Functions

1
Functions

• Computers take inputs and produce outputs, just
  like functions in math!
• Mathematical functions can be expressed in two
  ways
• We can represent logical functions in two
  analogous ways too
• A finite, but non-unique Boolean expression.
• A truth table, which will turn out to be unique
  and finite.

2
Basic Boolean operations

• There are three basic operations for logical
  values.

NOT (complement) on one input
AND (product) of two inputs
OR (sum) of two inputs
Operation
Expression
xy, or x?y
x y
x
Truth table
3
Boolean expressions

• We can use these basic operations to form more
  complex expressions
• f(x,y,z) (x y)z x
• Some terminology and notation
• f is the name of the function.
• (x,y,z) are the input variables, each
  representing 1 or 0. Listing the inputs is
  optional, but sometimes helpful.
• A literal is any occurrence of an input variable
  or its complement. The function above has four
  literals x, y, z, and x.
• Precedences are important, but not too difficult.
• NOT has the highest precedence, followed by AND,
  and then OR.
• Fully parenthesized, the function above would be
  kind of messy
• f(x,y,z) (((x (y))z) x)

4
Truth tables

• A truth table shows all possible inputs and
  outputs of a function.
• Remember that each input variable represents
  either 1 or 0.
• Because there are only a finite number of values
  (1 and 0), truth tables themselves are finite.
• A function with n variables has 2n possible
  combinations of inputs.
• Inputs are listed in binary orderin this
  example, from 000 to 111.

5
Primitive logic gates

• Each of our basic operations can be implemented
  in hardware using a primitive logic gate.
• Symbols for each of the logic gates are shown
  below.
• These gates output the product, sum or complement
  of their inputs.

NOT (complement) on one input
AND (product) of two inputs
OR (sum) of two inputs
Operation
Expression
xy, or x?y
x y
x
Logic gate
6
Expressions and circuits

• Any Boolean expression can be converted into a
  circuit by combining basic gates in a relatively
  straightforward way.
• The diagram below shows the inputs and outputs of
  each gate.
• The precedences are explicit in a circuit.
  Clearly, we have to make sure that the hardware
  does operations in the right order!

(x y)z x
7
Circuit analysis

• Circuit analysis involves figuring out what some
  circuit does.
• Every circuit computes some function, which can
  be described with Boolean expressions or truth
  tables.
• So, the goal is to find an expression or truth
  table for the circuit.
• The first thing to do is figure out what the
  inputs and outputs of the overall circuit are.
• This step is often overlooked!
• The example circuit here has three inputs x, y, z
  and one output f.

8
Write algebraic expressions...

• Next, write expressions for the outputs of each
  individual gate, based on that gates inputs.
• Start from the inputs and work towards the
  outputs.
• It might help to do some algebraic simplification
  along the way.
• Here is the example again.
• We did a little simplification for the top AND
  gate.
• You can see the circuit computes f(x,y,z) xz
  yz xyz

9
...or make a truth table

• Its also possible to find a truth table directly
  from the circuit.
• Once you know the number of inputs and outputs,
  list all the possible input combinations in your
  truth table.
• A circuit with n inputs should have a truth table
  with 2n rows.
• Our example has three inputs, so the truth table
  will have 23 8 rows. All the possible input
  combinations are shown.

10
Simulating the circuit

• Then you can simulate the circuit, either by hand
  or with a program like LogicWorks, to find the
  output for each possible combination of inputs.
• For example, when xyz 101, the gate outputs
  would be as shown below.
• Use truth tables for AND, OR and NOT to find the
  gate outputs.
• For the final output, we find that f(1,0,1) 1.

11
Finishing the truth table

• Doing the same thing for all the other input
  combinations yields the complete truth table.
• This is simple, but tedious.

12
Expressions and truth tables

• Remember that if you already have a Boolean
  expression, you can use that to easily make a
  truth table.
• For example, since we already found that the
  circuit computes the function f(x,y,z) xz yz
  xyz, we can use that to fill in a table
• We show intermediate columns for the terms xz,
  yz and xyz.
• Then, f is obtained by just ORing the
  intermediate columns.

13
Truth tables and expressions

• The opposite is also true its easy to come up
  with an expression if you already have a truth
  table.
• We saw that you can quickly convert a truth table
  into a sum of minterms expression. The minterms
  correspond to the truth table rows where the
  output is 1.
• You can then simplify this sum of minterms if
  desiredusing a K-map, for example.

f(x,y,z) xyz xyz xyz xyz m1 m2
m5 m7
14
Circuit analysis summary

• After finding the circuit inputs and outputs, you
  can come up with either an expression or a truth
  table to describe what the circuit does.
• You can easily convert between expressions and
  truth tables.

Find the circuits inputs and outputs
Find a Boolean expression for the circuit
Find a truth table for the circuit
15
Boolean operations summary

• We can interpret high or low voltage as
  representing true or false.
• A variable whose value can be either 1 or 0 is
  called a Boolean variable.
• AND, OR, and NOT are the basic Boolean
  operations.
• We can express Boolean functions with either an
  expression or a truth table.
• Every Boolean expression can be converted to a
  circuit.

16
Expression simplification

• Gates have important properties
• Cost
• Size
• Propagation delay
• Less is more
• We want our circuits to be as simple as possible

17
Formal definition of Boolean algebra

• A Boolean algebra requires
• A set of elements B, which needs at least two
  elements (0 and 1)
• Two binary (two-argument) operations OR and AND
• A unary (one-argument) operation NOT
• The axioms below must always be true (textbook,
  p. 33)
• The magenta axioms deal with the complement
  operation
• Blue axioms (especially 15) are different from
  regular algebra

18
Comments on the axioms

• The associative laws show that there is no
  ambiguity about a term such as x y z or xyz,
  so we can introduce multiple-input primitive
  gates
• The left and right columns of axioms are duals
• exchange all ANDs with ORs, and 0s with 1s
• The dual of any equation is always true

19
Are these axioms for real?

• We can show that these axioms are true, given the
  definitions of AND, OR and NOT
• The first 11 axioms are easy to see from these
  truth tables alone. For example, x x 1
  because of the middle two lines below (where y
  x)

20
Proving the rest of the axioms

• We can make up truth tables to prove (both parts
  of) DeMorgans law
• For (x y) xy, we can make truth tables for
  (x y) and for xy
• In each table, the columns on the left (x and y)
  are the inputs. The columns on the right are
  outputs.
• In this case, we only care about the columns in
  blue. The other outputs are just to help us
  find the blue columns.
• Since both of the columns in blue are the same,
  this shows that (x y) and xy are equivalent

21
Simplification with axioms

• We can now start doing some simplifications

xy xyz xy x(y y) xyz
Distributive xy xy x(y y) x?1
xyz Axiom 7 y y 1 x xyz Axiom
2 x?1 x (x x)(x yz)
Distributive 1 ? (x yz) Axiom 7 x x
1 x yz Axiom 2
22
Lets compare the resulting circuits

• Here are two different but equivalent circuits.
• In general the one with fewer gates is better
• It costs less to build
• It requires less power
• But we had to do some work to find the second
  form

23
Some more laws

• Here are some more useful laws (p. 37). Notice
  the duals again!
• We can prove these laws by either
• Making truth tables
• Using the axioms

x xy (x x)(x y) Distributive 1 ?
(x y) x x 1 x y Axiom 2
24
The complement of a function

• The complement of a function always outputs 0
  where the original function outputted 1, and 1
  where the original produced 0.
• In a truth table, we can just exchange 0s and 1s
  in the output column(s)

f(x,y,z) x(yz yz)
25
Complementing a function algebraically

• You can use DeMorgans law to keep pushing the
  complements inwards
• You can also take the dual of the function, and
  then complement each literal
• If f(x,y,z) x(yz yz)
• the dual of f is x (y z)(y z)
• then complementing each literal gives x (y
  z)(y z)
• so f(x,y,z) x (y z)(y z)

f(x,y,z) x(yz yz) f(x,y,z) ( x(yz
yz) ) complement both sides x (yz
yz) because (xy) x y x (yz)
(yz) because (x y) x y x (y
z)(y z) because (xy) x y, twice
26
Standard forms of expressions

• We can write expressions in many ways, but some
  ways are more useful than others
• A sum of products (SOP) expression contains
• Only OR (sum) operations at the outermost level
• Each term that is summed must be a product of
  literals
• The advantage is that any sum of products
  expression can be implemented using a two-level
  circuit
• literals and their complements at the 0th level
• AND gates at the first level
• a single OR gate at the second level
• This diagram uses some shorthands
• NOT gates are implicit
• literals are reused
• this is not okay in LogicWorks!

f(x,y,z) y xyz xz
27
Minterms

• A minterm is a special product of literals, in
  which each input variable appears exactly once.
• A function with n variables has 2n minterms
  (since each variable can appear complemented or
  not)
• A three-variable function, such as f(x,y,z), has
  23 8 minterms
• Each minterm is true for exactly one combination
  of inputs

xyz xyz xyz xyz xyz xyz xyz xyz
Minterm Is true when Shorthand xyz x0, y0,
z0 m0 xyz x0, y0, z1 m1 xyz x0, y1,
z0 m2 xyz x0, y1, z1 m3 xyz x1, y0,
z0 m4 xyz x1, y0, z1 m5 xyz x1, y1,
z0 m6 xyz x1, y1, z1 m7
28
Sum of minterms form

• Every function can be written as a sum of
  minterms, which is a special kind of sum of
  products form
• The sum of minterms form for any function is
  unique
• If you have a truth table for a function, you can
  write a sum of minterms expression just by
  picking out the rows of the table where the
  function output is 1.

f xyz xyz xyz xyz xyz m0
m1 m2 m3 m6 ?m(0,1,2,3,6)
f xyz xyz xyz m4 m5 m7
?m(4,5,7)
f contains all the minterms not in f
29
The dual idea products of sums

• Just to keep you on your toes...
• A product of sums (POS) expression contains
• Only AND (product) operations at the outermost
  level
• Each term must be a sum of literals
• Product of sums expressions can be implemented
  with two-level circuits
• literals and their complements at the 0th level
• OR gates at the first level
• a single AND gate at the second level
• Compare this with sums of products

f(x,y,z) y (x y z) (x z)
30
Maxterms

• A maxterm is a sum of literals, in which each
  input variable appears exactly once.
• A function with n variables has 2n maxterms
• The maxterms for a three-variable function
  f(x,y,z)
• Each maxterm is false for exactly one combination
  of inputs

x y z x y z x y z x y z x
y z x y z x y z x y z
Maxterm Is false when Shorthand x y z x0,
y0, z0 M0 x y z x0, y0, z1 M1 x y
z x0, y1, z0 M2 x y z x0, y1,
z1 M3 x y z x1, y0, z0 M4 x y
z x1, y0, z1 M5 x y z x1, y1,
z0 M6 x y z x1, y1, z1 M7
31
Product of maxterms form

• Every function can be written as a unique product
  of maxterms
• If you have a truth table for a function, you can
  write a product of maxterms expression function
  by picking out the rows of the table where the
  function is 0, and writing the maxterm for that
  row (the complement of the minterm for the row).

f (x y z)(x y z)(x y z) M4
M5 M7 ?M(4,5,7)
f (x y z)(x y z)(x y z) (x
y z)(x y z) M0 M1 M2 M3 M6
?M(0,1,2,3,6)
f contains all the maxterms not in f
See the next slide.
32
Product of maxterms form

• Find a row in the truth table for f for which the
  function is 0
• Write down the complement of the minterm for that
  row
• E.g., if the row is 110, then use DeMorgans law
  to change xyz into (x y z).
• The product of these maxterms is the original
  function f
• Why does this procedure works?
• F A B C D because we picked the rows
  for which f is 0
• F ABCD by DeMorgans law

33
Minterms and maxterms are related

• Any minterm mi is the complement of the
  corresponding maxterm Mi
• For example, m4 M4 because (xyz) x y
  z

Maxterm Shorthand x y z M0 x y z M1 x
y z M2 x y z M3 x y z M4 x
y z M5 x y z M6 x y z M7
Minterm Shorthand xyz m0 xyz m1 xyz m
2 xyz m3 xyz m4 xyz m5 xyz m6 xyz m
7
34
Converting between standard forms

• We can convert a sum of minterms to a product of
  maxterms
• In general, just replace the minterms with
  maxterms, using maxterm numbers that dont appear
  in the sum of minterms
• The same thing works for converting from a
  product of maxterms to a sum of minterms

From before f ?m(0,1,2,3,6) and f
?m(4,5,7) m4 m5 m7 complementing (f)
(m4 m5 m7) so f m4 m5 m7 DeMorgans
law M4 M5 M7 By the previous page
?M(4,5,7)
f ?m(0,1,2,3,6) ?M(4,5,7)
35
Summary

• So far
• A bunch of Boolean algebra trickery for
  simplifying expressions and circuits
• The algebra guarantees us that the simplified
  circuit is equivalent to the original one
• Introducing some standard forms and terminology
• Next
• An alternative simplification method
• Well start using all this stuff to build and
  analyze bigger, more useful, circuits