11.1 Boolean Functions

1. 11.1 Boolean Functions

2. Boolean Algebra • An algebra is a set with one or more operations defined on it. • A boolean algebra has three main operations, and, or, and not, (typically operating on the set {0,1}). • Rules of precedence for Boolean operators: PCoPS- parentheses, complements, products, sums

3. Examples:

4. Laws of Boolean Algebra Double Complement Idempotent Identity Domination

5. More Laws of Boolean Algebra Commutative Associative Distributive

6. Laws of Boolean Algebra Concluded DeMorgan’s Laws Absorption Laws Unit Property Zero Property

7. “Proving” the Laws of Boolean Algebra • If the underlying set is just {0,1}, we can prove these laws with truth tables, just as we did with propositional logic.

8. Duals and the Duality Principle • The dual of a boolean expression is obtained by replacing all sums by products, all products by sums, all 0’s by 1’s, and all 1’s by 0’s. • Example: • The duality principle says that if an equation in a boolean algebra is an identity, i.e. always true no matter what the values of the variables, then the equation obtained by replacing both sides by their duals is also an identity.

9. Boolean Functions of Degree n on the Boolean algebra {0, 1}

10. Example: Boolean Function Table

11. Using a 3-cube to represent a Boolean Function:

12. Lattices and Boolean Algebra • A lattice is a partially ordered set in which every pair of elements has both a least upper bound (lub) and a greatest lower bound (glb). • The supremum ab is defined as lub(a,b) and the infimum ab is defined as glb(a,b) • A lattice is said to be distributive if each of  and  distributes over the other. • For a distributive lattice to be a Boolean algebra, there must be (a) a largest element 1, (b) a least element 0, and (c) for each element x a complement with the property that

13. 11.2 Representing Boolean Functions ( n-ary functions on the set {0, 1} )

14. Disjunctive Normal Form • Pick out all the ones in the “function table” column corresponding to the function value. • Translate each to an “and” of n literals (a literal is a Boolean variable of the form x or ) • Each such product is called a “minterm”. • The desired function is the sum of these minterms, and is called the sum-of-products expansion or disjunctive normal form of the function.

15. Example

16. Another Example • Find the DNF expansion of

17. Example: Find the sum-of-products expansion of the Boolean function that equals 1 if and only if

18. Laws of Boolean Algebra Concluded DeMorgan’s Laws Absorption Laws Unit Property Zero Property

19. Functional Completeness • A set of operators on an algebra is said to be functionally complete if any function of any degree on that algebra can be expressed in terms of those operators • The set is functionally complete in any boolean algebra. • But since , so is • Also, since , is also functionally complete

20. Example: Express the Boolean function using only the operators and .

21. The NAND Operation

22. The NOR Operation

23. Example: Express the Boolean function using only the operator and using only the operator .

24. 11.3 Logic Gates • Boolean algebra is the algebra of circuits. • The elementary operations of the algebra correspond to circuit elements called gates.

25. Basic Types of Gates

26. Combinations of gates • Branching and multiple inputs

27. Example:

28. Example • Three switches x, y, and z, controlling a light

30. Half-Adder Circuit

31. Full Adder

32. 11.4 Minimization of Circuits • Minimizing a circuit is minimizing the number of gates necessary to achieve the required outputs • Equivalent to minimizing the Boolean function, i.e. to finding the least number of Boolean operations needed to compute the function • Recall that any such function can be expressed as a sum of minterms • Note that the number of possible minterms is exponential in the number n of variables. • Note also that the number of functions is exponential in the number of minterms.

33. Karnaugh Maps for Two Variables • Variables x and y • Label rows with x and x, columns with y and y

34. Another Example

35. Karnaugh Maps (K-Maps) for Three Variables • Variables x, y, and z • Use x and x as row labels • Use all possible products of y and z literals, arranged in a Gray code, as column labels • The geometric picture is that of a band, since the last and first cells in each row are to be considered adjacent

37. Examples

38. Implicants, Prime Implicants, and Essential Prime Implicants

39. Example

40. K-Maps for Four Variables • Variables w, x, y, and z • Use all possible products of w and x literals, arranged in a Gray code, as row labels • Use all possible products of y and z literals, arranged in a Gray code, as row labels • Geometric picture is that of a torus

41. Example

42. Another Example

43. A light controlled by 4 switches:

44. The Quine-McCluskey Method • Map each min-term into a bit string. E.g. map wxyz to 0110. • Generate implicants as bit strings with wild card characters, such as 0–01 • Prime implicants are those which are not generalized by any other implicant • Essential prime implicants are those which generalize a min-term not generalized by any other implicant

45. Example wxyz, wxyz, wxyz, wxyz, wxyz, wxyz, wxyz

46. Check with Karnaugh Map wxyz, wxyz, wxyz, wxyz, wxyz, wxyz, wxyz

47. Link http://www.mathcs.bethel.edu/~gossett/DiscreteMathWithProof/QuineMcCluskey.html