1. + CS 325: CS Hardware and Software Organization and Architecture Gates and Boolean Algebra Part 2

2. + Outline Sum of Products (SOP) Fan-in, Fan-out Cascading to Reduce Inputs Boolean Algebra Laws Gate Reduction using Boolean Algebra

3. + Circuits from SOP Functions Why simplify circuits? NAND and NOR gates are simpler (faster, smaller) than NOT AND and NOT OR. Reduction in complexity when using a small number of gate types. Goal: To implement circuit using a small complete set of operators. NAND and NOR are both complete since any Boolean function can be implemented with either. Faster to use small number of inputs to a gate (fan-in), and small number of gate inputs from a gate output (fan-out) Typically, fan-in and fan-out < 10.

4. + Logic Gate Fan-n and Fan-out

5. + Gate Cascading to Reduce Inputs Implementing 3-input AND and OR functions with 2-input gates ABC = (AB)C A+B+C = (A+B)+C Implementing a 3-input NAND function with 2-input gates. NO! Correct

6. + Basic Laws of Boolean Algebra Boolean Algebra follows many algebra rules which can be used to make simpler circuits. NameAND FormOR Form

7. + Basic Laws of Boolean Algebra Boolean Algebra follows many algebra rules which can be used to make simpler circuits. NameAND FormOR Form Identity Law

8. + Basic Laws of Boolean Algebra Boolean Algebra follows many algebra rules which can be used to make simpler circuits. NameAND FormOR Form Identity Law Null Law

9. + Basic Laws of Boolean Algebra Boolean Algebra follows many algebra rules which can be used to make simpler circuits. NameAND FormOR Form Identity Law Null Law Idempotent Law

10. + Basic Laws of Boolean Algebra Boolean Algebra follows many algebra rules which can be used to make simpler circuits. NameAND FormOR Form Identity Law Null Law Idempotent Law Commutative Law

11. + Basic Laws of Boolean Algebra Boolean Algebra follows many algebra rules which can be used to make simpler circuits. NameAND FormOR Form Identity Law Null Law Idempotent Law Commutative Law Associative Law

12. + Basic Laws of Boolean Algebra Boolean Algebra follows many algebra rules which can be used to make simpler circuits. NameAND FormOR Form Identity Law Null Law Idempotent Law Commutative Law Associative Law Distributive Law

13. + Basic Laws of Boolean Algebra Boolean Algebra follows many algebra rules which can be used to make simpler circuits. NameAND FormOR Form Identity Law Null Law Idempotent Law Commutative Law Associative Law Distributive Law Absorption Law

14. + Basic Laws of Boolean Algebra Boolean Algebra follows many algebra rules which can be used to make simpler circuits. NameAND FormOR Form Identity Law Null Law Idempotent Law Commutative Law Associative Law Distributive Law Absorption Law De Morgan’s Law

15. + Basic Laws of Boolean Algebra Boolean Algebra follows many algebra rules which can be used to make simpler circuits. Example: AB + ACThree gates = A(B + C), Distributive LawTwo gates NameAND FormOR Form Identity Law Null Law Idempotent Law Commutative Law Associative Law Distributive Law Absorption Law De Morgan’s Law

16. + Gate Reduction AB + ACThree gates = A(B + C), Distributive LawTwo gates

17. + Equivalent Gates/Symbols Using Boolean Laws (identities), alternative symbols for some gates can be derived:

18. + Functionally Complete Sets of Gates Not all gate types are typically implemented in circuit design. Simpler if only 1 or 2 types of gates are used. A functionally complete set of gates means that any Boolean function can be implemented using only the gates in that set. Examples of functionally complete sets: AND, OR, NOT AND, NOT OR, NOT NAND NOR

19. + NAND and NOR Completeness

20. + Implement XOR with NANDs Exclusive-OR (XOR) example: Step 1: build truth table Step 2: find SOP and build circuit using AND and OR. AB 000 011 101 110

21. + Implement XOR with NANDs

22. + Logic circuits implementing XOR:

23. + Simplification

24. + Checking Logic for Correctness 000100 011000 101111 111111

25. + 000100 011000 101111 111111

26. + Another Example

27. + Checking Logic for Correctness 00000000 00100000 01000000 01101111 10000000 10100000 11010011 11111111

28. + 00000000 00100000 01000000 01101111 10000000 10100000 11010011 11111111

29. + Another Example