61boolean - Computer Architecture 6.1 Combinational Circuits Earlier lectures TOY machine Next two lectures Digital circuits Culminating lecture

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6.1 Combinational Circuits George Boole (1815 – 1864) Claude Shannon (1916 – 2001) 3 Computer Architecture Earlier lectures. ! TOY machine. Next two lectures. ! Digital circuits. Culminating lecture. ! Putting it all together and building a TOY machine. (on paper, we mean) 4 Digital Circuits What is a digital system? ! Digital: signals are 0 or 1. ! Analog: signals vary continuously. Why digital systems? ! Accuracy and reliability. ! Staggeringly fast and cheap. Basic abstractions. ! On, off. ! Switch that can turn something on or off. Digital circuits and you. ! Computer microprocessors. ! Antilock brakes. ! Cell phones. ! Ipods ! . . . 5 Wires Wires. ! Propagate logical values from place to place. ! Signals "flow" from left to right. A drawing convention, sometimes violated Actually: flow from producer to consumer(s) of signal 0 1 Input Output
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6 Logic Gates Logical gates. ! Fundamental building blocks. 0 1 1 NOT 1 AND 1 1 1 1 1 1 1 1 1 1 1 OR x x' y xy x + y 7 Multiway AND Gates AND(x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ). ! 1 if all inputs are 1. ! 0 otherwise. 8 Multiway OR Gates OR(x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ). ! 1 if at least one input is 1. ! 0 otherwise. 9 Boolean Algebra History. ! Developed by Boole to solve mathematical logic problems (1847). ! Shannon master's thesis applied it to digital circuits (1937). Basics. ! Boolean variable: value is 0 or 1. ! Boolean function: function whose inputs and outputs are 0, 1. Relationship to circuits. ! Boolean variables: signals. ! Boolean functions: circuits. "possibly the most important, and also the most famous, master's thesis of the [20th] century" --Howard Gardner
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11 Truth Table Truth table. ! Systematic method to describe Boolean function. ! One row for each possible input combination. ! N inputs ! 2 N rows. AND AND(x, y) AND Truth Table y x 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 12 Truth Table for Functions of 2 Variables Truth table.
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This note was uploaded on 12/16/2008 for the course COS 126 taught by Professor Douglasw.clark during the Fall '08 term at Princeton.

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61boolean - Computer Architecture 6.1 Combinational Circuits Earlier lectures TOY machine Next two lectures Digital circuits Culminating lecture

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