Lecture 3 Boolean algebra

1. Lecture 3 Boolean algebra  Topics Error Correcting Codes Boolean algebra Combinational circuits Algebraic analysis, Truth tables, Logic Diagrams Sums-of-Products and Products-of-Sums

2. Overview Last Time: Readings 2.4-2.10, 3.1 BCD, excess-3 Ripple carry Adder Two�s complement Slides covered: Lecture 1 Slides 18-31, Lecture 2 slides Lec 2 slides: two�s complement overflow New: Boolean Algebra from Lecture 2 Slides Basic Gates: symbols and truth tables for : AND, OR, NOT, NOR, NAND, XOR, Half Adder: table and circuit Full Adder: Table and logic diagram from 2 Half-adders + ??? Gray Code, Error Correcting Codes Example Error Correcting Codes Boolean algebra Combinational circuits Algebraic analysis, Truth tables, Logic Diagrams Sums-of-Products and Products-of-Sums

3. Boolean Algebra George Boole (1854) invented a two valued algebra To �give expression � to the fundamental laws of reasoning in the symbolic language of a Calculus.� 1938 Claude Shannon at Bell Labs noted that this Boolean logic could be used to describe switching circuits. (Switching Algebra) In Shannon�s view a relay has two positions open and closed representing 1 and 0. Collections of relays satisfied the properties of Boolean algebra.

4. Basic Gates

5. Describing Circuits: Ex. Half-adder Boolean Expression Truth Table Block diagram symbol Logic Diagram

6. Full Adder Truth Table From last time the table for a full adder is shown at the right. In this the inputs are: Xi the ith bit of one of the input numbers Yi the ith bit of the other input Ci the carry into the ith stage And the outputs are: Si the sum from this stage and Ci+1 the carry to the (i+1)st stage

7. Full Adder From Half Adders Now one implementation of a full adder is to build one using two half-adders and an OR

8. Error Correcting codes Revisited For an n-bit code, consider the hypercube of dimension n Choose some subset of the nodes as code words. Suppose the distance between any two code words is at least 3. Now consider transmission errors. Then if there is an error in transmitting just one bit then the distance from the received word to one code word is one, distances to other code words are at least two. Single error correcting, double error detecting. Such codes are called Hamming codes after their inventor Richard Hamming.

9. Boolean Algebra George Boole (1854) invented a two valued algebra To �give expression � to the fundamental laws of reasoning in the symbolic language of a Calculus.� 1938 Claude Shannon at Bell Labs noted that this Boolean logic could be used to describe switching circuits. (Switching Algebra) In Shannon�s view a relay had two positions open and closed and collections of relays satisfied the properties of Boolean algebra.

10. Boolean Algebra Axioms The axioms of a mathematical system are a minimal set of properties that are assumed to be true. Axioms of Boolean Algebra X = 0 if X != 1 1�. X=1 if X!=0 If X = 0, then X� = 1 2�. if X=1 then X�=0 0 . 0 = 0 3�. 1 + 1 = 1 1 . 1 = 1 4�. 0 + 0 = 0 0 . 1 = 1 . 0 = 0 5�. 1 + 0 = 0 + 1 = 1 Axioms 1-5 and 1�-5� completely define what it means to be a Boolean algebra.

11. Boolean Algebra Theorems We can prove a new theorem: By directly applying axioms and other already proved theorems, or By perfect induction, i.e. considering all possible cases Truth tables

12. Consider a Proof from the axioms Prove Theorem �T5: X + X� = 1� from the axioms Proof: First suppose that X=1. Then by Axiom 2� (if X=1 then X�=0) we have X�=0 and thus X + X� = 1 + 0 and then by Axiom 5� (1+0 = 1) and so X + X� = 1. Now if X != 1 then by Axiom 1 (X = 0 if X != 1) and we have X = 0. Then by Axiom 2 (If X = 0, then X� = 1) we have X�=1 and so X + X� = 0 + 1 and again by axiom 5� (0 + 1 = 1) we have X + X� = 1

13. More Theorems N.B. T8�, T10, T11

14. Consider a Truth Table Proof Prove Theorem �T8�: (X+Y).(X+Z) = X + Y.Z� using a truth table

15. Consider a Truth Table Proof Prove Theorem �T8�: (X+Y).(X+Z) = X + Y.Z� using a truth table

16. Duals Given a boolean equation then we can take its dual by Replacing each 1 with 0, replacing each 0 with a 1, replacing each �+� (OR) with �.� (AND), and replacing each �.� (AND) with a �+� (OR) Example: The dual of X.Y + X.Z = X. (Y+Z) is �

17. Principle of Duality Given a boolean equation E that is a theorem if we take the dual then the resulting equation is also a theorem. Why? Each axiom (A1-A5) has a dual (A1�-A5�) Example: X + (Y + Z) = (X + Y) + Z (this is theorem T7)X � (Y � Z) = (X � Y) � Z (taking the dual yields T7�) Example 2: X + X� = 1 (Axiom A5) X � X� = 0 (taking the dual yields Axiom A5�)

18. Principle of Duality 2 Now consider the following argument using duals X + X � Y = X (this is theorem T9)X � X + Y = X (by taking the dual ???)X + Y = X (But then X � X = X by T3�) Counterexample? This is not true! (consider X=0, Y=1) Where did we go wrong?

19. N-variable Theorems Prove using finite induction Most important: DeMorgan theorems

20. Combinational Circuit Analysis A combinational circuit is one whose outputs are a function of its inputs and only its inputs. These circuits can be analyzed using: Truth tables Algebraic equations Logic diagrams

21. Boolean Algebra Proofs Axioms Statements (boolean equations) that are assumed to be true that form the basis of a mathematical system. Theorems Statements that can be �proved� from the axioms and earlier theorems. Lemmas, Corollaries, Postulates Proof by truth-table For a �possible theorem� with a small number of variables, we can exhaustively consider all possible cases. Algebraic Proofs Apply axioms and previously proven theorems to rewrite a �possible theorem� until it is reduced to an equation known to be true. Induction Basis case: P(1) Inductive hypothesis: Assuming P(n) show P(n+1)

22. Proof by truth-table Prove Demorgan�s Law: (X+Y)' = X ' . Y '

23. Algebraic Simplification Simplify F = A.B.C�.D + D.C.A + B.C.D + A�.B�.C.D

24. Proof by Induction Thereom 13 (X1 . X2 . X3 � . Xn)� = X1� + X2� + X3� � + Xn� Proof: Basis Step n = 2, (X1 . X2)� = X1� + X2� was proven using a truth-table. Now suppose as inductive hypothesis that (X1 . X2 . X3 � . Xn)� = X1� + X2� + X3� � + Xn� Then consider (X1 . X2 . � . Xn . Xn+1)� = ((X1 . X2 . X3 � . Xn ) . Xn+1)� by associativity = (X1 . X2 . X3 � . Xn )� + Xn+1� by the basis step = (X1� + X2� + X3� � + Xn�) + Xn+1� by the inductive hypothesis

25. Universal Sets of Gates A set of Gates(operators), S, is universal if every boolean function can be expressed using gates only from S. Examples {AND, OR, NOT} is a universal set {NAND} is a universal set

26. Universal Sets of Gates (cont.) Examples {AND, OR} is not {NOR}? {XOR}? HW

27. Combinational Circuit Analysis A combinational circuit is one whose outputs are a function of its inputs and only its inputs. These circuits can be analyzed using: Truth tables Algebraic equations Logic diagrams � timing considerations; graphical

28. Switching Algebra Terminology Literal � a variable or the complement of a variable Product term � a single literal or the AND of several literals Sum term � a single literal or the OR of several literals Sums-of-products Product-of-sums Normal term � a product (sum) term in which no variable appears twice Minterm � a normal product term with n literals Maxterm � a normal sum term with n literals

29. Logic Diagrams (Xilinx)

30. Timing Analysis We will do some extensive timing analysis in the labs but for right now we will assume the delay for and an AND-gate and an OR-gate is �d� When we fabricate circuits there are a couple special circumstances: Inverters (Not gates) cost nothing Circuits are usually fabricated from �NANDs�

31. Circuit Simplification Why would we want to simplify circuits? To minimize time delays To minimize costs To minimize area

32. Sums-of-Products What is the delay of sums-of-products circuit?

33. Circuit Simplification Minterms � a product term in which every variable occurs once either complemented or uncomplemented

34. Karnaugh Maps Tabular technique for simplifying circuits two variable maps three variable map

35. Karnaugh Map Simplification Simplify F(X,Y,Z) = S(0,2,6,4,7,5)

36. Karnaugh Map Terminology F(X,Y,Z) = S(1,4,5,6,7)

37. Karnaugh Map Terminology F(X,Y,Z) =

38. Karnaugh Map Terminology F(X,Y,Z) =

39. Karnaugh Map Simplification F(X,Y,Z) =

40. Karnaugh Map Simplification F(X,Y,Z) =

41. 4 Variable Map Simplification F(W,X,Y,Z) =

42. 4 Variable Map Simplification F(W,X,Y,Z) =

43. Products-of-Sums What is the delay of products-of-sums circuit?

44. PopQuiz Convert 37.0310 to Octal Convert 11011.010012 to hex