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>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 tha

>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. I get your point but can you prove it?

mannykannot 6y ago
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."
- crimsonalucard 6y ago
  >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.

mannykannot 6y ago

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."

crimsonalucard 6y ago
>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.