Let L be a language in NEXP. Since L is in NEXP, there is a non-deterministic Turing machine M that decides L in time for some constant c. Let
where 1 is a symbol not occurring in L. First we show that is in NP, then we will use the deterministic polynomial time machine given by P = NP to show that L is in EXP.
can be decided in non-deterministic polynomial time as follow. Given input , verify that it has the form and reject if it does not. If it has the correct form, simulate M(x). The simulation takes non-deterministic time, which is polynomial in the size of the input, . So, is in NP. By the assumption P = NP, there is also a deterministic machine DM that decides in polynomial time. We can then decide L in deterministic exponential time as follows. Given input , simulate . This takes only exponential time in the size of the input, .
The is called the "padding" of the language L. This type of argument is also sometimes used for space complexity classes, alternating classes, and bounded alternating classes.
- Arora, Sanjeev; Barak, Boaz (2009), Computational Complexity: A Modern Approach, Cambridge, p. 57, ISBN 978-0-521-42426-4
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