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PHILIP GREENSPUN: All right, folks. Welcome back. So now we'll learn about the process of learning to fly and some of the regulations around it that will be on the test. All right, so when you learn to fly, you'll pre-flight the aircraft with the instructor. Learn about that-- how to inspect everything. The first few lessons are really about learning to see the aircraft attitude, seeing whether you're pitched up, pitched down, banked left or right. Most people have a fairly easy time with. Then you learn to take off and land. As I said earlier, that takes about five or 10 hours. And in fact, people used to solo after four hours of flight training, especially young people. Now it takes a little longer, because they want you to know about more bureaucracy and how to talk to the control tower and so forth. A good trainer airplane-- why not start with the best and just learn how to fly your personal Gulfstream or whatever. If it's a very heavy airplane like a ...

MICHAEL SIPSER: Hi, folks. Why don't we get started? Welcome back. Good to see you all here. So I am going to first-- well, we'll recap what we did last time and what we're going to do today. I'll talk a little bit about the problem set. And we'll also have a break, as requested, halfway through. So why don't we jump in? What we did last time was besides introducing the course, we introduced finite automata in regular languages, which are the languages that the finite automata can recognize. We talked about these regular operations. Those allow us to build what we call our regular expressions. These are ways of describing languages. So we have finite automata can describe languages, and regular expressions can describe languages. And one of our goals is to show that those two systems are equivalent to one another, even though they look rather different at first glance. So to move in that direction, we're going to prove closure properties for th...