CS 61C L22 Representations of Combinatorial Logic Circuits (1) Garcia, Spring 2004 © UCB

Lecturer PSOE Dan Garcia

www.cs.berkeley.edu/~ddgarcia

inst.eecs.berkeley.edu/~cs61c CS61C : Machine Structures

Lecture 22 – Representations of Combinatorial Logic Circuits

2004-10-20

E-voting talk todayAt 4pm in 306 Soda SU Prof. David Dill will give a talk about

important issues in electronic voting. This affects us all! Get there early...

www.verifiedvoting.orgvotingintegrity.com

CS 61C L22 Representations of Combinatorial Logic Circuits (2) Garcia, Spring 2004 © UCB

Review…•We use feedback to maintain state•Register files used to build memories

•D-FlipFlops used for Register files

•Clocks usually tied to D-FlipFlop load•Setup and Hold times important

•Pipeline big-delay CL for faster clock•Finite State Machines extremely useful•You’ll see them again in 150, 152 & 164

CS 61C L22 Representations of Combinatorial Logic Circuits (3) Garcia, Spring 2004 © UCB

Representations of CL Circuits…•Truth Tables

•Logic Gates

•Boolean Algebra

CS 61C L22 Representations of Combinatorial Logic Circuits (4) Garcia, Spring 2004 © UCB

Truth Tables

0

CS 61C L22 Representations of Combinatorial Logic Circuits (5) Garcia, Spring 2004 © UCB

TT Example #1: 1 iff one (not both) a,b=1

a b y

0 0 0

0 1 1

1 0 1

1 1 0

CS 61C L22 Representations of Combinatorial Logic Circuits (6) Garcia, Spring 2004 © UCB

TT Example #2: 2-bit adder

HowManyRows?

CS 61C L22 Representations of Combinatorial Logic Circuits (7) Garcia, Spring 2004 © UCB

TT Example #3: 32-bit unsigned adder

HowManyRows?

CS 61C L22 Representations of Combinatorial Logic Circuits (8) Garcia, Spring 2004 © UCB

TT Example #3: 3-input majority circuit

CS 61C L22 Representations of Combinatorial Logic Circuits (9) Garcia, Spring 2004 © UCB

Logic Gates (1/2)

CS 61C L22 Representations of Combinatorial Logic Circuits (10) Garcia, Spring 2004 © UCB

And vs. Or review – Dan’s mnemonic

AND Gate

CA

B

Symbol

A B C0 0 00 1 01 0 01 1 1

Definition

AND

CS 61C L22 Representations of Combinatorial Logic Circuits (11) Garcia, Spring 2004 © UCB

Logic Gates (2/2)

CS 61C L22 Representations of Combinatorial Logic Circuits (12) Garcia, Spring 2004 © UCB

2-input gates extend to n-inputs

• N-input XOR is the only one which isn’t so obvious

• It’s simple: XOR is a 1 iff the # of 1s at its input is odd

CS 61C L22 Representations of Combinatorial Logic Circuits (13) Garcia, Spring 2004 © UCB

Truth Table Gates (e.g., majority circ.)

CS 61C L22 Representations of Combinatorial Logic Circuits (14) Garcia, Spring 2004 © UCB

Truth Table Gates (e.g., FSM circ.)

PS Input NS Output

00 0 00 0

00 1 01 0

01 0 00 0

01 1 10 0

10 0 00 010 1 00 1

or equivalently…

CS 61C L22 Representations of Combinatorial Logic Circuits (15) Garcia, Spring 2004 © UCB

Boolean Algebra

•George Boole, 19th Century mathematician

•Developed a mathematical system (algebra) involving logic• later known as “Boolean Algebra”

•Primitive functions: AND, OR and NOT

•The power of BA is there’s a one-to-one correspondence between circuits made up of AND, OR and NOT gates and equations in BA

+ means OR,• means AND, x means NOT

CS 61C L22 Representations of Combinatorial Logic Circuits (16) Garcia, Spring 2004 © UCB

Boolean Algebra (e.g., for majority fun.)

y = a • b + a • c + b • c

y = ab + ac + bc

CS 61C L22 Representations of Combinatorial Logic Circuits (17) Garcia, Spring 2004 © UCB

Boolean Algebra (e.g., for FSM)

PS Input NS Output

00 0 00 0

00 1 01 0

01 0 00 0

01 1 10 0

10 0 00 010 1 00 1

or equivalently…

y = PS1 • PS0 • INPUT

CS 61C L22 Representations of Combinatorial Logic Circuits (18) Garcia, Spring 2004 © UCB

BA: Circuit & Algebraic Simplification

BA also great for circuit verificationCirc X = Circ Y?use BA to prove!

CS 61C L22 Representations of Combinatorial Logic Circuits (19) Garcia, Spring 2004 © UCB

Laws of Boolean Algebra

CS 61C L22 Representations of Combinatorial Logic Circuits (20) Garcia, Spring 2004 © UCB

Boolean Algebraic Simplification Example

CS 61C L22 Representations of Combinatorial Logic Circuits (21) Garcia, Spring 2004 © UCB

Canonical forms (1/2)

Sum-of-products(ORs of ANDs)

CS 61C L22 Representations of Combinatorial Logic Circuits (22) Garcia, Spring 2004 © UCB

Canonical forms (2/2)

CS 61C L22 Representations of Combinatorial Logic Circuits (23) Garcia, Spring 2004 © UCB

Peer Instruction

A. (a+b)• (a+b) = b

B. N-input gates can be thought of cascaded 2-input gates. I.e., (a ∆ bc ∆ d ∆ e) = a ∆ (bc ∆ (d ∆ e))where ∆ is one of AND, OR, XOR, NAND

C. You can use NOR(s) with clever wiring to simulate AND, OR, & NOT

ABC1: FFF2: FFT3: FTF4: FTT5: TFF6: TFT7: TTF8: TTT

CS 61C L22 Representations of Combinatorial Logic Circuits (24) Garcia, Spring 2004 © UCB

“And In conclusion…”•Use this table and techniques we learned to transform from 1 to another