Day 29, Mat385

1. Proofs: Remember "Assume this; consider that...."
   1. HW problem 8.1.12b, p. 634
      • Read the problem; follow directions. So many of you neglected to state the dual result (why don't you like free stuff?:)....
      • Start with the more complicated side, and work to move to the less complicated side.

        Don't carry the whole theorem along for the ride, unless you like writing question marks over equalities (because, as far as you know, it's not true!).

      • Learn your vocabulary.

   2. Proof (8.1.13: a bit tricky, so I gave the hint)
      1. The easy direction is \[ x=0 \rightarrow x \cdot y' + x' \cdot y = y \]
      2. For the converse, \[ x \cdot y' + x' \cdot y = y \rightarrow x=0 \] use the fact that this holds for all \(y\), so, in particular, it holds for \(y=0\).

      3. One problem is that things like "cancellation" don't work: so, for example. Think about idempotence, which is true for every element in a Boolean algebra: \[ x + x = x = x + 0 \] If we could just cancel an \(x\) from both sides, then we would conclude that \[ x = 0 \] for every element in a Boolean algebra!

2. HW problem 8.2.14
   • You weren't asked to do any simplification, although if you did (and did so correctly) you didn't lose points.

3. Key

We will cover recognition -- another task of FSMs -- and then minimization: again, an optimization issue. We'll end today on a machine that we will want to optimize; the question is how do we do that?

As a discrete dynamical system

There are lots of things we still teach, even though there are "easier ways to do them". It's a good question as to when to drop a topic (e.g. cursive hand writing). In calculus class we still (try to) teach students how to graph functions, but surely no one graphs functions by hand anymore....

But we require an interesting example or two of Finite state machines, and the binary adder presents itself! It illustrates that, once one has the graph, then you can just follow your nose (and a directed graph) to sum up two binary numbers: we've turned it into a process that even a computer (or maybe especially a computer) can do.

Even little kids, unschooled in mathematics, can do it: one kid shouts out the binary digit pairs, and the other kid wanders the graph by moving to the appropriate next state, shouting out the output of each node. By the time they're done shouting at each other, they've written out the sum....

Computers have, for the most part, been that simplistic; until, perhaps, chatGPT and its kin come along! Soon computers will reflect, and reproduce, and .... There's already a 3D printer that can print itself -- only larger! The day that it can also assemble itself, we might be in trouble....