January 15, 2002
----------------
X = complement of x

(10 min) Review / Questions
EXAMPLE:  SP expansion for xyZ + xY
- F(x,y,z) = xyZ + xY
  SP exp. = xyZ + xYz + xYZ
EXAMPLE:  Build logic circuits for (simple) representation of F

(20 min) Karnaugh maps
- SP expansions useful as "black-box" Boolean function representations
- We built some simple logic circuits last time
- Visually minimize using several methods, most basic: a Karnaugh map
	- Save cost, power

Layout for degree 2		Layout for degree 3

	y	Y			yz	yZ	YZ	Yz

x	?	?		x	?	?	?	?

X	?	?		X	?	?	?	?

- Only responsible for K-maps of Boolean functions of degree 2 and 3
- 1's placed in each square corresponding to a minterm in SP expansion
- Isolate rectangular blocks of side length equal to a power of 2 and
  simplify by identifying variables that "don't change"
	- xyz + xyZ = xy since xy present in both minterms over all
	  possible values of z
CLASS EXAMPLES:	xyZ + xYZ + XyZ + XYZ (simple block example)
		xyz + xYz + Xyz + XYz (blocks wrap around edge)

(50 min.) Physical Realization of Circuits
- We can use Boolean algebra to model basic circuit design
	- In theoretical terms, a circuit designed in terms of logic
	  gates is equivalent to any decision problem composed of
	  *decidable* decision sub-problems (we will see why more
	  theoretically later)
- Physical implementation of gates could involve relays; although
  vacuum tubes are faster, and transistors even faster and smaller than
  that, they all work essentially the same way
	- Work by using a small current to switch another current
	  (input current runs through spring, causes an internal switch)
		- Relay - magnetic induction
		- Vacuum tube - Edison effect
		- Transistor - semiconductor conductivity change
	- In a NOT gate, an input voltage switches current OFF
	- In an AND gate, each input switches a relay, both relays
	  connected serially
	- In an OR gate, each input switches a relay, both relays
	  connected in parallel (or just hook inputs together)
- Logic circuits can form "feedback circuits" when the output from
  a circuit is used as an input to the circuit
	- EXAMPLE: RS NAND Latch
- One special feedback circuit is a flip-flop - used in SRAM
	- "Remembers" the last value shown
	- Achieved through use of a "feedback circuit"
	- Aside: DRAM achieved through an array of capacitors which
	  drain quickly and must be refreshed -- slow but cheap, while
	  SRAM is expensive but fast and reserved for critical
	  applications (such as registers, ROM)
- Clock signal provides consistency -- if various circuits only
  have a signal pushed through at regular intervals indicated by a
  clock cycle (square wave signal), improves internal consistency (data
  discipline for writing)
	- Common "heartbeat" is the falling (1-0) clock edge
	- D latches are a simple example of how data can be "latched"
	  and stored on a clock edge (don't care about D input until
	  the CLOCK input is 0)
	- D flip-flops can capture the falling edge of a clock signal
	  and trigger action on the edge by changing state