# 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.

## Example

The proof that P = NP implies EXP = NEXP uses "padding". $\mathrm {EXP} \subseteq \mathrm {NEXP}$  by definition, so it suffices to show $\mathrm {NEXP} \subseteq \mathrm {EXP}$ .

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

$L'=\{x1^{2^{|x|^{c}}}\mid x\in L\},$

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 $x'=x1^{2^{|x|^{c}}}$  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 $DM(x1^{2^{|x|^{c}}})$ . 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.