Padding argument

In computational complexity theory, the padding argument is a tool to conditionally prove that if some complexity classes are equal, then some other bigger classes are also equal.

ExampleEdit

The proof that P = NP implies EXP = NEXP uses "padding".   by definition, so it suffices to show  .

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 follows. 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.

ReferencesEdit

  • Arora, Sanjeev; Barak, Boaz (2009), Computational Complexity: A Modern Approach, Cambridge, p. 57, ISBN 978-0-521-42426-4