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.
The proof that P = NP implies EXP = NEXP uses "padding". E X P ⊆ N E X P by definition, so it suffices to show N E X P ⊆ E X P .
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 ∣ x ∈ 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 D M ( 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.
