# Dissertation versus thesis

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If the solution to a problem is easy to check for correctness, must the problem be easy to solve? The existence of problems within NP but outside both P and NP-complete, under that assumption, was established by Ladner’s theorem. The P versus NP problem is a major unsolved problem in computer science. The underlying issues were first discussed in the 1950s, in letters from John Forbes Nash Jr. National Security Agency, and from Kurt GĂ¶del to John von Neumann. Consider Sudoku, a game where the player is given a partially filled-in grid of numbers and attempts to complete the grid following certain rules.

Given an incomplete Sudoku grid, of any size, is there at least one legal solution? NP question would determine whether problems that can be verified in polynomial time, like Sudoku, can also be solved in polynomial time. NP, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be verified in polynomial time. Aside from being an important problem in computational theory, a proof either way would have profound implications for mathematics, cryptography, algorithm research, artificial intelligence, game theory, multimedia processing, philosophy, economics and many other fields. Although the P versus NP problem was formally defined in 1971, there were previous inklings of the problems involved, the difficulty of proof, and the potential consequences. In 1955, mathematician John Nash wrote a letter to the NSA, where he speculated that cracking a sufficiently complex code would require time exponential in the length of the key. The relation between the complexity classes P and NP is studied in computational complexity theory, the part of the theory of computation dealing with the resources required during computation to solve a given problem.

In such analysis, a model of the computer for which time must be analyzed is required. 8 believed the question may be independent of the currently accepted axioms and therefore impossible to prove or disprove. In 2012, 10 years later, the same poll was repeated. NP question, the concept of NP-completeness is very useful.

NP-complete problems are a set of problems to each of which any other NP-problem can be reduced in polynomial time, and whose solution may still be verified in polynomial time. That is, any NP problem can be transformed into any of the NP-complete problems. NP-hard problems are those at least as hard as NP problems, i. NP-hard problems need not be in NP, i. Levin theorem, so any instance of any problem in NP can be transformed mechanically into an instance of the Boolean satisfiability problem in polynomial time. NP-complete problem can be formulated as follows: given a description of a Turing machine M guaranteed to halt in polynomial time, does there exist a polynomial-size input that M will accept? The first natural problem proven to be NP-complete was the Boolean satisfiability problem, also known as SAT.

NP-complete contains technical details about Turing machines as they relate to the definition of NP. Versus, problems outside of P are known. Just as the class P is defined in terms of polynomial running time, the class EXPTIME is the set of all decision problems that have exponential running time. The problem of thesis the truth of a statement in Presburger arithmetic requires even more time. Hence, the problem dissertation known to need more than exponential run time.

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Even more difficult are the undecidable problems, such as the halting problem. It is also possible to consider questions other than decision problems. P: whereas an NP problem asks «Are there any solutions? P problem asks «How many solutions are there? NP then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems.

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. However, the best known quantum algorithm for this problem, Shor’s algorithm, does run in polynomial time, although this does not indicate where the problem lies with respect to non-quantum complexity classes.

All of the above discussion has assumed that P means «easy» and «not in P» means «hard», an assumption known as Cobham’s thesis. First, it is not always true in practice. A theoretical polynomial algorithm may have extremely large constant factors or exponents thus rendering it impractical. NP, there may still be effective approaches to tackling the problem in practice. Second, there are types of computations which do not conform to the Turing machine model on which P and NP are defined, such as quantum computation and randomized algorithms. It is also intuitively argued that the existence of problems that are hard to solve but for which the solutions are easy to verify matches real-world experience. NP, then the world would be a profoundly different place than we usually assume it to be.

There would be no special value in «creative leaps,» no fundamental gap between solving a problem and recognizing the solution once it’s found. NP is the total lack of fundamental progress in the area of exhaustive search. This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration.

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