Boolean Logic  NOT, AND, OR

Based on syyg,gygmbolic logic, designed by George Boole Boolean variables take values as 0 or 1. Boolean expressions created from: Boolean logic  NOT, AND, OR

Introduction to Computer Yung-Yu Chuang

with slides by Sedgewick & Wayne (introcs.cs.princeton.edu), Nisan & Schocken (www.nand2tetris.org) and Harris & Harris (DDCA) 2

NOT AND

XY XY X X’

Digital gate diagram for NOT: Digital gate diagram for AND:

NOT AND

3 4 OR Operator Precedence

Examples showing the order of operations: X+Y NOT > AND > OR

Digit a l ga te diagram for OR:

Use parentheses to avoid ambiguity

5 6

Defining a function Defining a function

Description: square of x minus 1 Description: number of days of the x-th month of a Algebraic form : x2-1 non-leappy year x f(x) Enumeration: Algebraic form: ? xf(x) 131 Enumeration: 228 1 0 331 23 430 38 531 415 630 731 524 831 : : 930 10 31 11 30 12 31 7 8 Truth Table Proving the equivalence of two functions

Truth table. Prove that x2-1=(x+1)(x-1)

 Systematic method to describe Boolean function.  One row for each possible input combination. Using algebra: (you need to follow some rules) N  N inputs  2 rows. (x+1)(x-1) = x2+x-x-1= x2-1

2 Using enumeration: x (x+1)(x-1) x -1 x y x y 1 0 0 0 0 0 23 3 0 1 0 1 0 0 38 8 1 1 1 4 15 15 AND truth table 52424 : : :

9 10

Important laws Simplifying Boolean Equations x + 1 = 1 x + y = y + x Example 1: x + 0 = x x + (y+z))( = (x+y) + z x + x = 1 • Y = AB + AB x．1 = x x．y = y．x x．0 = 0 x．(y．z) = (x．y)．z x．x = 0 x．(y+z) = xy + xz DMDeMorgan Law

x．y = x + y

Chapter 2 <12> 11 Simplifying Boolean Equations Simplifying Boolean Equations Example 1: Example 2: • Y = AB + AB • Y = A(AB + ABC) = B(A + A) = B(1) = B

Chapter 2 <13> Chapter 2 <14>

Simplifying Boolean Equations DeMorgan’s Theorem

A Example 2: • Y = AB = A + B Y B • Y = A(AB + ABC) A Y = A(AB(1 + C)) B = A(AB(1)) = A(AB) A (AA)B Y = (AA)B • Y = A + B = AB B

= AB A Y B

Chapter 2 <15> Chapter 2 <16> Bubble Pushing Bubble Pushing

• Backward: • What is the Boolean expression for this – Body changes circuit? – Adds bubbles to inputs A A Y Y A B B B Y • Forward: C – Body changes D – Adds bubble to output

A A Y Y B B

Chapter 2 <17> Chapter 2 <18>

Bubble Pushing Bubble Pushing Rules

• What is the Boolean expression for this • Begin at output, then work toward inputs circuit? • Push bubbles on final output back • Draw gates in a form so bubbles cancel A B A Y B C D C Y D Y = AB + CD

Chapter 2 <19> Chapter 2 <20> Bubble Pushing Example Bubble Pushing Example

no output A A bubble B B

C Y C Y D D

Chapter 2 <21> Chapter 2 <22>

Bubble Pushing Example Bubble Pushing Example

no output no output A bubble A bubble B B

C Y C Y D D

bubble on bubble on A input and output A input and output B B

C Y C Y D D no bubble on input and output A B

C Y D Y = ABC + D

Chapter 2 <23> Chapter 2 <24> Truth Tables (1 of 3) Truth Tables (2 of 3)

A Boolean function has one or more Boolean Example: X Y inputs, and returns a single Boolean output. A truth table shows all the inputs and outputs of a Boolean function

Example: X  Y

25 26

Truth Tables (3 of 3) S Truth Table for Functions of 2 Variables

When s=0, return x; otherwise, X mux Z Truth table. Y every 4-bit value represents one return y.  16 Boolean functions of 2 variables.

Example: (Y  S)  (X S) Two-input multiplexer

x y ZERO AND x y XOR OR 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1

Truth table for all Boolean functions of 2 variables

x y NOR EQ y' x' NAND ONE 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1

Truth table for all Boolean functions of 2 variables

27 28 All Boolean functions of 2 variables Truth Table for Functions of 3 Variables

Truth table. every 4-bit value represents one  16 Boolean functions of 2 variables. every 8-bit value represents one  256 Boolean functions of 3 variables. every 2n-bit value represents one  2^(2^n) Boolean functions of n variables!

x y z AND OR MAJ ODD 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 1 1 1 1 1

some functions of 3 variables

Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 29 30

Sum-of-Products Universality of AND, OR, NOT

Sum-of-products. Systematic procedure for Fact. Any Boolean function can be expressed reppgresenting a Boolean function using g, AND, usingg,, AND, OR, NOT. proves that { AND, OR, NOT } Notation Meaning OR, NOT. are universal  { AND, OR, NOT } are universal. x' NOT x  Form AND term for each 1 in Boolean function.  Ex: XOR(x,y) = xy' + x'y. xyx y x AND y  OR terms together. x + y x OR y x y z MAJ x'yz xy'z xyz' xyz x'yz + xy'z + xyz' + xyz Expressing XOR Using AND, OR, NOT 0 0 0 0 0 0 0 0 0 x y x' y' x'y xy' x'''y + xy' x XOR y 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 Exercise. Show {AND, NOT}, {OR, NOT}, {NAND}, 1 1 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 1 {NOR} are universal.

expressing MAJ using sum-of-products HH.int. DDMeMorg an's law w(: (x'y y)')' = x + y.

31 32 From Math to Real-World implementation Implementation of gates

We can implement any Boolean function using Fluid switch (http://www.cs.princeton.edu/introcs/lectures/fluid-computer.swf NAND ggyates only.

We talk about abstract Boolean algebra (logic) so far.

Is it possible to realize it in real world?

The tech nol ogy need s to permit swit c hing an d conducting. It can be built using magnetic, optica l, bio log ica l, hy drau lic an d pneuma tic mechanism.

Digital Circuits Wires

What is a digital system? Wires.

 Analoggg: signals vary continuously .  On (1): connected to power.

 Digital: signals are 0 or 1. 0 1 0  Off (0): not connected to power.

3.3V  If a wire is connected to a wire that is on, that 2.8V wire is also on. 0.5V  Typical drawing convention: "flow" from top, left 0.0V Whyyg digital sy stems? to bottom, ri ght.

 Accuracy and reliability. 1  Staggeringly fast and cheap. power connection 1 Basic abstractions. 1

 On, off.

 Wire: propagates on/off value. 0  Swwppgf/ffuitch: controls propagation of on/off values through wires.

35 36 Controlled Switch Relay

Controlled switch. [relay implementation]

 3 connections: inppput, output, control.

 Magnetic force pulls on a contact that cuts electrical flow.

 Control wire affects output wire, but output does not affect control; establishes forward flow of informati on over ti me.

X X

37 38

Circuit Anatomy Logic Gates: Fundamental Building Blocks

39 40 NOT NOT

41 42

OR Series relays = NOR

0 0 0 1

1 0

1 0 1 1

0 0

43 44 OR AND

45 46

AND Logic Gates: Fundamental Building Blocks

47 48 What about parallel relays? =NAND Can we implement AND/OR using parallel relays? Now we know how to implement AND,OR and 0 0 NOT. We can just use them as black boxes without knowing how they were implemented. Princippgle of information hiding.

1 1 0 1

49 50

Multiway Gates Multiway Gates

Multiway gates. Multiway gates.

 OR: 1 if anyyp input is 1; 0 otherwise.  OR: 1 if anyyp input is 1; 0 otherwise.

 AND: 1 if all inputs are 1; 0 otherwise.  AND: 1 if all inputs are 1; 0 otherwise.

 Generalized: negate some inputs.  Generalized: negate some inputs.

51 52 Multiway Gates Translate Boolean Formula to Boolean Circuit

Multiway gates. Sum-of-products. XOR.

 Can also be built from 2-wayyg gates (less efficient but implementation independent)

 Example: build 4-way OR from 2-way ORs

53 54

Translate Boolean Formula to Boolean Circuit Translate Boolean Formula to Boolean Circuit

Sum-of-products. XOR. Sum-of-products. XOR.

55 56 Gate logic ODD Parity Circuit Interface ODD(x, y, z). a  1 if odd number of inputs are 1. Xor out b  0 otherwise.

a btb out

0 0 0 0 1 1 101 0 1 IlImplemen ttitation 1 1 0 a And Not

Or out

Not And b

Xor(a, b) = Or (An d(a, Not (b)),An d(Not (a ), b)))

ODD Parity Circuit ODD Parity Circuit

ODD(x, y, z). ODD(x, y, z).

 1 if odd number of inputs are 1.  1 if odd number of inputs are 1.

 0 otherwise.  0 otherwise.

x y z ODD x'y'z x'yz' xy'z' xyz x'y'z+x'yz' + xyy'z' + xyyz 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1

Expressing ODD using sum-of-products

59 60 ODD Parity Circuit Expressing a Boolean Function Using AND, OR, NOT

ODD(x, y, z). Ingredients.

 1 if odd number of inputs are 1.  AND gates.

 0 otherwise.  OR gates.

 NOT gates.

 Wire.

Instructions.

 Step 1: represent input and output signals with Boolean variables.

 Step 2: construct truth table to carry out computation.

 Step 3: derive (simplified) Boolean expression using sum-of products.

 Steppp 4: transform Boolean expression into circuit.

61 62

Translate Boolean Formula to Boolean Circuit Translate Boolean Formula to Boolean Circuit

Sum-of-products. Majority. Sum-of-products. Majority.

63 64 Translate Boolean Formula to Boolean Circuit Translate Boolean Formula to Boolean Circuit

Sum-of-products. Majority. Sum-of-products. Majority.

65 66

Simplification Using Boolean Algebra Boolean expression simplification

Every function can be written as sum-of-product Karnaugh map

Many possible circuits for each Boolean function.

 Sum-of-products not necessarily optimal in: – number of switches (space) – depth of circuit (time)

67 68 Karnaugh Maps (K‐Maps) K‐Map

• Boolean expressions can be minimized by • Circle 1’s in adjjqacent squares combining terms • In Boolean expression, include only • K‐maps minimize equations graphically literals whose true and complement form • PA + PA = P are not in the circle Y A BCY AB 0 001 00 01 11 10 C A BCY Y Y 0 011 AB AB 0 1 0 0 0 0 0 1 00 01 11 10 00 01 11 10 0 1 0 0 0 0 011 C C 0 110 0 100 1 000 0 1 000 0 ABC ABC ABC ABC 0 110 1 010 1 000 1 1 0 0 1 1 0 0 0 1 010 1 1 000 1 ABC ABC ABC ABC 1 110 1 100 1 110 Y = AB

Chapter 2 <69> Chapter 2 <70>

3‐Input K‐Map 3‐Input K‐Map

Y Y AB AB C 00 01 11 10 C 00 01 11 10

0 ABC ABC ABC ABC 0 ABC ABC ABC ABC

1 ABC ABC ABC ABC 1 ABC ABC ABC ABC

Truth Table K-Map Truth Table K-Map Y Y A BCY AB A BCY AB 0 0 0 0 C 00 01 11 10 0 0 0 0 C 00 01 11 10 0 010 0 010 0 101 0 0 101 0 0 110 0 111 0 111 1 0 0 0 1 0 0 0 1 010 1 1 010 1 0 1 00 1 100 1 100 1 111 1 111 Y = AB + BC Chapter 2 <71> Chapter 2 <72> K‐Map Rules 4‐Input K‐Map

• Every 1 must be circled at least once A B C D Y Y • Each circle must span a power of 2 (i.e. 1, 2, 0 0 001 AB 0 0 010 CD 00 01 11 10 4) squares in each direction 0 0 1 0 1 0 0 111 00 0 1 000 • Each circle must be as large as possible 0 1 011 0 1 1 0 1 01 0 1 111 • A circle may wrap around the edges 1 0 001 1 0 011 11 • A “don' t care” (X) is circled only if it helps 1 0 1 0 1 1 0 110 1 1 000 10 minimize the equation 1 1 010 1 1 1 0 0 1 1 110

Chapter 2 <73> Chapter 2 <74>

4‐Input K‐Map 4‐Input K‐Map

A B C D Y Y A B C D Y Y 0 0 001 AB 0 0 001 AB 0 0 010 CD 00 01 11 10 0 0 010 CD 00 01 11 10 0 0 1 0 1 0 0 1 0 1 0 0 111 00 1 0 0 1 0 0 111 00 1 0 0 1 0 1 000 0 1 000 0 1 011 0 1 011 0 1 1 0 1 01 0 1 0 1 0 1 1 0 1 01 0 1 0 1 0 1 111 0 1 111 1 0 001 1 0 001 1 0 011 11 1100 1 0 011 11 1100 1 0 1 0 1 1 0 1 0 1 1 0 110 1 0 110 1 1 000 10 1101 1 1 000 10 1101 1 1 010 1 1 010 1 1 1 0 0 1 1 1 0 0 1 1 110 1 1 110 Y = AC + ABD + ABC + BD

Chapter 2 <75> Chapter 2 <76> K‐Maps with Don’t Cares K‐Maps with Don’t Cares

A B C D Y Y A B C D Y Y 0 0 001 AB 0 0 001 AB 0 0 010 CD 00 01 11 10 0 0 010 CD 00 01 11 10 0 0 1 0 1 0 0 1 0 1 0 0 111 00 0 0 111 00 1 0 X 1 0 1 000 0 1 000 0 1 01X 0 1 01X 0 1 1 0 1 01 0 1 1 0 1 01 0 X X 1 0 1 111 0 1 111 1 0 001 1 0 001 110 01 11 110 01 11 11XX 1 0 10X 1 0 10X 1 0 11X 1 0 11X 10 10 11XX 1 1 00X 1 1 00X 1 1 01X 1 1 01X 1 1 10X 1 1 10X 1 1 11X 1 1 11X

Chapter 2 <77> Chapter 2 <78>

K‐Maps with Don’t Cares Example

A B C D Y Y 0 0 001 AB CD 00 01 11 10 z 0 0 010 xy 0 0 1 0 1 0 0 111 00 1 0 X 1 0 1 000 0 1 01X 0 1 1 0 1 01 0 X X 1 0 1 111 1 0 001 110 01 11 11XX 1 0 10X 1 0 11X 1 1 00X 10 11XX 1 1 01X 1 1 10X 1 1 11X Y = A + BD + C

80 Chapter 2 <79> Simplification Using Boolean Algebra Layers of Abstraction

Many possible circuits for each Boolean function. Layers of abstraction.

 Sum-of-ppyproducts not necessarily optimal in:  Build a circuit from wires and switches. – number of switches (space) [implementation] – depth of circuit (time)  Define a circuit by its inputs and outputs. [API]

MAJ(x, y, z) = x'yz + xy'z + xyz' + xyz = xy + yz + xz.  To control complexity, encapsulate circuits. [ADT]

81 82

Layers of Abstraction Specification

Layers of abstraction.

 Build a circuit from wires and switches. [implementation]

 Define a circuit by its inputs and outputs. [API]

 To control complexity, encapsulate circuits.  Step 1: iden tify input and out pu t

[ADT]  Step 2: construct truth table

 Step 3: derive (simplified) Boolean expression using sum-of products.

 Step 4: transform Boolean expression into circuit/l/implement it using HDL.

Yo u wo uld like to test the gate before packagin g.

83 HDL Example: Building an And gate

Contract: And.cmp a b out When running your a 0 0 0 .hdl on our .tst, 0 1 0 And out your .out should b 1 0 0 1 1 1 be the same as our .cmp.

And.hdl And.tst

CHIP And load And.hdl, { IN a, b; output-file And.out, OUT out; compare-to And.cmp, // implementation missing output-list a b out; } seta0,set b 0,,,peval,output; set a 0,set b 1,eval,output; set a 1,set b 0,eval,output; seta1,setb1,eval,output;

Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 86

Building an And gate Building an And gate

Interface: And(a,b) = 1 exactly when a=b=1 Implementation: And(a,b) = Not(Nand(a,b))

a a And out out b b

And.hdl And.hdl

CHIP And CHIP And { IN a, b; { IN a, b; OUT out; OUT out; // implementation missing // implementation missing } }

Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 87 Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 88 Building an And gate Building an And gate

Implementation: And(a,b) = Not(Nand(a,b)) Implementation: And(a,b) = Not(Nand(a,b))

a a a out in out a out in out Nand x Not out NAND x NOT out b b b b

And.hdl And.hdl

CHIP And CHIP And { IN a, b; { IN a, b; OUT out; OUT out; // implementation missing Nand(a = a, } b = b, out = x); Not(in = x, out = out) }

Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 89 Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 90

Hardware simulator (demonstrating Xor gate construction) Hardware simulator

test HDL HDL script program program

Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 91 Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 92 Hardware simulator Project materials: www.nand2tetris.org

Project 1 web site

HDL program

output file

And. hdl , And.tst , And.cmp files

Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 93 Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 94

Project 1 tips Gates for project #1 (Basic Gates)

 Read the Introduction + Chapter 1 of the book  Download the book’s software suite  Go through the hardware simulator tutorial  Do Project 0 (optional)  You’re in business.

Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 95 Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 96 Gates for project #1 Gates for project #1 (Multi-bit version)

Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 97 Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 98

Gates for project #1 (Multi-way version) Gates for project #1 (Multi-way version)

Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 99 100 Gates for project #1 (Multi-way version) Gates for project #1 (Multi-way version)

Perspective End notes: Canonical representation

 Each Boolean function has a canonical representation Whodunit story: Each suspect may or may not have an alibi (a), a motivation to commit the crime (m), and a relationship to the weapon found in the scene of the crime (w). The pppolice decides to focus attention only on suspects  The canonical representation is expressed in terms of And, Not, Or for whom the proposition Not(a) And (m Or w) is true.

 And, Not, Or can be expressed in terms of Nand alone Truth table of the "suspect" function s(a,m, w)  a  (m  w)  Ergo, every Boolean function can be realized by a standard PLD consisting of Nand gates only

 Mass production b and c

. f(a,b,c)  or Universal building blocks, . unique topology

and  Gates, neurons, atoms, … Canonical form: s(a, m, w)  am w  amw  amw

Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 103 104 End notes: Canonical representation (cont.) End notes: Programmable Logic Device for 3-way functions

a s(a, m, w)  a (m  w) legend: active fuse b and blown fuse a s c or m and 8 and terms . f(a,b,c) w connected to the . or s(a,m, w)  am w  amw  amw same 3 inputs .

a m and w single or term connected to the and outputs of 8 and terms s and or

_ _ _ PLD implementation of f(a,b,c)= a b c + a b c and (the on/off states of the fuses determine which gates participate in the computation)

End notes: Programmable Logic Device for 3-way functions

 Two-level logic: ANDs followed by ORs  Example: Y = ABC + ABC + ABC A B C

ABC minterm: ABC

minterm: ABC

Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org , Chapter 1: Boolean Logic slide 107