CS310, Computer Architecture

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Boolean Algebra and Logic Gates

• CS310, Computer Architecture

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Digital Logic Gates

• Composed of one or a few transistors
• Implement Boolean functions, such as
• And
• Or
• Not
• etc.
• These, in turn, compose integrated circuits

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Digital vs. Analog Circuits

• Analog
• Controls voltage
• Controls current
• Operates continuously
• Digital
• Controls boolean logic
• Concerned with true/false values
• Operates discretely

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Types of Digital Circuits

• Combinatorial The output at some time t is a
  function of the inputs at the same time t
• Biggest point These cannot have cycles (i.e.
  signals cannot feedback into previous portions of
  the circuit)
• So you can think of this as almost a snapshot
• Sequential Outputs at time t are a function of
  inputs at the same time t and all previous inputs
• This introduces the concept of memory
• You can have cycles and feedback

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Which are actually used?

• Some parts of an IC are combinatorial
• ALUs, buses, some I/O
• Other parts of an IC are sequential
• Registers, memory, some I/O
• The circuit as a whole is __________ .

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Gates

• Currently the most cost effective way to build
  computers
• Have one or two inputs, generate a single output
• Operate conceptually as truth tables
• Remember these?

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The AND gate
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The OR gate
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The NOT gate
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The NAND gate

• Actually, its an AND and a NOT.
• NOT is abbreviated with a circle before the
  output
• So you first take the AND, then negate it

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The NOR gate

• Analogously, an OR and a NOT

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The XOR gate

• True if inputs are different
• False if inputs are the same

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NAND Equivalencies
Lets look at truth tables to see why!
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NOR Equivalencies
Lets look at truth tables to see why!
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Boolean Algebra Rules

• Involution
• Complement
• Identity
• Null
• Idempotent
• Commutative
• Associative
• Distributive
• Absorption
• DeMorgans

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Involution

• The easiest
• Applying NOT twice to a boolean value, is the
  value
• NOT(NOT x) x

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Complement

• A value OR its complement is 1
• x OR (NOT x) 1
• A value AND its complement is 0
• x AND (NOT x) 0

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Identity

• Any value OR 0, or AND 1 is the value itself
• x OR 0 x
• x AND 1 x
• For these reasons, OR is commonly represented by
  a PLUS (), and AND is commonly represented by a
  TIMES ()
• Since any number plus zero is itself
• And any number times one is itself
• Well do this from now on

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Null

• Any value OR 1 is 1
• x 1 1
• Any value AND 0 is 0
• x 0 0

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Idempotent

• Any value AND (or OR) itself, is itself
• x x x
• x x x

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Commutative Law

• Binary boolean operations AND, OR and XOR are
  commutative
• x y y x
• x y y x
• x XOR y y XOR x
• XOR is commonly represented by an encircled plus
  sign

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Associative Law

• These same binary operators are also associative
• (x y) z x (y z)
• (x y) z x (y z)
• (x XOR y) XOR z x XOR (y XOR z)

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Distributive Law

• Works with AND and OR (not XOR)
• (x y) z (x z) (y z)
• (x y) z (x z) (y z)

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Absorption Law

• This ones more interesting, and used very often
  to simplify digital circuits
• x (x y) x
• x (x y) x
• x (NOT x y) x y
• x (NOT x y) x y
• We can either remove
• An input and two operations
• A single operation and a NOT gate

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DeMorgans Law

• (NOT x) (NOT y) NOT (x y)
• Replaces one AND and two NOT gates with a NOR
• (NOT x) (NOT y) NOT (x y)
• Replaces one OR and two NOT gates with a NAND

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Combinatorial Circuits

• Design a circuit that has three inputs and one
  output, such that if the three inputs have more
  ones than zero bits, the circuit output is one.
• Actually has a name, the majority circuit
• Steps
• 1. Make a truth table (already know how to do
  this)
• 2. Represent the output as a sum of minterms
  (dont know this yet)
• 3. Construct the circuit using AND, NOR and NOT
  gates
• We can get this easily from step 2.

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Truth Table

• Lets do this part together on the board.
• Three inputs a, b, and c and output z.

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Sum of Minterms

• Take each record in the truth table which yields
  an output of 1.
• Form a product of the inputs, where if their
  value is zero negate and if their value is one
  keep them as is.
• Sum them.
• So in this case

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What does this mean?

• Remember is AND and is OR
• So essentially this says
• z is 1 if
• a is zero and b is one and c is one, or
• a is one and b is zero and c is one, or
• etc.

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Building the circuit

• The first step is to place the input signals on
  the left hand side

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Building the circuit

• Then, add NOT gates so we have signals for all
  input possibilities

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Building the circuit

• Use an AND gate to get the signal for the first
  minterm

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Building the circuit

• Get the rest of the minterms the same way

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Building the circuit

• Finally, OR all of the minterms together, and you
  have z.

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You can also use maxterms

• You can also construct the circuit using a
  product of the maxterms (complete reverse)
• Maxterms are formed by finding input sequences
  that result in z0 instead of z1
• Instead each term contains a sum of input values
• And 0 inputs are left as is, while 1s are negated
• And they are ANDed together (as opposed to OR)
• So for the majority circuit
• Why does this work? Lets view the truth table.

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Circuit

• What then does our circuit look like? Lets do
  it on the board.

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When to use which?

• Theres no rule as to which is better
• People think one is easier than the other
• Also may base the decision on how many records in
  the truth table generate 0 outputs and how many
  generate 1.
• So if theres more 0s, a product of maxterms MAY
  be simpler, and if theres more 1s, a sum of
  minterms MAY be simpler
• We say MAY, because as we saw there are ways to
  simplify Boolean expressions, which you can apply
  sometimes.
• Also, you can simplify using Karnaugh Maps

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Simplifying Example

• Lets look at our minterm sum for the majority
  circuit again
• We can group the last terms together using the
  distributive property
• Then use the complement law to simplify
• We now have a simpler circuit!

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Karnaugh Maps

• Karnaugh Maps are a graphical way to capitalize
  upon these simplifications
• An alternative to using algebra
• A two-dimensional table
• Rows and columns represent input values
• Key point each successive column or row can only
  differ from the previous by one value. Thus

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Grouping

• Simplify by grouping adjacent entries, i.e.
• Represents ab
• Represents ac
• Represents bc

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Why does this work?

• What does this equate to?
• So we eliminate the value that is different
  between the entries using the commutative property

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The full picture

• Sum all groupings, and you get the simplest form
  possible

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Rules on Grouping

• You can only form groups of powers of 2 elements
• Thus you can group one entry (doesnt change
  anything)
• These are called isolated cells. You just have
  to use the minterm sum, like usual.
• You can group two entries, like we saw
• You CANNOT group three entries
• You can group four entries. Lets do an example.
• You can group eight entries.
• With three inputs (23 eight entries in the
  map), solution is trivial
• With four inputs (sixteen), its more useful
• Groups can also overlap across edges. Lets look
  at an example.

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Other things with Karnaugh Maps

• We used it with minterms and produce a minterm
  sum
• Can also use it for a product of maxterms
• One factor generated for each covering of 0s (not
  1s)
• Group by 0s and not by 1s
• Variables get complemented
• Fewer coverings result in fewer terms
• In general, a term with n 1s reduces the size of
  the term
• by log n (i.e. a group of 2 will eliminate
  one input, a group of 4 will eliminate two, etc.)

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The mysterious NAND

• A NAND gate is interesting, since given any
  circuit you can represent the entire thing with
  just NANDs
• Step 1 Apply the law of involution (double
  negative)

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The mysterious NAND

• 2. Migrate bubbles, changed NOT to NAND, and use
  DeMorgans to change OR