Com S 631: Lower bounds and Separation Results Lecture 1 Scribe: Pavan Aduri Summary: Com S 631: Lower bounds and Separation Results Lecture 1 Scribe: Pavan Aduri 1. Hierarchy Theorems Today we will talk about deterministic and nondeterministic time hierarchy theorems. Given a string x, let Mx be a deterministic Turing machine defined as follows: If x encodes a valid Turing machine N, then Mx is the same as N. If x does not encode a valid Turing machine, then Mx is a machine that rejects every string. Theorem 1. There is a Universal Turing machine U such that for every x and w U(x, w) = Mx(w). For every x, there is a constant Cx such that U simulates t steps of Mx within Cxt log t steps. Now we are ready to state the deterministic time-hierarchy theorem. Theorem 2. Let f(n) log f(n) o(g(n)). There is a language in DTIME(g(n)) that is not in DTIME(f(n)). Proof. The proof works by diagonalization. Consider the following machine N. (1) Input w, |w| = n. (2) If w is not of the form x, 0i , then reject. (3) Run U(x, w) for g(n) steps. (4) If U does not halt within g(n) steps, reject input. (5) If U accepts, then Reject. If U rejects, then accept. Collections: Computer Technologies and Information Sciences