Notes on Complexity Theory Lecture 3

Proof We give a detailed proof sketch. (Note that the proof we give here is different from the one in [1]; in particular, we do not rely on the existence of oblivious Turing machines.) Let L be a language in NP. This means there is a Turing machine M and a polynomial p such that (1) M(x,w) runs in time p(|x|), and (2) x ∈ L if and only if there exists a w for which M(x,w) = 1. Note that we may assume that any such w, if it exists, has length exactly p(|x|)− |x| − 1. We also assume for simplicity (and without loss of generality) that M has a single tape (that is used as both its input tape and work tape) and a binary alphabet. A simple observation is that we can represent the computation of M(x,w) (where |x| = n) by a tableau of p(n) + 1 rows, each O(p(n)) bits long. Each row corresponds to the entire configuration of M at some step during its computation; there are p(n) + 1 rows since M always halts after at most p(n) steps. (If M(x,w) halts before p(n) steps, the last rows may be duplicates of each other. Or we may assume that M(x,w) always runs for exactly p(|x|) steps.) Each row can be represented using O(p(n)) bits since a configuration contains (1) the contents of M ’s tape (which can be stored in O(p(n)) bits — recall that space(p(n)) ⊆ time(p(n))); (2) the location of M ’s head on its tape (which can be stored in p(n) bits1); and (3) the value of M ’s state (which requires O(1) bits). Moreover, given a tableau that is claimed to correspond to an accepting computation of M(x,w), it is possible to verify this via a series of “local” checks. (This notion will become more clear below.) Specifically, letting p = p(n) and assuming we are given some tableau, do: