# Lecture 1

In this lecture we talked about diagnolization to motivate the study of circuits. We saw the Deterministic Time Hierarchy Theorem (Hartmanis and Stearns ’65) that shows that adding (enough) time to deterministic Turing machines adds to their power. We then proved the Non-Determinstic Time Hierarchy Theorem (Cook ’72). The latter has a more involved diagonalization-based proof . We saw this theorem for two reasons (1) we will actually use it in the proof of Williams’ result, and (2) it demonstrates that one can have some fairly non-straightforward diagnolization-based proofs.

Then came the beautiful result by Baker, Gill and Solovay ’75, that shows that resolving the P vs. NP problem, in some sense, cannot be proven using only black-box simulations of Turing machines, a property of diagnolization-based proofs. This result gives a central motivation for the study of circuits, which are more amendable to white-box inquiry than Turing machines. The cool thing about BGS is that its proof uses diagnolization!

The formal statement of BGS uses the notion of Oracle Turing Machines, a notion useful in other contexts as well. In fact, Cook originally defined NP-completeness using this notion. Tom asked a great question which I promised to answer at the end of the class, and forgot to do so: What is the probability that P^C = NP^C, for a ”random” oracle C? The answer to this question, when formally stated, is tending to zero (see Exercises 3.8, 3.9 from Arora-Barak book). Moreover, one might suggest that this can be considered as an evidence for P not equal to NP. This hypothesis was suggested by Bennett and Gill and is called ”The Random Oracle Hypothesis”,. It later turned out to be false (not for P vs. NP of course, but e.g., for IP vs. PSPACE, complexity classes that we will meet later on), see the paper by Chang et al from ’92. It will be one of the papers for the final talks.

Today’s lecture covered most of Chapter 3 from Arora-Barak Book. Another cool theorem that appears in this chapter, which I only briefly mentioned, is Ladner’s Theorem.

Daniel will scribe this lecture and we will post it online in a few weeks. Next lecture we will talk about the Polynomial Hierarchy (Chapter 5 in Arora-Barak book) and start to talk about circuits (partially based on Chapter 6 of the same book).