and not if x >= 0 and x < 10: print("x is a single digit") &

Transcription

1 LOGIC OPERATIONS

2 Logic operations We have alread seen kewords or, and, not used in Pthon Had a specific purpose Boolean epressions. For eample: if >= and < : print(" is a single digit") Pthon has a set of operators for bitwise computations: & : bitwise AND : bitwise OR ~ : bitwise NOT >>: shift to the right <<: shift to the left

3 A bit of intuition 2 << 5 & ?? 2 >>

4 A bit of intuition Sometimes the bits are almost visible: 2 << 5 & >> ~

5 A bit of intuition Sometimes the bits are almost visible: 2 << 5 & >> ~

6 However ~ gives -??? >>> = >>> ~ - We will see a thorough discussion on this later. Here is a quick illustration of the idea. Binar Decimal Take an eample of 3-bit binar number. We can represent 8 different values. If we want to represent negative numbers, we would tpicall (for good reasons!) to have (n+) to n. In this case, -4 to 3. In this sstem, the binar pattern of - is, which is ~.

7 Truth tables input output AND(,) input output input output OR(,) NOT() AND outputs onl if ALL inputs are OR outputs if ANY input is NOT reverses its input AND OR NOT

8 All computation consists of functions of bits, or boolean values Boolean inputs,, can onl be or (False or True). Boolean functions can output onl or (False, True). inputs output fn(,) Truth table

9 Lots of bits! inputs z Truth table output fn(,,z)

10 LOGIC GATES AND CIRCUITS

11 Reviewing We have: eplored a wide range of data tpes learned how different encodings are used for different tpes learned that, at the core of all data stored in the computer are bits observed different operations that can be performed on these bits (AND, OR, NOT) We have one BIG QUESTION remaining HOW IS COMPUTATION ACTUALLY CARRIED OUT?

12 In a computer, each bit is represented as a voltage ( is +5v and is v) 42 Computation is simpl the deliberate combination of those voltages! Fenman: Computation is just a phsics eperiment that alwas works! ADDER circuit 5 9

13 In a computer, each bit is represented as a voltage ( is +5v and is v) 42 Computation is simpl the deliberate combination of those voltages! Fenman: Computation is just a phsics eperiment that alwas works! () set input voltages ADDER circuit 9

14 In a computer, each bit is represented as a voltage ( is +5v and is v) Computation is simpl the deliberate combination of those voltages! 42 () set input voltages ADDER circuit 5 (2) read output voltages 9

15 In a computer, each bit is represented as a voltage ( is +5v and is v) 42 Computation is simpl the deliberate combination of those voltages! HOW? The focus for this week: Learn to design circuits that can accomplish simple computations! ADDER circuit 5 (2) read output voltages 9

16 We need onl three building circuits to compute anthing at all input output AND(,) input output OR(,) input output NOT() input input " and " output + " or " output "not "

17 Circuits from logic gates...? What are all of the inputs that make this circuit output? Note the three input gates, both OR and AND gates.

18 Logisim HW 5 Use Logisim (a free circuit simulation package) to design circuits to perform simple computations Hard? Well, let's recall our claim we onl need AND, OR and NOT to compute anthing at all

19 Constructive Proof! i Specif a truth table defining an function ou want input output fn(,) iii ii For each input row whose output needs to be, build an AND circuit that outputs onl for that specific input! OR them all together Formula! + minterm Minterm Epansion Principle algorithm for building epressions from truth tables

20 Minterm epansion readil converts into logic gates

21 Constructive Proof! i Specif a truth table defining an function ou want input output fn(,) iii ii For each input row whose output needs to be, build an AND circuit that outputs onl for that specific input! OR them all together + NOT AND Does this work for the red row? Does this work for an other rows?

22 Constructive Proof! i Specif a truth table defining an function ou want input output fn(,) iii ii For each input row whose output needs to be, build an AND circuit that outputs onl for that specific input! OR them all together + How did we handle the rows outputting zero? NOT AND OR NOT AND

23 Constructive Proof! i Specif a truth table defining an function ou want input output fn(,) iii ii For each input row whose output needs to be, build an AND circuit that outputs onl for that specific input! OR them all together How did we handle the rows outputting zero? NOT AND OR NOT AND

24 Constructive Proof! i Specif a truth table defining an function ou want input output fn(,) iii ii For each input row whose output needs to be, build an AND circuit that outputs onl for that specific input! OR them all together How did we handle the rows outputting zero? NOT AND OR NOT AND

25 EXAMPLE

26 the less than or equal circuit (<=) Tr it! We usuall know what we want to do We just have to determine how to build it! "English" Truth Table Formula f(,) should output when <= otherwise, output input bits output bit <= Circuit

27 the less than or equal circuit (<=) Tr it! We usuall know what we want to do We just have to determine how to build it! "English" f(,) should output when <= otherwise, output input bits Truth Table output bit <= Formula + + Circuit

28 <=