inst.eecs.berkeley.edu/~cs61c
CS61C : Machine Structures
Lecture 28 –
Representations of Combinatorial Logic Circuits
2004-04-02
Lecturer PSOE Dan Garcia
www.cs.berkeley.edu/~ddgarcia
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CS 61C L28 Representations of Combinatorial Logic Circuits (1)
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 L28 Representations of Combinatorial Logic Circuits (2)
Garcia, Spring 2004 © UCB
Finite State Machine Example: 3 ones…
Draw the FSM…
Truth table…
PS Input NS Output
00
0
00
0
00
1
01
0
01
0
00
0
01
1
10
0
10
0
00
0
10
1
00
1
CS 61C L28 Representations of Combinatorial Logic Circuits (3)
Garcia, Spring 2004 © UCB
Hardware Implementation of FSM
+
= ?
CS 61C L28 Representations of Combinatorial Logic Circuits (4)
Garcia, Spring 2004 © UCB
General Model for Synchronous Systems
CS 61C L28 Representations of Combinatorial Logic Circuits (5)
Garcia, Spring 2004 © UCB
Peer Instruction (last time)
A. HW feedback akin to SW recursion
B. We can implement a D-Q flipflop
as simple CL (And, Or, Not gates)
C. You can build a FSM to signal
when an equal number of 0s and
1s has appeared in the input.
CS 61C L28 Representations of Combinatorial Logic Circuits (6)
1:
2:
3:
4:
5:
6:
7:
8:
ABC
FFF
FFT
FTF
FTT
TFF
TFT
TTF
TTT
Garcia, Spring 2004 © UCB
Representations of CL Circuits…
• Truth Tables
• Logic Gates
• Boolean Algebra
CS 61C L28 Representations of Combinatorial Logic Circuits (7)
Garcia, Spring 2004 © UCB
Truth Tables
0
CS 61C L28 Representations of Combinatorial Logic Circuits (8)
Garcia, Spring 2004 © UCB
TT Example #1: 1 iff one (not both) a,b=1
a
0
0
1
1
b
0
1
0
1
CS 61C L28 Representations of Combinatorial Logic Circuits (9)
y
0
1
1
0
Garcia, Spring 2004 © UCB
TT Example #2: 2-bit adder
How
Many
Rows?
CS 61C L28 Representations of Combinatorial Logic Circuits (10)
Garcia, Spring 2004 © UCB
TT Example #3: 32-bit unsigned adder
How
Many
Rows?
CS 61C L28 Representations of Combinatorial Logic Circuits (11)
Garcia, Spring 2004 © UCB
TT Example #3: 3-input majority circuit
CS 61C L28 Representations of Combinatorial Logic Circuits (12)
Garcia, Spring 2004 © UCB
Logic Gates (1/2)
CS 61C L28 Representations of Combinatorial Logic Circuits (13)
Garcia, Spring 2004 © UCB
And vs. Or review – Dan’s mnemonic
AND Gate
Symbol
A
B AN
D
Definition
C
CS 61C L28 Representations of Combinatorial Logic Circuits (14)
A
0
0
1
1
B
0
1
0
1
C
0
0
0
1
Garcia, Spring 2004 © UCB
Logic Gates (2/2)
CS 61C L28 Representations of Combinatorial Logic Circuits (15)
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 L28 Representations of Combinatorial Logic Circuits (16)
Garcia, Spring 2004 © UCB
Truth Table  Gates (e.g., majority circ.)
CS 61C L28 Representations of Combinatorial Logic Circuits (17)
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
0
10
1
00
1
CS 61C L28 Representations of Combinatorial Logic Circuits (18)
or equivalently…
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 L28 Representations of Combinatorial Logic Circuits (19)
Garcia, Spring 2004 © UCB
Boolean Algebra (e.g., for majority fun.)
y=a•b+a•c+b•c
y = ab + ac + bc
CS 61C L28 Representations of Combinatorial Logic Circuits (20)
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
0
10
1
00
1
or equivalently…
y = PS1 • PS0 • INPUT
CS 61C L28 Representations of Combinatorial Logic Circuits (21)
Garcia, Spring 2004 © UCB
BA: Circuit & Algebraic Simplification
BA also great for
circuit verification
Circ X = Circ Y?
use BA to prove!
CS 61C L28 Representations of Combinatorial Logic Circuits (22)
Garcia, Spring 2004 © UCB
Laws of Boolean Algebra
CS 61C L28 Representations of Combinatorial Logic Circuits (23)
Garcia, Spring 2004 © UCB
Boolean Algebraic Simplification Example
CS 61C L28 Representations of Combinatorial Logic Circuits (24)
Garcia, Spring 2004 © UCB
Canonical forms (1/2)
Sum-of-products
(ORs of ANDs)
CS 61C L28 Representations of Combinatorial Logic Circuits (25)
Garcia, Spring 2004 © UCB
Canonical forms (2/2)
CS 61C L28 Representations of Combinatorial Logic Circuits (26)
Garcia, Spring 2004 © UCB
Peer Instruction
A. (a+b)• (a+b) = b
1:
B. N-input gates can be thought of
2:
3:
cascaded 2-input gates. I.e.,
4:
(a ∆ bc ∆ d ∆ e) = a ∆ (bc ∆ (d ∆ e))
where ∆ is one of AND, OR, XOR, NAND 5:
6:
7:
C. You can use a NOR with clever wiring
8:
to simulate AND, OR, & NOT
CS 61C L28 Representations of Combinatorial Logic Circuits (27)
ABC
FFF
FFT
FTF
FTT
TFF
TFT
TTF
TTT
Garcia, Spring 2004 © UCB
“And In conclusion…”
• Use this table and techniques we
learned to transform from 1 to another
CS 61C L28 Representations of Combinatorial Logic Circuits (28)
Garcia, Spring 2004 © UCB