1. CS 300 – Lecture 3 Intro to Computer Architecture / Assembly Language Digital Design

2. Text The chapter we'll work out of is available in my pickup folder – hardware.pdf.

3. How Logic Gates Work A gate has “inputs” and “outputs” (wires). The outputs are determined by the inputs. For a relay: input is voltage on the coil, outputs are connections between terminals Transistors are similar except a lot smaller Key characteristics: Delay: how soon the output has the “right” answer Power dissipation: how much heat is generated Size: how much silicon is needed on the chip

4. Building Logic Gates Many different sorts of logic gates are used in computing A Pentium has about 50,000,000 transistors

5. Combinatorial Circuits We'll start with "Combinatorial" circuits – these are ordinary functions over booleans. This is the "expression" part of the hardware programming language. A combinatorial circuit is a function from a set of boolean inputs to a set of boolean outputs.

6. Truth Tables We can specify the behavior of a combinatorial circuit using a "Truth Table" that enumerates the output for all possible inputs (note that this isn't how we usually specify functions – there is no way to enumerate all integers or reals, for example) A truth table for a circuit with N inputs and M outputs has 2 N rows and M output columns. Two circuits compute the same function if they have the same truth table (but these circuits are not necessary equivalent – why?)

7. Adding Two Numbers Truth table for 2-bit addition: x y Sum Carry 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 Inputs Outputs

8. Gates and Boolean Expressions We have two ways of expressing a combinatorial circuit: * As boolean expressions that define each output in terms of the inputs (and maybe a temp or two) * As a circuit diagram, expressed in gates and wires. These are equivalent – you can go back and forth between these representations.

9. Boolean Expressions Variables are either true or false X ·Y = the 'and' of X, Y (also just XY) X+Y = the 'or' of X a, Y X' = the inverse of X (also overbar) This is no different than a programming language expression over booleans involving and, or, and not.

10. Boolean Algebra X(Y+Z) = XY + XZ Distribution X+X' = 1 (True) Inverse law X+(Y+Z) = (X+Y)+Z Associativity XY = YZ, X+Y = Y+X Commutivity X+X = X, XX = X Absorption X+0 = X, X+1 = 1, X0 = 0, X1 = X Removal of constants X+Y = (X'Y')' DeMorgans Law And many others! It's very easy to rewrite boolean equations to make them easier to understand or more efficient when implemented in hardware.

11. A Canonical Representation All boolean equations can be converted to "sum of products" form. That is, a term in which all of the or operators are at the top, all ands are below, and all nots are at the leaves. Repeated application of DeMorgan and distributivity put any term in this form. There is a direct link between truth tables and SOP form. Note that: * You can remove all constants (unless the expression collapses completely) * You can remove repeated terms * Ordering within a + or * is unimportant

12. Examples Convert these to SOP: X(Y+Z') (A+B)' (A+B)'(C+D)' (A+0)(B+1)

13. Truth Table Every line of a truth table corresponds to a product of the inputs – if the input is 1, use the input variable. If 0, use the inverse of the variable. The result is a sum of all products that correspond to a "1". Example: Turn the adder into SOP equations.

14. Circuit Diagrams We can express equations graphically as lines (wires) and gate symbols. And: AB Or: A+B Not: A'

15. Why Wires? The visual depiction of boolean function directly corresponds to the physical circuit. From the diagram, we can easily see: * Number of gates * Overall delay (depth) * Fan out (number of inputs connected to an output) * Layout (sort of!) We'll generally use diagrams instead of equations.

16. Exercises Convert the boolean equations from the previous exercise into diagrams Create a circuit for the adder (actually a "Half Adder")

17. Essential Skills * Rewriting of boolean equations * Creating SOP equations * Conversion between truth tables, equations, and circuits