If the solution to a problem is easy to check for theory of computation michael sipser solutions pdf, is the problem easy to solve? 1,000,000 prize for the first correct solution.
For some questions, there is no known way to find an answer quickly, but if one is provided with information showing what the answer is, it is possible to verify the answer quickly. Given a partially filled-in Sudoku grid, of any size, is there at least one legal solution? Thousands of other problems seem similar, fast to check but slow to solve. Decades of searching have not yielded a fast solution to any of these problems, so most scientists suspect that none of these problems can be solved quickly.
However, this has never been proven. Sudoku, can also be solved in polynomial time. 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. In such analysis, a model of the computer for which time must be analyzed is required. In 2012, 10 years later, the same poll was repeated. Boolean satisfiability problem in polynomial time.
M guaranteed to halt in polynomial time, does there exist a polynomial-size input that M will accept? M that takes the solution to be verified as input. Then the question of whether the instance is a yes or no instance is determined by whether a valid input exists. So a polynomial time solution to Sudoku leads, by a series of mechanical transformations, to a polynomial time solution of satisfiability, which in turn can be used to solve any other NP-complete problem in polynomial time. Using transformations like this, a vast class of seemingly unrelated problems are all reducible to one another, and are in a sense “the same problem”.
Hence, the problem is known to need more than exponential run time. It is also possible to consider questions other than decision problems. How many solutions are there? For these problems, it’s very easy to tell whether solutions exist, but thought to be very hard to tell how many. First, it is not always true in practice. A theoretical polynomial algorithm may have extremely large constant factors or exponents thus rendering it impractical. 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.