Computer Organization: Boolean Logic
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1 Computer Organization: Boolean Logic
2 Representing and Manipulating Data Last Unit How to represent data as a sequence of bits How to interpret bit representations Use of levels of abstraction in representing more complex information (music, pictures) using simpler building blocks (numbers) This Unit How sequences of bits are implemented using electrical signals, and manipulated by circuits Use of levels of abstraction in designing more complex computer components from simpler components
3 3 Foundations Boolean logic is the logic of digital circuits
4 Implementing Bits Computers compute by manipulating electricity according to specific rules. We associate electrical signals inside the machine with bits. Any electrical device with two distinct states (e.g. on/off switch, two distinct voltage or current levels) could implement our bits. The rules are implemented by electrical circuits.
5 5 Conceptualizing bits and circuits ON or 1: true OFF or 0: false circuit behavior: expressed in Boolean logic or Boolean algebra
6 6 Boolean Logic (Algebra) Computer circuitry works based on Boolean Logic (Boolean Algebra) : operations on True (1) and False (0) values. A B A Λ B (A B) (conjunction) A A (NOT A) (negation) A and B in the table are Boolean variables, and are operations (also called functions).
7 Foundations of Digital Computing Boolean Algebra was invented by George Boole in 1854 (before digital computers) Variables and functions take on only one of two possible values: True (1) or False (0). The correspondence between Boolean Logic and circuits was not discovered until 1930s Shannon s thesis: A Symbolic Analysis of Relay and Switching Circuits argued that electrical applications of Boolean Algebra could construct any logical, numerical relationship. We forget about the logical (truth and falsehood) aspect of Boolean logic and just manipulate symbols. 7
8 8 Boolean Logic & Truth Tables Example: You can think of A Ù B below as is fun and is useful where A stands for the statement is fun, B stands for the statement is useful. A B A Λ B (A B) (conjunction) A A (NOT A) (negation)
9 9 Logic gates the basic elements of digital circuits
10 10 Logic Gates A gate is a physical device that implements a Boolean operator by performing basic operations on electrical signals. Nowadays, gates are built from transistors. Physical behavior of circuits is beyond the scope of our course. A B A Ù B A B A Ú B A NOT
11 A Mechanical Implementation Push-pull logic gate For an input pushed-in lever represents 1 For an output pushed-in lever represents 0 Source: randomwraith.com by Martin Howard
12 12 Combinational circuits combinations of logic gates
13 13 Combinational Circuits The logic states of inputs at any given time determine the state of the outputs. A Ù B B Ú C C Ù B (B Ú C) Ù (C Ù B) What is Q? (A Ù B) Ú ((B Ú C) Ù (C Ù B))
14 14 Truth Table of a Circuit Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q How do I know that there should be 8 rows in the truth table? Describes the relationship between inputs and outputs of a device
15 15 Truth Table of a Circuit Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q Describes the relationship between inputs and outputs of a device
16 16 Truth Table of a Circuit Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q Describes the relationship between inputs and outputs of a device
17 17 Truth Table of a Circuit Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q Describes the relationship between inputs and outputs of a device
18 18 Truth Table of a Circuit Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q Describes the relationship between inputs and outputs of a device
19 19 Truth Table of a Circuit Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q Describes the relationship between inputs and outputs of a device
20 20 Truth Table of a Circuit Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q Describes the relationship between inputs and outputs of a device
21 21 Truth Table of a Circuit Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q Describes the relationship between inputs and outputs of a device
22 22 Truth Table of a Circuit Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q Describes the relationship between inputs and outputs of a device
23 23 Describing Behavior of Circuits Boolean expressions Circuit diagrams Equivalent notations Truth tables
24 24 Manipulating circuits Boolean algebra and logical equivalence
25 25 Why manipulate circuits? The design process simplify a complex design for easier manufacturing, faster or cooler operation, Boolean algebra helps us find another design guaranteed to have same behavior
26 26 Logical Equivalence Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q Can we come up with a simpler circuit implementing the same truth table? Simpler circuits are typically cheaper to produce, consume less energy etc.
27 Logical Equivalence Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) Q = B Ù (A Ú C) A B C Q This smaller circuit is logically equivalent to the one above: they have the same truth table. By using laws of Boolean Algebra we convert a circuit to another equivalent circuit. 27
28 28 Laws for the Logical Operators Ù and Ú (Similar to and +) Commutative: A Ù B = B Ù A A Ú B = B Ú A Associative: A Ù B Ù C = (A Ù B) Ù C = A Ù (B Ù C) A Ú B Ú C = (A Ú B) Ú C = A Ú (B Ú C) Distributive: A Ù (B Ú B) = (A Ù B) Ú (A Ù C) A Ú (B Ù C) = (A Ú B) Ù (A Ú C) Identity: A Ù 1 = A A Ú 0 = A Dominance: A Ù 0 = 0 A Ú 1 = 1 Idempotence: A Ù A = A A Ú A = A Complementation: A Ù A = 0 A Ú A = 1 Double Negation: A = A
29 Laws for the Logical Operators Ù and Ú (Similar to and +) Commutative: A Ù B = B Ù A A Ú B = B Ú A Associative: A Ù B Ù C = (A Ù B) Ù C = A Ù (B Ù C) A Ú B Ú C = (A Ú B) Ú C = A Ú (B Ú C) Distributive: A Ù (B Ú C) = (A Ù B) Ú (A Ù C) A Ú (B Ù C) = (A Ú B) Ù (A Ú C) Not true for + and Identity: A Ù 1 = A A Ú 0 = A The A s and B s here are schematic variables! You can instantiate them with any expression that has a Boolean value: (x Ú y) Ù z = z Ù (x Ú y) (by commutativity) A Ù B = B Ù A
30 Applying Properties for Ù and Ú Showing (x Ù y) Ú ((y Ú z) Ù (z Ù y)) = y Ù (x or z) Commutativity A Ù B = B Ù A (x Ù y) Ú ((z Ù y) Ù (y Ú z)) Distributivity A Ù (B Ú C) = (A Ù B) Ú (A Ù C) (x Ù y) Ú (z Ù y Ù y) Ú(z Ù y Ù z) Associativity, Commutativity, Idempotence (x Ù y) Ú ((z Ù y) Ú (y Ù z)) Commutativity, idempotence A Ù A = A ( y Ù x) Ú (y Ù z) Distributivity (backwards) (A Ù B) Ú (A Ù C) = A Ù (B Ú C) y Ù (x Ú z) Conclusion: (x Ù y) Ú ((y Ú z) Ù (z Ù y)) = y Ù (x Ú z)
31 31 Extending the system more gates and DeMorgan s laws
32 More gates (N, N, X) A B A nand B A nor B nand ( not and ): A nand B = not (A and B) A xor B A B (A Ù B) nor ( not or ): A nor B = not (A or B) xor ( exclusive or ): A xor B = (A and not B) or (B and not A) A B A B (A Ú B) A Å B 32
33 A curious fact Functional Completeness of N and N Any logical circuit can be implemented using N gates only Same applies to N
34 34 DeMorgan s Law Nand: (A Ù B) = A Ú B Nor: (A Ú B) = A Ù B
35 35 DeMorgan s Law Nand: (A Ù B) = A Ú B if not (x > 15 and x < 110):... is logically equivalent to if (not x > 15) or (not x < 110):... Nor: (A Ú B) = A Ù B if not (x < 15 or x > 110):... is logically equivalent to if (not x < 15) and (not x > 110):...
36 36 A circuit for parity checking Boolean expressions and circuits
37 37 A Boolean expression that checks parity 3-bit odd parity checker F: an expression that should be true when the count of 1 bits is odd: when 1 or 3 of the bits are 1s. P = ( A Ù B Ù C) Ú ( A Ù B Ù C) Ú (A Ù BÙ C) Ú (A Ù B Ù C) A B C P There are specific methods for obtaining canonical Boolean expressions from a truth table, such as writing it as a disjunction of conjunctions or as a conjunction of disjunctions. Note we have four subexpressions above each of them corresponding to exactly one row of the truth table where P is 1.
38 38 The circuit 3-bit odd parity checker P = ( A Ù BÙ C) Ú ( A Ù B Ù C) Ú (A Ù BÙ C) Ú (A Ù B Ù C) A B C P A B P = (A Å B) Å C C P
39 39 Summary You should be able to: Identify basic gates Describe the behavior of a gate or circuit using Boolean expressions, truth tables, and logic diagrams Transform one Boolean expression into another given the laws of Boolean algebra
40 40 Next Time How circuits are combined to form a computer Von Neumann architecture revisited Fetch Decode - Execute Cycle
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