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# 21_boolean - Computer Architecture 6 Combinational Circuits...

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4/14/08 Copyright 2000, Kevin Wayne 1 Introduction to Computer Science Sedgewick and Wayne Copyright © 2007 http://www.cs.Princeton.EDU/IntroCS 6. Combinational Circuits George Boole (1815 – 1864) Claude Shannon (1916 – 2001) 2 Computer Architecture TOY lectures. von Neumann machine. This lecture. Boolean circuits. 3 Digital Circuits Q. What is a digital system? A. Digital: signals are 0 or 1. Q. Why digital systems? A. Accurate, reliable, fast, cheap. Basic abstractions. On, off. Wire: propagates on/off value. Switch: controls propagation of on/off values through wires. Applications. Cell phone, iPod, antilock brakes, microprocessors , … analog: signals vary continuously 4 Building Blocks

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4/14/08 Copyright 2000, Kevin Wayne 2 5 Wires Wires. On (1): connected to power. Off (0): not connected to power. If a wire is connected to a wire that is on, that wire is also on. Typical drawing convention: "flow" from top, left to bottom, right. 0 power connection 1 1 1 6 Logic Gates 7 Logic Gates 8 Multiway Gates Multiway gates. OR: 1 if any input is 1; 0 otherwise. AND: 1 if all inputs are 1; 0 otherwise. Generalized: negate some inputs.
4/14/08 Copyright 2000, Kevin Wayne 3 9 Multiway Gates Multiway gates. OR: 1 if any input is 1; 0 otherwise. AND: 1 if all inputs are 1; 0 otherwise. Generalized: negate some inputs. 10 Boolean Algebra 11 History. Developed by Boole to solve mathematical logic problems (1847). Shannon master's thesis applied it to digital circuits (1937). Boolean algebra. Boolean variable: value is 0 or 1. Boolean function: function whose inputs and outputs are 0, 1. Relationship to circuits. Boolean variable: signal. Boolean function: circuit. Boolean Algebra “possibly the most important, and also the most famous, master's thesis of the [20th] century” — Howard Gardner 12 Copyright 2004, Sidney Harris http://www.sciencecartoonsplus.com Boole Orders Lunch

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4/14/08 Copyright 2000, Kevin Wayne 4 13 Truth Table Truth table. Systematic method to describe Boolean function. One row for each possible input combination. n inputs 2 n rows. AND truth table 0 0 0 1 1 0 1 1 0 0 0 1 x y x y 14 Truth Table for Functions of 2 Variables Truth table. 16 Boolean functions of 2 variables. ZERO truth table for all Boolean functions of 2 variables y 0 0 0 1 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 x 0 0 1 1 AND 0 0 0 1 y 0 1 0 1 XOR 0 1 1 0 OR 0 1 1 1 x 0 NOR truth table for all Boolean functions of 2 variables y 0 1 0 1 0 1 0 0 1 1 0 y' 1 0 1 0 x' 1 1 0 0 1 0 1 1 EQ 1 0 0 1 1 1 0 1 NAND 1 1 1 0
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21_boolean - Computer Architecture 6 Combinational Circuits...

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