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Can all problems be solved by computing?

November 9, 2018

Can all problems be solved by computing?

This post is also available in: Spanish

P vs np. Can IT solve any issue? Let’s find out

Today, many people think that, at some point, computing will be able to solve all human problems.

The rise of machine learning and artificial intelligence make us dream of a future in which the algorithms of intelligent machines will be able to answer any question and, therefore, we will be able to obtain solutions for all our setbacks. We’ll just have to ask the question and lie back and wait for the solution.

However, from a mathematical point of view, it is not so clear that things are that way. In this article we are going to talk about a question that mathematicians have been asking themselves for decades and about this answer which has not yet been found. Is it p vs np?

Is it p vs np?

Of course, if you’re not a mathematician or a computer scientist, or if you haven’t been interested at some point in this topic, you won’t have a clue what we’re talking about. And rightly so.

In computing, there is a rough distinction between two types of problems: solvable problems (those that can be solved using sufficient computing power) and unsolvable problems, known as not decidable problems (those that could never be solved, no matter how much computing power was used).

Within the solvable problems we find another category, which is called the unrivalled problems, that is, those that theoretically can be solved, but for which it would be necessary to use so much calculation time that they are at the same level as the unsolvable problems. To understand it better, it is not about a human being taking charge of the calculations, but of whether or not there is an algorithm that is capable of performing the calculations in a reasonable period of time.

Having made these distinctions, let’s keep moving forward. The problems that we can solve in a reasonable period of time are called polynomials, represented by the letter “P”. On the other hand, there is a type of problems that groups all the P’s, to which many others are added whose category is, for the moment, unknown (we could say that we still do not know if we can include them in class P or the difficulty of their resolution supposes that they are intractable problems).

Thus, and having laid these foundations we arrive at the question that gives title to this epigraph? Is it p vs np? Can we include that entire category that we still do not know if they are resolvable within the P category or should we place them all within the intractable problems?

At this point, you may be wondering yourself. Well, it’s not that the world is going to collapse according to the sense of the answer, but if it were P≠NP, the computer scientists would stop looking for a solution for a certain (and numerous) type of problems when they consider them intractable, that is, unsolvable in practice.

However, if the answer were that P=NP this could be good news for many industries and services, which would have the possibility of solving many problems whose solution we do not know today.

Returning to the initial question. Is it p vs np?

The truth is that if we “ascend” to our “dirty” real world from the complex and exact world of mathematics, things are not always so precise.

Although our entire reality has a mathematical basis (or, at least, that’s what many mathematicians say), the real world is so complex that deciphering it accurately is too often an impossible task.

If we translate this into the algorithms we talked about at the beginning of this article (machine learning and artificial intelligence), what we find is that when these are faced with a problem that they have to solve in the real world, they usually do not offer an exact answer.

What’s the meaning of this? Of course, it does not mean that algorithms use an intuitive or “non-mathematical” way of thinking. On the contrary, mathematics is its “language” and calculation its favourite tool. However, the complexity and even irresolubility of many problems means that algorithms do not always offer exact solutions, but rather estimate probabilities (often deciding between the option that seems most likely).

A simple example that can be seen by everyone. When, in 2011, the artificial intelligence program Watson (developed by IBM) participated in – and won – the question-and-answer television show “Jeopardy”, it did not offer categorical answers to the questions posed by the host of the show. On the contrary, it drew up a list of possible answers, to which it gave a percentage of probability of being correct, and offered as a definitive answer the one that obtained the highest probability of success.

Many of the “weak” artificial intelligence algorithms we know today work in a similar way. The problems that arise in the real world can be so complex and require so many variables to be taken into account that it is impossible to give an exact mathematical answer, so you have to “trust” the probabilities. But to answer some problems, it’s better than relying on our limited human brains!

So, if we go back to the question that gives title to this article, we could say no, that computing cannot give an exact answer to any problem, although it is possible that it can help us (it already does today and will do so even better in the future) to give a more or less satisfactory solution to most of the problems we have to face (if we are still here to see it!).

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