1. Arithmetic and Decisions
CS/COE 0447
Jarrett Billingsley

2. Class announcements
hooooooOO!
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3. Binary addition, in logic
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4. A B C S 1 1 1 1 1 1 1 The Tables of Truth
let's try to come up with a truth table for adding two bits
each column will hold 1 bit
let's name the outputs C and S, for Carry and Sum
let's name the inputs A and B A B C S
now let's fill in the output values
for the input values, we count up in binary 1 1
hey, this C column looks familiar… what about S? 1 1 1 1 1
great! But this is wrong.
- whenever making a truth table, ALWAYS count up in binary – so easy to miss cases if you jump around
- C and S form a 2-bit number
- C just looks like A & B
- S looks like A xor B
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5. 00111 110 1011 0010 +0010 1111 1110 0001 Co Ci A B S Half-truth tables
what we just made was a half-adder
it has a carry output but not a carry input
(which might be useful for the lowest bit)
to make a full adder, we need 3 input bits Ci A B Co S 1 1 Co Ci 1 1 1 1 1 A 1 1 B 1 1 1 1 1 1 S 1 1 1 1 1
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6. The logic of it all Ci A B Co S 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
it looks a little messy, but it kinda makes sense if you think of it like this:
it counts how many input bits are "1"
if we look at the outputs in isolation:
S is 1 if we have an odd number of "1s"
Co is 1 if we have 2 or 3 "1s"
there's a full adder example circuit for you Ci A B Co S 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
- Again, Co and S are a 2-bit number
- it might not be obvious, but we can come up with a collection of gates that output each of these columns 1 1 1 1 1
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7. Sweeping that under the rug…
in programming, we use functions to be able to reuse code
in logic diagrams, we can group these 5 gates into a component
here's the symbol for a one-bit full adder + B A Ci Co S
outputs are like return values
now we don't have to care how it adds, just that it does
inputs are like parameters
unfortunately Logisim flips this upside down but whatever.
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8. Making a component in Logisim
we can use Project > Add Circuit… to make a new component
then we can edit it by double-clicking it on the left
this is like editing a function's code
a component has pins: its inputs and outputs
one of the labs will go into more detail on how to make components.
now we can place multiple instances of the component on the main circuit, like making function calls.
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9. Adding longer numbers
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10. 1 0 1 1 0 0 1 0 +0 0 1 0 1 1 1 1 Where do the carries go?
when you add one place, you might get a carry out
that becomes the carry in for the next higher place
Bit Bucket..?
- what is this called?
- how can we handle it?
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11. + + + + A2 A1 A0 B2 B1 B0 S2 S1 S0 Ripple Carry A2 S2 B2 A1 S1 B1 A0
if we want to add two three-bit numbers, we'll need three one-bit adders
we chain the carries from each place to the next higher place, like we do on paper
we have to split the numbers up like so: + B2 A2 S2 + B1 A1 S1 A2 A1 A0 B2 B1 B0 S2 S1 S0 + B0 A0 S0 +
- it's called ripple carry because the carry "ripples" from the LSB to the MSB one place at a time
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12. Gate Delay electrical signals can't move infinitely fast
transistors can't turn on and off infinitely fast
since each digit must wait for the next smaller digit to compute its carry…
ripple carry is linear in the number of digits
this is a diagram of how the outputs of a 16-bit ripple carry adder change over time
they added 0xFFFF + 1
it's measured in picoseconds! so ~100ps total
but if we went to 32 bits, it'd take 200ps
and 64 bits, 400ps...
there are more efficient ways of adding
details, schmetails
- look-ahead carry is faster but requires O(n^2) space
300
400
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(courtesy of Kate Temkin)

13. + + Flip side A1 S1 ~B1 A0 S0 ~B0 1 remember how we do subtraction?
x - y = x + (-y)
negation means NOT and add 1
NOT uses… NOT gates
how do we add 1 without any extra circuitry?
we use a full adder for the LSB, and when we're subtracting, set the "carry in" to 1
now we're doing x + (~y) + 1
which is x + (-y) ! +
~B1 A1 S1 +
~B0 A0 S0 1
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14. Detecting unsigned overflow in hardware
how did we detect unsigned overflow?
if the sum is smaller than either of the addends.
but there's an easier way…
if we add two n bit numbers, and get an n + 1 bit number, what will the last carry be? 1!
so we can look at the MSB's carry-out.
if it's 1, it's an overflow.
1 if overflow! + B2 A2 S2 + B1 A1 S1 + B0 A0 S0
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15. Detecting signed overflow in hardware
how did we detect signed overflow?
same input signs; different output sign.
that feels truth-table-y. SA SB So O 1 1
when the input signs differ, we can't have an overflow, so those rows are all 0. 1 1 1
when the input signs match, overflow happens when the output sign differs. 1 1 1 1 1 1 1 1 1
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16. What makes a good word size?
can you think of an example of…
100 of something?
a million of something? a billion?
a quintillion (1018)? more?
28 = 256, 216 ≅ 65,000, 232 ≅ 4 billion, 264 ≅ 18 quintillion
for a given word size, all the circuitry has to be built to support it
64 1-bit adders
128 wires going in
64 wires coming out
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17. A new way of thinking about making decisions
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18. How you're used to doing it
how do we make decisions in programming?
with if-elses, conditional loops, switches…
if(mode == 1)
return A + B;
else
return A - B;
if mode == 1, is the else clause run?
No.
when you make a decision in software, only one code path is run; the others never happen.
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19. How hardware does it sum = A + B; diff = A – B; if(mode == 1)
instead of making the decision first and then doing one thing…
hardware does all possible things, then decides which to use.
sum = A + B;
diff = A – B;
if(mode == 1)
return sum;
else
return diff;
this seems wasteful, but: these two paths can be run at the same time!
when you make a decision in hardware, you do all possible things, then pick only the thing you need, and ignore the rest.
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20. Hardware that makes decisions
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21. Y A B S Y A B S=0 A Y B S=1 Multiplexers
a multiplexer (mux) outputs one of its inputs based on a select. Y A B S Y A B
S=0 A Y B
S=1
the unselected input is ignored!
This is the select input.
- muxes are universal too! you can build any logic gate – and therefore an entire computer – out of muxes
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22. Big muxes
a mux can have any (power-of-2) number of inputs
here's one with 4 inputs!
to choose among 4 values… how big does the select signal have to be now?
2 bits. A 00 B 01 Y C 10 D 11 2 S
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23. Mux anti-patterns A 1 A 1 ~A A B
just like with gates, there are some "pointless" ways to use muxes. A 1 A 1 ~A A B
this is an overcomplicated wire.
this is an overcomplicated NOT gate.
the select signal should never be a constant.
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24. they are pretty much useless for this class.
Demultipliexers
a demultiplexer is the opposite of a multiplexer, but... In S In
S=0 In
S=1
they are pretty much useless for this class.
- please don't use them. okay? okay.
95% of the times I've seen people using them, they're pointless.
so avoid them.
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25. Implementing this if-else
now we have enough information to do this:
if(mode == 1)
return A + B;
else
return A - B; A - B
mode
result
else if +
the select signal is basically the condition.
we send the input values to all possible branches.
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