Is There a Simple Solution to All Complicated Problems?

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In summary: It's actually hard because it is NP-complete.In summary, the conversation discusses the idea that all problems are both "hard" and "easy" depending on the individual's familiarity with them. The question at hand is whether "hard" problems have simple answers, and if we can find a simple answer to one complicated question, then all complicated questions will have simple answers. The conversation also touches on complexity theory and how it relates to problem-solving. It is suggested that our methods of solving problems may need to change in order to make progress.
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magpies
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Ok basicaly it boils down to this... Two types of problems exist, simple problems and complicated problems. The current question is do the "hard" problems have a simple answer.

Ok so here is my take on this question. All problems are "hard" problems in the fact that if you run into a problem that you have not seen before it will be hard to find the solution. But also all problems are also "easy" problems in the fact that when you run into a type of problem you have done before it will be easy for you to find the solution. So what am I trying to say? Well the question is basicaly if we can find a simple answer to one of the complicated questions then all complicated questions will have simple answers. I think this is true. I think what we lose site of is the fact that the simple problems we have today where not always simple and a lot of work went into solving them. Also I would like to note that we have never been so good at solving problems that we did not need to show our work when solving them. This seems to suggest to me that while at the same time we are forgetting that all problems are simple in nature. We also have forgotten that all problems are complex or NP complete in other words. And the only reason we can solve any problems at all is because we did the slow work of putting the puzzle together to get to that point.

Now it seems to me that we are possibly making a mistake in our way of solving problems. If the goal is to be able to solve problems without having to do the work required for solving them then we will at some point need to change our methods of learning how to solve them. Because after 4000 years of advancement we are basicaly right where we left off putting 1 and 1 together. If it is not possible to just get the answer without doing the work then I suppose we will have to be ok with that but if it is possible we should not waste time by not even being wrong by not even trying to figure it out.
 
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It looks like someone just discovered complexity theory :)

What you said about "complex problems", that if there is an easy solution for one of them then all complicated questions will have simple answers is true. I think you're talking about the Cook–Levin theorem. If any NP complete problems are in P, then all of them are, and P = NP.
 
  • #3
Nonono, you got it all wrong. Two types of problems exist, computable problems and non-computable problems!

In any case, complexity theory doesn't mean "hard" or "complex" in the way humans use the word every day. It specifically refers to "how fast" (polynomial time?) a non-deterministic turing machine can solve a decision problem.
 
  • #5
magpies said:
Ok basicaly it boils down to this... Two types of problems exist, simple problems and complicated problems. The current question is do the "hard" problems have a simple answer.

Ok so here is my take on this question. All problems are "hard" problems in the fact that if you run into a problem that you have not seen before it will be hard to find the solution. But also all problems are also "easy" problems in the fact that when you run into a type of problem you have done before it will be easy for you to find the solution. So what am I trying to say? Well the question is basicaly if we can find a simple answer to one of the complicated questions then all complicated questions will have simple answers. I think this is true. I think what we lose site of is the fact that the simple problems we have today where not always simple and a lot of work went into solving them. Also I would like to note that we have never been so good at solving problems that we did not need to show our work when solving them. This seems to suggest to me that while at the same time we are forgetting that all problems are simple in nature. We also have forgotten that all problems are complex or NP complete in other words. And the only reason we can solve any problems at all is because we did the slow work of putting the puzzle together to get to that point.

Now it seems to me that we are possibly making a mistake in our way of solving problems. If the goal is to be able to solve problems without having to do the work required for solving them then we will at some point need to change our methods of learning how to solve them. Because after 4000 years of advancement we are basicaly right where we left off putting 1 and 1 together. If it is not possible to just get the answer without doing the work then I suppose we will have to be ok with that but if it is possible we should not waste time by not even being wrong by not even trying to figure it out.

If you want to see a hard problem, look up the traveling salesmen problem.
 

1. What does it mean for a problem to be NP complete?

NP complete problems are a type of problem in computer science and mathematics that are considered to be among the most difficult to solve. They are characterized by the fact that any other NP problem can be reduced to them in polynomial time.

2. Can all problems be classified as NP complete?

No, not all problems can be classified as NP complete. Only certain types of problems, specifically decision problems, can be classified as NP complete. Other types of problems, such as optimization problems, may have a different classification.

3. What makes it difficult to solve NP complete problems?

NP complete problems are difficult to solve because they require a significant amount of time and resources to find a solution. They often involve exploring a large number of possible solutions, which can be time-consuming and computationally expensive.

4. How do researchers determine if a problem is NP complete?

Researchers typically use a method called reduction to determine if a problem is NP complete. This involves showing that the problem in question can be reduced to a known NP complete problem in polynomial time. If this can be done, then the problem is classified as NP complete.

5. Are there any real-world applications for NP complete problems?

Yes, there are many real-world applications for NP complete problems. Some examples include scheduling problems, routing problems, and resource allocation problems. These types of problems are commonly found in industries such as transportation, manufacturing, and logistics.

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