R. Johnsonbaugh Discrete Mathematics 5 th edition, 2001 Chapter 9 Boolean Algebras and Combinatorial Circuits

9.1 Combinatorial circuits A bit is a 0 or a 1. Input x 1, x 2 can be 0 or Conjunction: AND x 1  x 2 = 1 if x 1 = x 2 = 1 0 otherwise  Logical gates

Inclusive disjunction OR x 1 v x 2 = 1 if x 1 = 1 or x 2 = 1 0 otherwise

Negation NOT ~ x = 1 if x = 0 0 if x = 1

Logic Tables A logic table of a combinatorial circuit is a table with all possible inputs and all its resulting outputs x1x1 x2x2 x1^x2x1^x2 x 1 v x x~x 10 01

Boolean expressions A Boolean expression such as  y = ~((x 1 ^ x 2 )vx 3 ) is composed of  literals like x 1, x 2, x 3 and  symbols like ^, v, ~, ) and (  We write y = y(x 1, x 2, x 3 ) The value of a Boolean expression is the bit obtained when the literals are replaced by bits, e.g. y(0,1,0) = 1.

Combinatorial circuits

Logic table The logic table corresponding to the Boolean expression y = ~((x 1 ^ x 2 ) v x 3 ). x1x1 x2x2 x3x3 y

9.2 Properties of Combinatorial Circuits 4. Identity laws av0 = a a^1 = a 5. Complement laws av(~a) = 1 a^(~a) = 0 For a, b, c  {0, 1} 1. Associative laws av(bvc) = (avb)vc a^(b^c) = (a^b)^c 2. Commutative laws avb = bva a^b = b^a 3. Distributive laws a^(bvc) = (a^b) v (a^c) av(b^c) = (avb) ^ (avc)

Boolean expressions Two Boolean expressions X = X(x 1, x 2,…,x n ) and Y = Y(x 1, x 2,…, x n ) are equal (write X = Y) if and only if X(a 1,a 2,…, a n ) = Y(a 1, a 2,…, a n ) for all a i  {0, 1}, 1 < i < n.  Example: X(x 1,x 2, x 3 ) = ~x 1 ^ (x 2 vx 3 ) and Y(x 1,x 2, x 3 ) = (~x 1 ^ x 2 ) v (~x 1 ^ x 3 ) are equal.

Equivalent combinatorial circuits  Two combinatorial circuits, each having inputs x 1,…, x n and a single output for each set of inputs, are equivalent if, given the same inputs, the two circuits produce the same output.  Theorem 9.2.7: Let C and D be two combinatorial circuits with X and Y the Boolean expressions that represent C and D, respectively. C and D are equivalent if and only if X = Y,

9.3 Boolean algebras  Let A be a nonempty set containing distinct elements 0 and 1,  endowed with two binary operations + and  and a unary operation ~ (complement)  that satisfy properties 1 to 5.  Then A is called a Boolean algebra. 1. Associative laws a + (b + c) = (a + b) + c a (b c) = (a b) c 2. Commutative laws a + b = b + a a b = b a 3. Distributive laws a (b + c) = (a b) + (a c) a + (b c) = (a + b) (a + c) 4. Identity laws a + 0 = a a 1 = a 5. Complement laws a + (~a) = 1 a (~a) = 0

Example 1 of a Boolean algebra Example 1: Z Z 2 = {0,1}, with the operations  and  defined by the tables below, is a Boolean algebra.  

Example 2: Power set  Let U be a universal set  and S = P(U) = power set of U.  For any elements X, Y  S define X + Y = X  Y X·Y = X  Y ~X = U - X  Let 0 =  and 1 = U.  Then properties 1-5 hold and S is a Boolean algebra.

Complement in a Boolean algebra  In a Boolean algebra A, given x  A, an element x' such that x' + x = 1 and x' x = 0 is called the complement of x. Theorem 9.3.4: The complement x' of each x in a Boolean algebra is unique. Specifically, if x + y = 1 and x y = 0 then y = x'.

Dual statements  Equations in Boolean algebras come in pairs.  For example, the identity laws x+0 = x and x1 = x.  These are called dual statements.  The dual of a statement in Boolean algebras is obtained by making the following replacements: 0 replaced by 1, 1 replaced by 0, + replaced by, replaced by +

A duality theorem The following is an important theorem about duality statements in a Boolean algebra:  Theorem : The dual of a theorem about Boolean algebras is also a theorem.

9.4 Boolean functions and synthesis of circuits Let X(x 1,…, x n ) be a Boolean expression.  A Boolean function is a function of the form f(x 1,…, x n ) = X(x 1,…, x n )  Its domain is Z (Z 2 ) n = {(x 1, x 2,…, x n ) | x i  {0, 1}, 1 < i < n} Z and its range is Z 2 = {0, 1} ZZ X : (Z 2 ) n  Z 2

Example of a Boolean function Exclusive OR, in symbol  The binary operation  can be expressed as a Boolean function x 1  x 2 = X(x 1,x 2 ), Z with domain = {(1,1),(1,0),(0,1),(0,0)} = (Z 2 ) 2 ; Z and range = Z 2 = {0, 1}. x1x1 x2x2 x 1  x

Minterm A minterm in the symbols x 1,…, x n is a Boolean expression of the form m = y 1 ^ y 2 ^ … ^ y n where each y k is either x k or its complement x k ', 1 < k < n.

Disjunctive normal form Theorem 9.4.6: Let f: (Z 2 ) n  Z 2 be a Boolean function not identically zero.  Let A 1, A 2,…, A k denote the elements of (Z 2 ) n such that f(A i ) = 1.  For each A i = (a 1,…, a n ) set m i = y 1 ^…^ y n  where y j = x j (if a j = 1) OR ~x j (if a j = 0).  Then f(x 1,…, x n ) = m 1 v…v m k.  The disjunctive normal form of a Boolean function is a representation of f as a disjunction of m k 's as in the theorem.

Use of disjunctive normal form  Let f(x 1, x 2 ) = (~x 1 )vx 2. Its truth table is  A 1 = (1,1)  A 2 = (0,1)  A 3 = (0,0)  Then m 1 = x 1 ^ x 2  m 2 = (~x 1 ) ^ x 2  m 3 = (~x 1 ) ^ (~x 2 )  It can be verified that  f(x 1, x 2 ) = m 1 v m 2 v m 3 x1x1 x2x2 ~x 1 vx

9.5 Applications Definition of functionally complete set of gates:  A gate is a function f : (Z 2 ) n  Z 2, for n > 1  A set of gates S = {g 1, g 2,…, g k } is said to be functionally complete if given any positive integer n and a function f : (Z 2 ) n  Z 2, it is possible to construct a combinatorial circuit that computes f using only the set of gates S.

NAND  The NAND gate x 1  x 2 is equivalent to ~(x 1 ^ x 2 ), and is defined by the table and the diagram x1x1 x2x2 x1x2x1x

NAND circuits  Using NAND we can write  x' = x  x  x v y = (x  x)  (y  y)

Functionally complete sets of gates Theorem 9.4.6: The set of gates {AND, OR, NOT} is functionally complete. Theorem 9.5.5: The sets of gates S 1 = {AND, NOT} and S 2 = {OR, NOT} are functionally complete. Theorem 9.5.7: The set {NAND} is functionally complete.

Minimization problem The minimization problem consists of finding the circuit that represents a Boolean function with the minimal number of gates. Example: The following circuit represents the Boolean function f(x,y,z) = xyz v xyz' v xy'z' = x(y v z')

Half adder  Input: two bits x and y  Two outputs: binary sum cs, where c is the carry bit and s is the sum bit. xycs

Half adder circuit Two outputs: s = (x v y) ^ [~(x ^ y)] = x  y c = x ^ y

Full adder xyzcs Full adder table

A circuit to add two 3-bit numbers