EEA051 - Digital Logic 數位邏輯 吳俊興高雄大學資訊工程學系 September 2004 Boolean Algebra (formulated by E.V. Huntington, 1904) A set of elements B={0,1} and two binary operators + and Huntington postulates 1. Closure w.r.t. the operator + ( ) x, y B x+y B;x, y B x y B 2. Associative w.r.t. + ( ) (x+y)+z = x + (y + z); 3. Commutative w.r.t. + ( ) x+y = y+x; x y = y x (x y) z = x (y z) 4. An identity element w.r.t. + ( ) 0+x = x+0 = x; 1 x = x 1= x 5. x B, x' B (complement of x) x+x'=1; x x'=0 6. is distributive over + : x (y+z)=(x y)+(x z) + is distributive over : x+ (y z)=(x+ y) (x+ z) Duality principle: remains valid if the operators and identity elements are interchanged 2-1 Basic Definitions 2-2 Axiomatic Definition of Boolean Algebra 2-3 Basic Theorems and Properties 2-4 Boolean Functions 2-5 Canonical and Standard Forms 2-6 Other Logic Operations 2-7 Digital Logic Gates 2-8 Integrated Circuits = AND + = OR = NOT Distributive law: x (y+z)=(x y)+(x z)
Operator Precedence 1. parentheses 2. NOT 3. AND 4. OR A table of all possible combinations of the variables showing the relation between the variable values and the result of the operation Theorem 6(a) Absorption Theorem 5. DeMorgan
Logic Circuit Boolean Function Boolean Functions F 1 = x + (y z) F 2 = x y z + x yz + xy 2-variable DeMorgan s Theorem (x + y) = x y and (xy) = x + y 3-variable DeMorgan s Theorem Generalized DeMorgan s Theorem F2 = x y z + x yz + xy
Complement of a variable x is x (0 1 and 1 0) The complement of a function F is F and is obtained from an interchange of 0 s for 1 s and 1 s for 0 s in the value of F The dual of a function is obtained from the interchange of AND and OR operators and 1 s and 0 s Finding the complement of a function F Applying DeMorgan s theorem as many times as necessary complementing each literal of the dual of F Minterms and Maxterms Expressing combinations of 0 s and 1 s with binary variables (normal form x or complement form x ) Logic circuit Boolean function Truth table Any Boolean function can be expressed as a sum of minterms Any Boolean function can be expressed as a product of maxterms Canonical and Standard Forms Minterm (or standard product): = n variables combined with AND n variables can be combined to form 2 n minterms two variables: x y, x y, xy, and xy A variable of a minterm is primed if the corresponding bit of the binary number is a 0, and unprimed if a 1 Maxterm (or standard sum): = n variables combined with OR A variable of a maxterm is unprimed if the corresponding bit is a 0 and primed if a 1 001 => x y z m j = M j 100 => xy z 111 => xyz
Expressing Truth Table in Boolean Function Any Boolean function can be expressed as a sum of minterms or a product of maxterms (either 0 or 1 for each term) said to be in a canonical form n variables 2 n minterms 2 2n possible functions Expressing Boolean Function in Sum of Minterms (Method 2 Truth Table) (x+y +z ) Expressing Boolean Function in Sum of Minterms (Method 1 - Supplementing) F(A, B, C) = Σ(1, 4, 5, 6, 7) = Π(0, 2, 3) F (A, B, C) = Σ(0, 2, 3) = Π(1, 4, 5, 6, 7) Expressing Boolean Function in Product of Maxterms
Canonical conversion procedure Consider: F(A, B, C) = Σ(1, 4, 5, 6, 7) F : Complement of F = F (A, B, C) = Σ(0, 2, 3) = m 0 + m 2 + m 3 Compute complement of F by DeMorgan s Theorem F = (F ) = (m 0 + m 2 + m 3 ) = (m 0 + m 2 + m 3 ) = m 0 m 2 m 3 = M 0 M 2 M 3 = Π(0, 2, 3) Summary m j = M j Conversion between product of maxterms and sum of minterms Σ(1, 4, 5, 6, 7) = Π(0, 2, 3) Shown by truth table (Table 2-5) Canonical forms: each minterm or maxterm must contain all the variables Standard forms: the terms that form the function may contain one, two, or any number of literals (variables) Two types of standard forms (2-level) sum of products F 1 = y + xy + x yz product of sums F 2 = x(y + z)(x + y + z ) Canonical forms Standard forms Sum of minterms, Product of maxterms Sum of products, Product of sums Boolean expression: F(x, y, z) = xy + x z Deriving the truth table Expressing in canonical forms F(x, y, z) = Σ(1, 3, 6, 7) = Π(0, 2, 4, 5) F 1 = y + xy + x yz F 2 = x(y + z)(x + y + z )
Nonstandard form: F 3 = AB + C(D+E) Standard form: F 3 = AB + CD + CE A two-level implementation is preferred: produces the least amount of delay through the gates when the signal propagates from the inputs to the output There are 2 2n functions for n binary variables For n=2 there are 16 possible functions AND and OR operators are two of them: x y and x+y Subdivided into three categories: Figure 2-5 Digital Logic Gates 1. Two are equal to a constant 2. Four are repeated twice 3. Two, Inhibition and implication, are impractical The gates can be extended to have more than two inputs except for the inverter and buffer
NAND and NOR functions are communicative but not associative Define multiple NOR (or NAND) gate as a complemented OR (or AND) gate (Section 3-6) XOR and equivalence gates are both communicative and associative uncommon, usually constructed with other gates XOR is an odd function (Section 3-8) Digital ICs are often categorized according to their circuit complexity as measured by the number of logic gates in a single package Small-scale integration (SSI) the inputs and outputs of the gates are connected directly to the pins in the package usually fewer than 10 gates, limited by the number of pins available Medium-scale integration (MSI) 10 to 1,000 gates in a package usually perform specific elementary digital operations Large-scale integration (LSI) Thousands of gates Include digital systems such as processors, memory chips, and programmable logic devices Very large-scale integration (VLSI) Hundred of thousands of gates Chapter 2 Boolean Algebra and Logic Gates 2-1 Basic Definitions 2-2 Axiomatic Definition of Boolean Algebra 2-3 Basic Theorems and Properties 2-4 Boolean Functions 2-5 Canonical and Standard Forms 2-6 Other Logic Operations 2-7 Digital Logic Gates 2-8 Integrated Circuits