Padding argument explained

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.

Example

The proof that P = NP implies EXP = NEXP uses "padding".

EXP \subseteq NEXP by definition, so it suffices to show NEXP \subseteq EXP .

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

2 ^{n ^{c}} for some constant c. Let L' = \{x 1 ^{2 ^{|x| ^{c}}} \mid x \in L\}, where '1' is a symbol not occurring in L. First we show that L' is in NP, then we will use the deterministic polynomial time machine given by P = NP to show that L is in EXP.L' can be decided in non-deterministic polynomial time as follows. Given input x' , verify that it has the form x' = x 1 ^{2 ^{|x| ^{c}}} and reject if it does not. If it has the correct form, simulate M(x). The simulation takes non-deterministic 2 ^{|x| ^{c}} time, which is polynomial in the size of the input, x' . So, L' is in NP. By the assumption P = NP, there is also a deterministic machine DM that decides L' in polynomial time. We can then decide L in deterministic exponential time as follows. Given input x , simulate DM(x 1 ^{2 ^{|x| ^{c}}} ) . This takes only exponential time in the size of the input, x .

The

1 ^{d} 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.

See also

• Paddable language