding. When someone makes a point, it does not necessarily carry the implication that you said otherwise.
First phrase was in response to your statement. Second phrase was a followup to my sentence that you dotted out (...). And you accused me of modifying your wording with bad intentions.
>It does not prove that. There are programs that one can prove will halt, and also others that you can prove will not halt.
Semantic error, The halting problem only proves that a program can never CALCULATE or PROVE that a given program will fail to halt. That was what I meant.
>This seems to be demonstrating the same sort of misunderstanding as over what the halting theorem proves. Even where we cannot compute the Kolmogorov complexity, it does not follow that there is not a minimal one. As I mentioned before, the whole idea of Kolmogorov complexity is predicated on there being programs of minimal complexity.
>Going back to that Wikipedia article, we see that "The theorem says that, among algorithms that decode strings from their descriptions (codes), there exists an optimal one."
No. Not a demonstration of misunderstanding of halting. It is a demonstration of misunderstanding of Kolmogorov as you only just introduced me to it today.
Either way given any code, you cannot determine whether it is the most optimal one even if there Does exist an optimal one as the theorem states. So the intent still stands, let me modify the statement to get the semantics correct while preserving the intent:
"For any program you can determine whether Complexity for that program can be reduced or destroyed"
The above statement is False.
So in all practicality even though a simplified statement always exist we can never know if we found it. So the possibility is always there and the intent of what I said is still preserved if not stronger , see initial reply of this post:
"The examples are obvious but from the examples you can deduce that their are many instances in software engineering where such reductions can exist but are not obvious at all."
Kolmogorov still offers axiomatic support for my statement. In fact it makes the statement stronger and changes it too:
"In all instances of software engineering reductions can exist, you can never determine otherwise."
I can't say for sure but I believe that the statement above applies not only to Kolmogorov complexity, but algorithmic complexity as well.
Let me clarify. If simplicity of a solution cannot be verified then a probability of the existence of a simpler solution will ALAWYS be greater than zero. This is in line with what I wrote but hopefully more clear.
>Furthermore, you were originally objecting to the statement that "the solution can be as simple as the problem but no simpler", but this is a reasonable statement given that, as we both seem to agree,
No I don't agree. The statement is easily disproven with an example of a problem that is more complex than the solution:
>there is no formal metric that captures the sort of complexity we are talking about hereI disagree, fuzzy human intuition can always be formalized given enough thought. If the intuition is flawed and irrational, formalizations of such intuitions will make this evident and further our overall understanding.
But either way even with this fuzzy definition of complexity my example of Newtons laws of motion, seemingly the simplest solution to the laws of motion was not only not the most general or simple solution but ultimately "less correct" than relativity. It always seems as if it's the simplest solution but you actually never know. This is in line with my experience.
>In being pedantic about the lack of a formal definition of such terms, you are avoiding (or, at least, unnecessarily complicating) the sort of issues he was discussing (and we are here).
I'm not trying to be pedantic. I am just trying to say that there are tons of instances where something looks like it's the simplest solution but it is actually not and you can never really know either.
Formal proof that something cannot ever be verified to be in it's most simplest form is just the ultimate supporting pillar. We can talk in terms of opinions and vague human definitions but this kind of argument leads to conversations that are never ending. A formal proof moves the argument towards a final resolution. I actually wasn't expecting a formal proof to be introduced into this argument, but it was introduced from your wikipedia source.
Before you introduced that source, I always knew that verifying that a solution is the simplest possible solution is an impossible endeavor in terms of science. Your source introduced to me that it is also an impossible endeavor in terms of logic.
Of course the strategy to win an argument that has already been verified by formal proof is to move the argument out of the realm of formal proof which you are doing now. It's a valid strategy but I still disagree, even on those fuzzy informal grounds.
So to keep in line with the overall topic. Can more complexity always be destroyed for a given solution? Can complexity always be reduced for a given statement?
The formal answer is: We can absolutely never know.