Propositional Calculus

1
Propositional Calculus

• Objective To provide students with the concepts
  and techniques from propositional calculus so
  that they can use it to codify logical statements
  and to reason about these statements. To
  illustrate how a computer can be used to carry
  out formal proofs and to provide a framework for
  logical deduction.
• Topics
• Boolean functions and expressions
• Rules of Boolean Algebra
• Logic Minimization
• Tautologies and automatic verification of
  tautologies
• Application to Circuit Design

2
Programming Example

• Boolean expressions arise in conditional
  statements. It is possible to abstract the
  relations with boolean variables (propositions
  that are either true or false). Using this
  abstraction one can reason and simplify
  conditional statements.
• if ((a lt b) ((a gt b) (c d)) then
  else
• Let p denote the relation (altb) and q denote the
  relation (c d). The above expression is then
  equal to
• p !p q

3
Programming Example (cont)

• The previous expression is equivalent (two
  expressions are equivalent if they are true for
  the same values of the variables occurring in the
  expressions) to a simpler expression
• (p !p q) ? p q
• We can see this since if p is true both
  expressions are true, and if p is false, then !p
  is true and (!p q) is true exactly when q is
  true.

4
Limitations of Propositional Calculus

• Propositions hide the information in the
  predicates they abstract.
• Sometimes properties of the hidden information is
  required to make further deductions.
• E.G. for integers a,b, and c, (a lt b) (b lt c)
  implies that a lt c however, this can not be
  deduced without using the order properties of the
  integers.
• The predicate calculus allows the use of
  predicates to encode this additional information.
• E.G. we can introduce a parameterized predicate
  lt(a,b) to encode the predicate a lt b.
  Properties such as lt(a,b) lt(b,c) ? lt(a,c)
  can be asserted. This type of notation and
  deduction is discussed in chapter 14.

5
Boolean Functions

• A Boolean variable has two possible values
  (true/false) (1/0).
• A Boolean function has a number of Boolean input
  variables and has a Boolean valued output.
• A Boolean function can be described using a truth
  table.
• There are 22n Boolean function of n variables.

s x0 x1 f 0 0 0 0 0 0 1
0 0 1 0 1 0 1 1 1 1 0
0 0 1 0 1 1 1 1 0 0 1 1
1 1
Multiplexor function
6
Boolean Expressions

• An expression built up from variables, and, or,
  and not.

x y x ? y 0 0 0 0 1 0 1 0
0 1 1 1
x y x y 0 0 0 0 1 1 1 0
1 1 1 1
x x 0 1 1 0
and
or
not
7
Boolean Expressions

• A Boolean expression is a Boolean function.
• Any Boolean function can be written as a Boolean
  expression
• Disjunctive normal form (sums of products)
• For each row in the truth table where the output
  is true, write a
  product such that the corresponding
  input is the only input
  combination that is true
• Not unique
• E.G. (multiplexor function)
• s ? x0 ? x1 s ? x0 ? x1 s ? x0 ? x1
  s ? x0 ? x1

s x0 x1 f 0 0 0 0 0 0 1
0 0 1 0 1 0 1 1 1 1 0
0 0 1 0 1 1 1 1 0 0 1
1 1 1
8
Boolean Logic

• Boolean expressions can be simplified using rules
  of Boolean logic
• Identity law A 0 A and A ? 1 A.
• Zero and One laws A 1 1 and A ? 0 0.
• Inverse laws A A 1 and A ? A 0.
• Commutative laws A B B A and A ? B B ?
  A.
• Associative laws A (B C) (A B) C and
  A ? (B ? C) (A ? B) ? C.
• Distributive laws A ? (B C) (A ? B) (A ?
  C) and
• A (B ? C)
  (A B) ? (A C)
• DeMorgans laws A B A ? B and A ? B
  A B
• The reason for simplifying is to obtain shorter
  expressions, which we will see leads to simpler
  logic circuits.

9
Simplification of Boolean Expressions

• Simplifying multiplexor expression using Boolean
  algebra
• s ? x0 ? x1 s ? x0 ? x1 s ? x0 ? x1 s
  ? x0 ? x1
• s ? x0 ? x1 s ? x0 ? x1 s ? x1 ? x0
  s ? x1 ? x0 (commutative law)
• s ? x0 ? (x1 x1) s ? x1 ? (x0 x0)
  (distributive
  law)
• s ? x0 ? 1 s ? x1 ? 1
  (inverse law)
• s ? x0 s ? x1
  (identity law)
• Verify that the boolean function corresponding to
  this expression as the same truth table as the
  original function.

10
Additional Notation

• Several additional Boolean functions of two
  variables have special meaning and are given
  special notation. By our previous results we
  know that all boolean functions can be expressed
  with not, and, and or so the additional notation
  is simply a convenience.

x y x ? y 0 0 1 0 1 1 1 0
0 1 1 1
x y x ? y 0 0 1 0 1 0 1 0
0 1 1 1
implication
equivalence
11
Tautologies

• A tautology is a boolean expression that is
  always true, independent of the values of the
  variables occurring in the expression. The
  properties of Boolean Algebra are examples of
  tautologies.
• Tautologies can be verified using truth tables.
  The truth table below shows that x ? y ? x y

x y x ? y x y 0 0 1
1 0 1 1 1 1 0 0
0 1 1 1 1
12
Exercise

• Derive the tautology x ? y ? x x from the sum
  of products expression obtained from the truth
  table for x ? y. You will need to use properties
  of Boolean algebra to simplify the sum of
  products expression to obtain the desired
  equivalence.

13
Tautology Checker

• A program can be written to check to see if a
  Boolean expression is a tautology.
• Simply generate all possible truth assignments
  for the variables occurring in the expression and
  evaluate the expression with its variables set to
  each of these assignments. If the evaluated
  expressions are always true, then the given
  Boolean expression is a tautology.
• A similar program can be written to check if any
  two Boolean expressions E1 and E2 are equivalent,
  i.e. if E1 ? E2. Such a program has been
  provided.

14
Karnaugh Map

• A Karnaugh map is a two dimensional version of a
  truth table. It can be used to simplify Boolean
  expressions expressed as sums of products.
• The example below shows the Karnaugh table for
  the truth table defining implication. There is a
  1 in each box corresponding to each value of p
  and q where x ? y is true and a 0 where it is
  false.

y0 y1
x0 1 1
x1 0 1
15
Logic Minimization

• We want a sum of products that is true for all of
  the boxes with 1s (a cover). One such cover is
  obtained using a product for each individual box.
  A simpler expression can be obtained using the
  literals !x and y which cover the first row and
  the second column respectively.
• This shows that x ? y ? x x
• This can be generalized to more the one variable
  (Sec. 12.5)

16
Logic Circuits

• A single line labeled x is a logic circuit. One
  end is the input and the other is the output. If
  A and B are logic circuits so are
• and gate
• or gate
• inverter (not)

A
B
A
B
A
17
Logic Circuits

• Given a boolean expression it is easy to write
  down the corresponding logic circuit
• Here is the circuit for the original multiplexor
  expression

18
Logic Circuits

• Here is the circuit for the simplified
  multiplexor expression

x0
x1
s
19
Nand Gates

• A nand gate is an inverted and gate
• All boolean functions can be implemented using
  nand gates (and and not can be implemented
  using nand)

x y x y 0 0 1 0 1 1 1 0
1 1 1 0
nand
x

x
x
20
Decoder

• A decoder is a logic circuit that has n inputs
  (think of this as a binary number) and 2n
  outputs. The output corresponding to the binary
  input is set to 1 and all other outputs are set
  to 0.

d0
b0
d1
b1
d2
d3
21
Encoder

• An encoder is the opposite of a decoder. It is a
  logic circuit that has 2n inputs and n outputs.
  The output equal to the input line (in binary)
  that is set to 1 is set to 1.

d0
d1
b0
d2
b1
d3
22
Multiplexor

• A multiplexor is a switch which routes n inputs
  to one output. The input is selected using a
  decoder.

d0
d1
d2
d3
s0
s1
23
Exercise

• Derive a truth table for the output bits (Sum and
  CarryOut) of a full adder.
• Using the truth table derive a sum of products
  expression for Sum and CarryOut. Draw a circuit
  for these expressions.
• Using properties of Boolean algebra and Karnaugh
  Maps to simplify your expressions. Draw the
  simplified circuites.

24
Full Adder

• Sum parity(a,b,CarryIn)
• a xor b xor c a?b?c ? a xor b xor c
• CarryOut majority(a,b,CarryIn)
• b?CarryIn a?CarryIn a?b a?b?CarryIn ?
• b?CarryIn a?CarryIn a?b