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". ${\displaystyle \mathrm {EXP} \subseteq \mathrm {NEXP} }$ by definition, so it suffices to show ${\displaystyle \mathrm {NEXP} \subseteq \mathrm {EXP} }$.

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 ${\displaystyle 2^{n^{c}}}$ for some constant c. Let

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

where 1 is a symbol not occurring in L. First we show that ${\displaystyle L'}$ is in NP, then we will use the deterministic polynomial time machine given by P = NP to show that L is in EXP.

${\displaystyle L'}$ can be decided in non-deterministic polynomial time as follows. Given input ${\displaystyle x'}$, verify that it has the form ${\displaystyle x'=x1^{2^{|x|^{c}}}}$ and reject if it does not. If it has the correct form, simulate M(x). The simulation takes non-deterministic ${\displaystyle 2^{|x|^{c}}}$ time, which is polynomial in the size of the input, ${\displaystyle x'}$. So, ${\displaystyle L'}$ is in NP. By the assumption P = NP, there is also a deterministic machine DM that decides ${\displaystyle L'}$ in polynomial time. We can then decide L in deterministic exponential time as follows. Given input ${\displaystyle x}$, simulate ${\displaystyle DM(x1^{2^{|x|^{c}}})}$. This takes only exponential time in the size of the input, ${\displaystyle x}$.

The ${\displaystyle 1^{d}}$ 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.

References

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