LOGIC OPERATIONS Logic operations We have already seen

LOGIC OPERATIONS

Logic operations • We have already seen keywords or, and, not used in Python – Had a specific purpose – Boolean expressions. For example: if x >= 0 and x < 10: print("x is a single digit") • Python has a set of operators for bitwise computations: – – – & : bitwise AND | : bitwise OR ~ : bitwise NOT >>: shift to the right <<: shift to the left

A bit of intuition… 21 << 1 42 21 >> 1 10 5 & 6 ? 5 | 6 ?

A bit of intuition… Sometimes the bits are almost visible: 10101 21 << 1 10101 21 >> 1 101 110 5 & 6 101 110 5 | 6 000 ~0

A bit of intuition… Sometimes the bits are almost visible: 10101 21 << 1 101010 101 110 5 & 6 100 101 110 5 | 6 111 000 10101 21 >> 1 1010 ~0 111

However ~0 gives -1 ? ? ? >>> x = 0 >>> ~x -1 We will see a thorough discussion on this later. Here is a quick illustration of the idea. Binary Decimal 000 0 001 1 010 2 011 3 100 -4 101 -3 110 -2 111 -1 Take an example of 3 -bit binary number. We can represent 8 different values. If we want to represent negative numbers, we would typically (for good reasons!) to have – (n+1) to n. In this case, -4 to 3. In this system, the binary pattern of -1 is 111, which is ~0.

Truth tables input x y 0 0 1 1 0 1 output AND(x, y) 0 0 0 1 input x y 0 0 1 1 0 1 output OR(x, y) 0 1 1 1 AND outputs 1 only if ALL inputs are 1 OR outputs 1 if ANY input is 1 AND OR input output x NOT(x) 1 0 0 1 NOT reverses its input NOT

All computation … consists of functions of bits, or boolean values Boolean inputs x, y, … can only be 0 or 1 (False or True). inputs x 0 0 1 1 y 0 1 Truth table Boolean functions can output only 0 or 1 (False, True). output fn(x, y) 0 1 1 0

Lots of bits! output inputs x 0 0 1 1 y 0 0 1 1 z 0 1 0 1 Truth table fn(x, y, z) 0 1 1 0 0 1

LOGIC GATES AND CIRCUITS

Reviewing… • We have: – explored a wide range of data types – learned how different encodings are used for different types – 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?

42 101010 ADDER circuit 9 001001 Computation is simply the deliberate combination of those voltages! Feynman: Computation is just a physics experiment that always works! 110011 In a computer, each bit is represented as a voltage (1 is +5 v and 0 is 0 v) 51

In a computer, each bit is represented as a voltage (1 is +5 v and 0 is 0 v) (1) set input voltages ADDER circuit 9 Feynman: Computation is just a physics experiment that always works! 001001 000000 42 101010 Computation is simply the deliberate combination of those voltages!

In a computer, each bit is represented as a voltage (1 is +5 v and 0 is 0 v) 101010 (1) set input voltages ADDER circuit 110011 42 Computation is simply the deliberate combination of those voltages! 51 (2) read output voltages 9 001001

42 101010 ADDER circuit Computation is simply the deliberate combination of those voltages! HOW? The focus for this week: Learn to design circuits that can accomplish simple computations! 110011 In a computer, each bit is represented as a voltage (1 is +5 v and 0 is 0 v) 51 (2) read output voltages 9 001001

We need only three building circuits to compute anything at all input x y 0 0 1 1 0 1 output input AND(x, y) 0 0 0 1 x y 0 0 1 1 0 1 output OR(x, y) 0 1 1 1 input x xy "x and y" y x+y output "x or y" output x NOT(x) 1 0 0 1 input x y input output x x "not x"

Circuits from logic gates. . . ? What are all of the inputs that make this circuit output 1? Note three input gates, both OR and AND gates.

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 only need AND, OR and NOT to compute anything at all…

Constructive Proof ! i Specify a truth table defining any function you want input x 0 0 1 1 y 0 1 ii output fn(x, y) 0 1 1 0 Formula ! xy + xy minterm iii For each input row whose output needs to be 1, build an AND circuit that outputs 1 only for that specific input! OR them all together Minterm Expansion Principle – algorithm for building expressions from truth tables

Minterm expansion readily converts into logic gates…

Constructive Proof ! i Specify a truth table defining any function you want input x 0 0 1 1 y 0 1 output fn(x, y) 0 1 1 0 iii y OR them all together xy + xy NOT x ii For each input row whose output needs to be 1, build an AND circuit that outputs 1 only for that specific input! AND Does this work for the red row? Does this work for any other rows?

Constructive Proof ! i Specify a truth table defining any function you want input x 0 0 1 1 y 0 1 ii output fn(x, y) 0 1 1 0 iii For each input row whose output needs to be 1, build an AND circuit that outputs 1 only for that specific input! OR them all together xy + xy How did we handle the rows outputting zero? NOT AND OR NOT x y AND

Constructive Proof ! i Specify a truth table defining any function you want f o o r p y e fn(x, y) v i t c u r T t 0 0 s O n N iii OR them all together 1 1 a co d n a , 0 is is 1 R O h , y T 1 0 ND n a d A l i t u a b ! h t n o o t i t e c NOT c i n f AND f u ) s n a e l o (bo NOT input x 0 0 1 1 x ii For each input row whose output needs to be 1, build an AND circuit that outputs 1 only for that specific input! output How did we handle the rows outputting zero? y AND OR

i Constructive Proof ! Bu t. . . For each input row whose A Specify a truth table b defining o L o needs to be 1, build ii any function you want lea L output f u. AND an circuit that outputs 1 nforctthat n f only f o oiospecific input! u r p n n x y e fn(x, y) c v i s t t i c a o u r r n T t 0 0 0 e s O s n N : j iii u OR them all together 0 1 1 a co d b n s i a n t s i , 1 0 is 1 a R r O h , y y T 1 1 0 ND n ! a d A l i t u ! tha e to b n o i t c NOT c i n f AND f u ) s n a e l OR o o b ( input output How did we handle the rows outputting zero? NOT x y AND

EXAMPLE

Try it! the less than or equal circuit (<=) We usually know what we want to do… We just have to determine how to build it! Truth Table "English" f(x, y) should output 1 when x <= y otherwise, output 0 input bits x 0 0 1 1 y 0 1 Formula output bit x <= y 1 1 0 1 Circuit

Try it! the less than or equal circuit (<=) We usually know what we want to do… We just have to determine how to build it! Truth Table "English" f(x, y) should output 1 when x <= y otherwise, output 0 input bits x 0 0 1 1 y 0 1 output bit x <= y 1 1 0 1 Formula xy + xy Circuit