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Boolean Algebra

Boolean Algebra. Chapter 10. What is Boolean Algebra?. A minor generalization of propositional logic. In general, an algebra is any mathematical structure satisfying certain standard algebraic axioms. Such as associative/commutative/transitive laws, etc.

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Boolean Algebra

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  1. Boolean Algebra Chapter 10

  2. What is Boolean Algebra? • A minor generalization of propositional logic. • In general, an algebra is any mathematical structure satisfying certain standard algebraic axioms. • Such as associative/commutative/transitive laws, etc. • General theorems that are proved about an algebra then apply to any structure satisfying these axioms. Claude Shannon’sMaster’s thesis! (c)2001-2003, Michael P. Frank

  3. What is Boolean Algebra? • Boolean algebra just generalizes the rules of propositional logic to sets other than {T,F}. • E.g., to the set {0,1} of base-2 digits, or the set {VL, VH} of low and high voltage levels in a circuit. • We will see that this algebraic perspective lends itself to the design of digital logic circuits. Claude Shannon’sMaster’s thesis! (c)2001-2003, Michael P. Frank

  4. Boolean Algebra Operations • Or (+) • 1 + 1 = 1 1 + 0 = 1 0 + 1 = 1 0 + 0 = 0 • AND (· or no operator like multiplication) • 1 · 1 = 1 1 · 0 = 0 0 · 1 = 0 0 · 0 = 0 • Complement or NOT • 1 = 0 0 = 1

  5. Complement, Sum, Product • Correspond to logical NOT, OR, and AND. • We will denote the two logic values as0:≡F and 1:≡T, instead of False and True. • Using numbers encourages algebraic thinking. • New, more algebraic-looking notation for the most common Boolean operators: Precedence order→ (c)2001-2003, Michael P. Frank

  6. Example 1 • 1 · 0 + (0 + 1) = • 0 + (1) = • 0 + 0 = • 0

  7. Boolean equivalents, operations on Boolean expressions • Two Boolean expressions e1 and e2 that represent the exact same function f are called equivalent. We write e1e2, or just e1=e2. • Implicitly, the two expressions have the same value for all values of the free variables appearing in e1 and e2. (c)2001-2003, Michael P. Frank

  8. Boolean equivalents, operations on Boolean expressions • The operators ¯, +, and · can be extended from operating on expressions to operating on the functions that they represent, in the obvious way. (c)2001-2003, Michael P. Frank

  9. Double complement: x = x Idempotent laws: x + x = x, x · x = x Identity laws: x + 0 = x, x · 1 = x Domination laws: x + 1 = 1, x · 0 = 0 Commutative laws: x + y = y + x, x · y = y · x Associative laws: x + (y + z) = (x + y) + z x· (y · z) = (x· y) · z Distributive laws: x + y·z = (x + y)·(x + z) x · (y + z) = x·y + x·z De Morgan’s laws: (x · y) = x + y, (x + y) = x · y Absorption laws: x + x·y = x, x · (x + y) = x Some popular Boolean identities ← Not truein ordinaryalgebras. also, the Unit Property: x + x = 1 and Zero Property: x· x = 0 (c)2001-2003, Michael P. Frank

  10. Duality • The dualed of a Boolean expression e representing function f is obtained by exchanging + with ·, and 0 with 1 in e. • The function represented by ed is denoted fd. (c)2001-2003, Michael P. Frank

  11. Duality • Duality principle: If e1e2 then e1de2d. • Example: The equivalence x(x+y) = ximplies (and is implied by) x + xy = x. (c)2001-2003, Michael P. Frank

  12. Representing Boolean Functions 10.2

  13. Example 1 • Find Boolean expressions to represent F(x,y,z) and G(x,y,z) • Table 1, page 709 • F(x,y,z) = xyz • G(x,y,z) = xyz + x’yz’

  14. Sum-of-Products Expansions • Theorem: Any Boolean function can be represented as a sum of products of variables and their complements. • Proof: By construction from the function’s truth table. For each row that is 1, include a term in the sum that is a product representing the condition that the variables have the values given for that row. Show an example on the board. (c)2001-2003, Michael P. Frank

  15. Sum-of-Products Expansions • Theorem: Any Boolean function can be represented as a sum of products of variables and their complements. • Proof: By construction from the function’s truth table. For each row that is 1, include a term in the sum that is a product representing the condition that the variables have the values given for that row. (c)2001-2003, Michael P. Frank

  16. See Table 1 page 709 • F(x,y,z) = xy’z • G(x,y,z) = xyz’ + x’yz’ (c)2001-2003, Michael P. Frank

  17. Literals, Minterms, DNF • A literal is a Boolean variable or its complement. • A minterm of Boolean variables x1,…,xn is a Boolean product of n literals y1…yn, where yi is either the literal xi or its complement xi. • Note that at most one minterm can have the value 1. • ab' c is a minterm • a+b'+c is a maxterm (c)2001-2003, Michael P. Frank

  18. Literals, Minterms, DNF • The disjunctive normal form (DNF) of a degree-n Boolean function f is the unique sum of minterms of the variables x1,…,xn that represents f. • A.k.a. the sum-of-products expansion of f. (c)2001-2003, Michael P. Frank

  19. Find DNF of F(x,y,z) = (x+y)z’ F(x,y,z) = (x+y)z’ = xz’ + yz’ Distributive = x1z’ + 1yz’ Identity = x(y + y’)z’ + (x + x’)yz’ Unit = xyz’ + xy’z’ + xyz’ + x’yz’ Distributive = xyz’ + xy’z’ + x’yz’ Idemp (1&3) (c)2001-2003, Michael P. Frank

  20. Conjunctive Normal Form • A maxterm is a sum of literals. • CNF is a product-of-maxterms representation. • To find the CNF representation for f, • take the DNF representation for complement f, f = ∑i∏j yi,j • and then complement both sides & apply DeMorgan’s laws to get: f = ∏i∑jyi,j Can also get CNF moredirectly, using the 0rows of the truth table; Get DNF, then negate (c)2001-2003, Michael P. Frank

  21. Find CNF of F(x,y,z) = (x+y)z’ F(x,y,z) = (x+y)z’ Table 1 p 710 = xyz + xy’z + x’yz + x’y’z + x’y’z’ DNF of 0’s = (xyz + xy’z + x’yz + x’y’z + x’y’z’)’ Negate = (x’+y’+z’)(x’+y+z’)(x+y’+z’)(x+y+z’)(x+y+z) Distribute / DeMorgans (c)2001-2003, Michael P. Frank

  22. Logic Gates 10.3

  23. §10.3 – Logic Gates • Inverter, Or, And gate symbols. • Multi-input gates. • Logic circuits and examples. • Adders, “half,” “full,” and n-bit. (c)2001-2003, Michael P. Frank

  24. Logic Gate Symbols x • Inverter (logical NOT,Boolean complement). • AND gate (Booleanproduct). • OR gate (Boolean sum). • XOR gate (exclusive-OR,sum mod 2). x x·y y x x+y y x x⊕y y (c)2001-2003, Michael P. Frank

  25. Basic logic gates • Not • And • Or • Nand • Nor • Xor

  26. NAND, NOR, XNOR • Just like the earlier icons,but with a small circle onthe gate’s output. • Denotes that output is complemented. • The circles can also be placed on inputs. • Means, input is complementedbefore being used. x y x y x y (c)2001-2003, Michael P. Frank

  27. Multi-input AND, OR, XOR • Can extend these gates to arbitrarilymany inputs. • Two commonlyseen drawing styles: • Note that the second style keeps the gate icon relatively small. x1 x1x2x3 x2 x3 x1⋮ x5 x1…x5 (c)2001-2003, Michael P. Frank

  28. Rosen, §10.3 question 1 • Find the output of the following circuit • Answer: (x+y)y • Or (xy)y __

  29. (x+y)y y Rosen, §10.3 question 1 • Find the output of the following circuit • Answer: (x+y)y • Or (xy)y x+y __

  30. ___ _ _ Rosen, §10.3 question 2 • Find the output of the following circuit • Answer: xy • Or (xy) ≡ xy

  31. ___ _ _ x y x y x y Rosen, §10.3 question 2 • Find the output of the following circuit • Answer: xy • Or (xy) ≡ xy

  32. Rosen, §10.3 question 6 • Write the circuits for the following Boolean algebraic expressions • x+y __

  33. x+y x Rosen, §10.3 question 6 • Write the circuits for the following Boolean algebraic expressions • x+y __

  34. Rosen, §10.3 question 6 • Write the circuits for the following Boolean algebraic expressions • (x+y)x _______

  35. (x+y)x x+y Rosen, §10.3 question 6 • Write the circuits for the following Boolean algebraic expressions • (x+y)x _______ x+y

  36. Writing xor using and/or/not • p  q  (p  q)  ¬(p  q) • x  y  (x + y)(xy) ____ x+y (x+y)(xy) xy xy

  37. How to add binary numbers • Consider adding two 1-bit binary numbers x and y • 0+0 = 0 • 0+1 = 1 • 1+0 = 1 • 1+1 = 10 • Carry is x AND y • Sum is x XOR y • The circuit to compute this is called a half-adder

  38. The half-adder • Sum = x XOR y • Carry = x AND y

  39. Using half adders • We can then use a half-adder to compute the sum of two Boolean numbers 1 0 0 1 1 0 0 + 1 1 1 0 ? 0 1 0

  40. The full adder • The “HA” boxes are half-adders

  41. The full adder • The full circuitry of the full adder

  42. Adding bigger binary numbers • Just chain full adders together

  43. S-R Flip-flops • Consider the following circuit: • Set • Reset • What does it do? Set High/ Reset Low Q (output) Q’

  44. §10.4 – Minimizing Circuits • Karnaugh Maps • Don’t care conditions • The Quine-McCluskey Method (c)2001-2003, Michael P. Frank

  45. Goals of Circuit Minimization • (1) Minimize the number of primitive Boolean logic gates needed to implement the circuit. • Ultimately, this also roughly minimizes the number of transistors, the chip area, and the cost. • Also roughly minimizes the energy expenditure • among traditional irreversible circuits. • This will be our focus. • (2) It is also often useful to minimize the number of combinational stages or logical depth of the circuit. • This roughly minimizes the delay or latency through the circuit, the time between input and output. (c)2001-2003, Michael P. Frank

  46. Minimizing DNF Expressions • Using DNF (or CNF) guarantees there is always some circuit that implements any desired Boolean function. • However, it may be far larger than needed! • We would like to find the smallest sum-of-products expression that yields a given function. • This will yield a fairly small circuit. • However, circuits of other forms (not CNF or DNF) might be even smaller for complex functions. (c)2001-2003, Michael P. Frank

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