inst.eecs.berkeley.edu/~cs61c
CS61C : Machine Structures
Lecture 22 –
Representations of Combinatorial Logic Circuits
Lecturer PSOE Dan Garcia
www.cs.berkeley.edu/~ddgarcia
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CS61C L22 Representations of Combinatorial Logic Circuits (1)
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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
CS61C L22 Representations of Combinatorial Logic Circuits (2)
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Representations of CL Circuits…
• Truth Tables
• Logic Gates
• Boolean Algebra
CS61C L22 Representations of Combinatorial Logic Circuits (3)
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Truth Tables
0
CS61C L22 Representations of Combinatorial Logic Circuits (4)
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TT Example #1: 1 iff one (not both) a,b=1
a
0
0
1
1
b
0
1
0
1
CS61C L22 Representations of Combinatorial Logic Circuits (5)
y
0
1
1
0
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TT Example #2: 2-bit adder
How
Many
Rows?
CS61C L22 Representations of Combinatorial Logic Circuits (6)
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TT Example #3: 32-bit unsigned adder
How
Many
Rows?
CS61C L22 Representations of Combinatorial Logic Circuits (7)
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TT Example #3: 3-input majority circuit
CS61C L22 Representations of Combinatorial Logic Circuits (8)
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Logic Gates (1/2)
CS61C L22 Representations of Combinatorial Logic Circuits (9)
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And vs. Or review – Dan’s mnemonic
AND Gate
Symbol
A
B
AN
D
Definition
C
CS61C L22 Representations of Combinatorial Logic Circuits (10)
A
0
0
1
1
B
0
1
0
1
C
0
0
0
1
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Logic Gates (2/2)
CS61C L22 Representations of Combinatorial Logic Circuits (11)
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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 
CS61C L22 Representations of Combinatorial Logic Circuits (12)
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Truth Table  Gates (e.g., majority circ.)
CS61C L22 Representations of Combinatorial Logic Circuits (13)
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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
CS61C L22 Representations of Combinatorial Logic Circuits (14)
or equivalently…
Garcia © 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
CS61C L22 Representations of Combinatorial Logic Circuits (15)
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Boolean Algebra (e.g., for majority fun.)
y=a•b+a•c+b•c
y = ab + ac + bc
CS61C L22 Representations of Combinatorial Logic Circuits (16)
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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
CS61C L22 Representations of Combinatorial Logic Circuits (17)
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BA: Circuit & Algebraic Simplification
BA also great for
circuit verification
Circ X = Circ Y?
use BA to prove!
CS61C L22 Representations of Combinatorial Logic Circuits (18)
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Laws of Boolean Algebra
CS61C L22 Representations of Combinatorial Logic Circuits (19)
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Boolean Algebraic Simplification Example
CS61C L22 Representations of Combinatorial Logic Circuits (20)
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Canonical forms (1/2)
Sum-of-products
(ORs of ANDs)
CS61C L22 Representations of Combinatorial Logic Circuits (21)
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Canonical forms (2/2)
CS61C L22 Representations of Combinatorial Logic Circuits (22)
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Administrivia
• Midterm Regrades
• If you want a regrade…
• Explain your reasoning in a paragraph on
a piece of paper along with the
• Staple that to the front of your exam
• Return your exam to your TA
• We will regrade your entire exam
• Your score MAY go down
CS61C L22 Representations of Combinatorial Logic Circuits (23)
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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:
C. You can use NOR(s) with clever wiring 7:
8:
to simulate AND, OR, & NOT
CS61C L22 Representations of Combinatorial Logic Circuits (24)
ABC
FFF
FFT
FTF
FTT
TFF
TFT
TTF
TTT
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Peer Instruction Answer
A.
(a+b)•(a+b) = aa+ab+ba+bb = 0+b(a+a)+b = b+b = b TRUE
B. (next slide)
C. You can use NOR(s) with clever wiring to
simulate AND, OR, & NOT.
°
°
°
NOR(a,a)= a+a = aa = a
Using this NOT, can we make a NOR an OR? An And?
TRUE
ABC
A. (a+b)• (a+b) = b
1: FFF
B. N-input gates can be thought of
2: FFT
3: FTF
cascaded 2-input gates. I.e.,
4: FTT
(a ∆ bc ∆ d ∆ e) = a ∆ (bc ∆ (d ∆ e))
TFF
where ∆ is one of AND, OR, XOR, NAND 5:
6: TFT
C. You can use NOR(s) with clever wiring 7: TTF
8: TTT
to simulate AND, OR, & NOT
CS61C L22 Representations of Combinatorial Logic Circuits (25)
Garcia © UCB
A.
Peer Instruction Answer (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…FALSE
Let’s confirm!
CORRECT 3-input
XYZ|AND|OR|XOR|NAND
000| 0 0|0 0| 0 0| 1 1
001| 0 0|1 1| 1 1| 1 1
010| 0 0|1 1| 1 1| 1 1
011| 0 0|1 1| 0 0| 1 1
100| 0 0|1 1| 1 1| 1 0
101| 0 0|1 1| 0 0| 1 0
110| 0 0|1 1| 0 0| 1 0
111| 1 1|1 1| 1 1| 0 1
CORRECT 2-input
YZ|AND|OR|XOR|NAND
00| 0 |0 | 0 | 1
01| 0 |1 | 1 | 1
10| 0 |1 | 1 | 1
11| 1 |1 | 0 | 0
CS61C L22 Representations of Combinatorial Logic Circuits (26)
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“And In conclusion…”
• Use this table and techniques we
learned to transform from 1 to another
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