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.