What is the purpose of the concept of the non-deterministic Turing machine?
The Turing machine is apparently an important concept in theoretical computer science. I myself am a complete novice in computer science, but I've heard of it repeatedly in other contexts. Until now, however, I've never understood the purpose of the concept of a " non-deterministic Turing machine " (NTM): As far as I understand from a layman's perspective, you pretend that this NTM can essentially guess the results for an input (oracle) and then quickly check for acceptance. What does "guess" mean here? What is the conceptual difference from deterministic TMs? There, too, I can split it into any number of computation paths (memory is unlimited, after all) and essentially perform "parallel" computations. NTMs aren't about how efficient they are—but then I don't understand what's so interesting about them.
Can someone give me a concrete example so I can see the benefit of the concept?
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In fact, deterministic and non-deterministic turbine machines are the same – that is, they can recognize the same languages and calculate the same functions. By the way, this property has many automatic models, but not all (for example, the cellar machines and the buses).
These non-deterministic turbine machines (NTM) are crucial in the big question whether P = NP, from which one has certainly already heard as an interested layman. In class P, all problems that can be solved by a deterministic turing machine (DTM) in polynomial time with regard to the size of the input are: can be solved efficiently. In the class NP, on the other hand, all the problems that can be solved by an NTM in polynomial time (i.e. “efficient”) are those.
The big question is: are the two classes the same? So far you don’t know, but you guess: No.
The question can also be formulated: Can you build an efficient DTM from any efficient NTM?
I think I understood it now: I thought that the equality between NTMs and DTMs would automatically follow that both classes would have to be equally efficient. This is not the case, however: Equatorialism means only that one can solve any problem with an NTM, even with a DTM, but this is an ad hoc nicjhts about efficiency. The existence of an NTM that efficiently solves a certain problem does not imply that there is also an efficient DTM – is this so?
Exactly! This conclusion:
that unresolved P = NP? Problem is – so he’s just right probably.
I’m going to ask a few more questions about complexity theory and hope you can help me back up.
An NTM is not an OracleTM – the latter is also available.
https://de.wikipedia.org/wiki/non-deterministic_turning machine#%C3%84quivalenz_zu_Deterministic_turning machines
Yes, of course, you can also use a DTM, but it may be simpler to use.
I would like to say something about intuitive importance here.
The question P =?=NP is, in principle, whether any problem for which one can efficiently verify a possible solution (polynomiell) (so it can determine whether it is correct) can also be solved efficiently algorithmically.
The former is the interpretation of an OTM which is equivalent to the NTM, and the latter is the interpretation of the DTM.
At first glance, that makes no sense. But I believe that the concept has now been understood: The NTM assumes that it can carry out all possible branchings in parallel and it is only about how long it needs for one of the final branches. This has more to do with a real machine nicjhts. It is interesting for quantum computers…
A more suitable view is that the NTM only passes through the fastest path without being generally known which is. The connection to the OTM is then disclosed. She “rathers” the fastest path. This is relevant from the point of view of complexity, since the running of all paths generally takes much longer.
It is true, however, that it only results from the theoretical one. The purpose of this is not to be able to apply this in real terms, but to have a formal solution mechanism with which considerations can be made in principle and helps to lead evidence and to understand the “nature” of certain problems, i.e. what makes them so difficult, for example.