CMPT 407/710 - Complexity Theory: Lecture 20

Do we have non-relativizing techniques? Yes, we do! In fact, already the proof of the CookLevin theorem is non-black-box: to reduce the computation of a nondeterministic TM on a given input x to a 3-cnf φx, we had to “open up” the TM and look inside at the sequence of configurations the TM goes through, and moreover, exploit a very special property of a TM computation: that the computation is locally checkable. The latter simply means that to decide if a certain symbol in position i of a configuration at time t is correct, we need only to check the symbols in positions i− 1, i, i+ 1 in the configuration at time t− 1. This “local checkability” is used to get a 3-cnf formula which is satisfiable iff the computation is accepting. Another example is NP-completeness of 3-COLORING. Again, to reduce computation to a graph problem, we essentially relied on the local checkability of computation. This reduction completely breaks down if we allow NP machines to have arbitrary oracles.