Why do equalities between complexity classes translate upwards and not downwards? Hey Guys, I understand that the padding trick allows us to translate complexity classes upwards - for example $P=NP \rightarrow EXP=NEXP$. Padding works by "inflating" the input, running the conversion (say from say $NP$ to $P$), which yields a "magic" algorithm which you can run on the padded input. While this makes technical sense, I can't get a good intuition of how this works. What exactly is going on here? Is there a simple analogy for what padding is? 

Can provide a common sense reason why this is the case?
I think the best way to get intuition for this issue is to think of what the complete problems for exponential time classes are.  For example, the complete problems for NE are the standard NP-complete problems on succinctly describable inputs, e.g., given a circuit that describes the adjacency matrix of a graph, is the graph 3-colorable?  Then the problem of
whether E=NE becomes equivalent to whether NP problems are solvable in polynomial time on the
succinctly describable inputs, e.g., those with small effective Kolmogorov complexity.   This is obviously no stronger than whether they are solvable on all inputs.  The larger the time bound, the smaller the Kolmogorov complexity of the relevant inputs, so collapses for larger time bounds are in effect algorithms that work on smaller subsets of inputs.

Russell Impagliazzo