Motivating Example. Lecture 10: How does a computer act so logically? Simple Example: Boolean Variables

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1 9/27/ C 22 Introduction to Computation " UNIVRITY of WICONIN-MAION Computer ciences epartment Lecture : How does a computer act so logically? Professor Andrea Arpaci-usseau Fall 2 Motivating xample Ben only rides his bike to class if he overslept, but even then if it is raining he ll walk and show up late (he hates to bike in the rain). But if there s an exam that day he ll bike if he overslept, even in the rain. It s raining, Ben overslept, and there s an exam Will Ben bike today???? Propositional Logic: History Aristotle Law of contradiction..it will not be possible to be and not to be the same thing Law of excluded middle verything must either be or not be Man is mortal, ocrates is a man, therefore, ocrates is mortal toic Philosophers (3 rd century BC) Basic inference rules If p then q; not q; therefore not p e Morgan and Boole (9 th century) ymbolic logic automated, mechanical C. hannon (93s) Proposal to use digital hardware = True; = False imple xample: Boolean Variables d goes to the party if and only if tella does Choose Boolean variables for 2 events ach boolean variable is either TRU or FAL : d goes to party : tella goes to party Relationship between and? =

2 9/27/ imple xample: Logical OR d goes to the party if and only if an goes or tella goes Choose Boolean variables for 3 events: : d goes to party : an goes to party : tella goes to party = OR is TRU if one or both of and are TRU WARNING: In nglish OR has additional meaning! xample: You can eat an orange OR an apple Use term xclusive OR or XOR for this usage imple xample: Logical NOT, AN d goes to the party if and only if an does not and tella does. Choose Boolean variables for 3 events: : d goes to party : an goes to party : tella goes to party = (NOT ) AN Alternately: = AN s Composed of Boolean variables (True=, False=) Three Basic Operators: AN, OR, and NOT AN ( P OR (NOT Q)) C OR OR Boolean Algebra horthand A AN B A B (AB) = = = = A OR B A + B + = + = + = + = NOT A A = = What does Boolean Logic have to do with Computers? ) Reasoning in many algorithms uses boolean logic How can one use boolean logic in cratch programs? 2) Modern computers are themselves built from boolean logic Basic building block, logic gates, perform boolean logic; 2

3 9/27/ Boolean Logic in cratch: = (NOT ) AN Three Variables:,, No real booleans et and to true or false Value of depends on, Will d go to the party =False, =False? No =False, =True? Yes =True, =False? No =True, =True? No Order of Operations Are these two expressions the same? = (NOT ) AN = NOT ( AN ) In first case, when does d go to party? does not go AN goes In second case, when does d go to party? does not go AN goes goes AN does not go does not go AN does not go How would you construct in cratch? Variables are NOT formulas! Boolean gates hannon (939) What is output of script? Lesson: Variables hold VALU Value of variable determined when set Variables do not change except when set Variables do not change when subexpressions change x y x y High voltage = Low voltage = x y x + y AN: Output voltage is high if both of the input voltages are high; otherwise output voltage low. OR: Output voltage is high if either of the input voltages are high; otherwise output voltage low. x x NOT: Output voltage is high if the input voltage is low; otherwise output voltage high. (implicit extra wires for power) 3

4 9/27/ Combinational Circuits Boolean gates connected by wires with no cycles Wires transmit voltage (and hence values) NO! Circuits compute functions very combinational circuit computes a Boolean function of its inputs ots are shorthand for inverter Inputs Outputs Crossed wires that are not connected are sometimes drawn like this. Combinational circuits and control How would you express: If data has arrived and packet has not been sent, send a signal Need 3 boolean variables:, P, Truth Table Lists the truth value of the Boolean expression for all combinations of values for the variables. = AN = AN (NOT P) P ata arrived? Packet sent? end signal Truth table Write for all possible values of,. 4

5 9/27/ Truth table Truth Table xample Lists the truth value of the Boolean expression for all combinations of values for the variables. = AN = OR Truth table Write for all possible values of,. What is?!? Truth Table xample Three quivalent Representations What is?!? = OR Boolean Circuit = AN Truth table: Value of for every possible,. TRU=; FAL=. 5

6 9/27/ Ben Revisited Ben only rides to class if he overslept, but even then if it is raining he ll walk and show up late (he hates to bike in the rain). But if there s an exam that day he ll bike if he overslept, even in the rain. What boolean variables do we need? B: Ben Bikes (Output) R: Raining : xam today O: Overslept Can you create Boolean expression for B in terms of R, and O? Ben Revisited Ben only rides to class if he overslept, but even then if it is raining he ll walk and show up late (he hates to bike in the rain). But if there s an exam that day he ll bike if he overslept, even in the rain. What boolean variables do we need? B: Ben Bikes (Output) R: Raining : xam today O: Overslept B = O R + O Ben s truth table Ben only rides to class if he overslept, but even then if it is raining he ll walk and show up. But if there s an exam that day he ll bike if he overslept, even in the rain. Ben s truth table Ben only rides to class if he overslept, but even then if it is raining he ll walk and show up. But if there s an exam that day he ll bike if he overslept, even in the rain. O R B O R B B = O R + O B = O R + O 6

7 9/27/ Truth table Boolean expression Gates um of Products: Use OR of all input combinations that lead to TRU output O R B B = O NOT R + O O B = O R + O R + O R Can simplify expression: B = O R + O R O B AN, OR, and NOT gates can implement every Boolean function! How to determine hidden function? Reverse ngineer Approach: xplore all input combinations Today s ummary Today s topics Boolean logic: Operates on True () and False () Operators: AN, OR, NOT Three equivalent representations: Boolean expressions Combinational circuits Truth Tables Reading of Invitation to Computer cience Announcements Homework 4 available: pencil and paper Lab sections as always: MW 2-2, TT:4-6 7