Computer Architecture & Operations I Instructor: Ryan Florin

Morgan Kaufmann Publishers April 18, 2018 0s and 1s Modern Computers are Digital 1 Corresponding to a high voltage Signal Asserted Logical True Corresponding to low voltage Deasserted False 0s and 1s are complimentary 0’s inverse is 1 1’s inverse is 0 Chapter 1 — Computer Abstractions and Technology

Units Bit Byte (B) Kilo (KB) Mega (MB) Giga (GB) Tera (TB) 0 or 1 8 bits (00101010) Kilo (KB) 1024 bytes Mega (MB) 1,048,576 bytes Giga (GB) 1,073,741,824 bytes Tera (TB) 1,099,511,628,000 bytes

Binary Representation of Positive Integers 0 : 0000 0000 1 : 0000 0001 2 : 0000 0010 3 : 0000 0011 4 : 0000 0100 5 : 0000 0101 6 : 0000 0110 7 : 0000 1111 8 : 0000 1000 16: 0001 0000 32: 0010 0000 64: 0100 0000 128:1000 0000

Binary Representation of Positive Integers 100 100 = 64 + 32 + 4 0 1 1 0 0 1 0 0

Binary Representation of Positive Integers 45 45 = 32 + 8 + 4 + 1 0 0 1 0 1 1 0 1

Combinational Logic and Sequential Logic A logic system whose blocks do not contain memory and hence compute the same output given the same input Sequential Logic A group of logic elements that contain memory and hence whose value depends on the inputs as well as the current contents of the memory

Boolean Logic -- AND AND (Logical Product) Its output = 1, only if both inputs are 1 Truth table A B A·B 1

Boolean Logic -- OR OR (Logical Sum) Its output = 1 if either input = 1 Truth table A B A+B 1

Boolean Logic -- NOT NOT (Logical Inversion) or ~A The output is the opposite of the input Truth Table A ~A 1

Order of Precedence Precedence Rule Example Parentheses (Highest) NOT AND OR Example

Boolean Logic Any Boolean Logic function can be implemented with only NOT, AND, OR functions NOT, AND, OR functions are the basic logic functions Others can be implemented by the basic logic functions NOT, AND, OR

Truth Table Example from the book:

Answer

Boolean Logic Laws Identity Law Zero and One Law Inverse Law Commutative Law

Boolean Logic Laws (cont.) Associative Laws Distributive Laws De Morgan’s Laws

How to prove a logical law? One approach: Truth table Truth table for de Morgan Laws

Gates Gates basic digital building blocks which correspond to and perform the basic logical functions AND OR NOT Complex digital functions that make up a computer are built from these basic digital building blocks

Simplification of NOT Gate

In Class Exercise Design a Combinational Logic to implement the following logical expression

In Class Exercise Design a Combinational Logic to implement the following logical expression

NAND NAND Its output = 1, only if both inputs are not 1 Boolean Expression: A • B Truth Table The NAND functions has traditionally been the universal gate in digital circuits. It is simple to implement in hardware and can be used to construct the other gates. A B C 1

NOR NOR Its output = 1, only if no inputs are not 1 Boolean Expression: A + B Truth Table A B C 1

XOR XOR is EXCLUSIVE-OR Its output = 1 if the inputs are different and equal 0 if all are the same. Boolean Expression: A Å B Truth Table Equivalent to (A•B) + (A•B) = C A C B A B C 1

Summary 0s and 1s in Computer Binary Boolean Logic Truth Table Gates NOT, AND, OR Boolean Logic Laws Truth Table Gates Basic Gates Other Gates NAND, NOR, XOR

What I want you to do Review Chapter 1 Review Appendix B Finish assignment 1