Lectures Engin112 6-8
Binary Logic and Boolean Algebra
Maciej Ciesielski Department of Electrical and Computer Engineering 09/15/2008
Recap from last lecture
Codes
Assignment of bit sequence to represent discrete elements BCD code and addition Weighted code Error detecting codes
Today's lecture
Binary storage and registers
Register transfer operations
Binary logic Boolean algebra
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Error-Detecting Code
Communication between systems can be "noisy"
Environmental conditions can cause bit flips
Error-detecting codes
If one bit is flipped, the code becomes invalid Communication system can detect that and retransmit
"Parity bit" is added to binary data
"Even parity": choose parity bit such that # of 1's is even "Odd parity": choose parity bit such that # of 1's is odd
Example:
Data=1000001, even parity=0, odd parity=1 Data=1010100, even parity=1, odd parity=0
Only one parity bit is used
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Binary Storage and Registers
How is information stored on a digital system?
Bits are stored in "binary cells" Binary cell can have two stable states: `0' and `1'
Binary cells are grouped into registers
n cells make up n-bit register
Size of registers is typically predefined
Simple microcontroller: 8 bits = 1 byte Pentium: 32 bits = 4 bytes Mac G5: 64 bits = 8 bytes
Digital system can usually process entire registers
"Register transfer" operation specify processing
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Register Example
Reading from keyboard:
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Register Transfer Operations
Example: addition
Designing digital logic is focus of rest of course
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Binary Logic
Binary logic uses two possible values
`1' and `0' `yes' and `no' `true' and `false' Can be represented by variables: A, B, C, x, y, z, ...
Logic functions modify input values
What is the minimum number of inputs?
1 input -> pretty uninteresting, inverter or wire only 2 inputs -> basic logic functions
What are possible logic functions?
NOT, AND, OR Others can be derived from those (NAND, XOR, etc.)
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George Boole
Father of Boolean algebra
He came up with a type of linguistic algebra, the three most basic operations of which were (and still are) AND, OR and NOT. It was these three functions that formed the basis of his premise, and were the only operations necessary to perform comparisons or basic mathematical functions. Boole's system (detailed in his 'An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities', 1854) was based on a binary approach, processing only two objects - the yes-no, true-false, on-off, zero-one approach. Surprisingly, given his standing in the academic community, George Boole (1815 - 1864) Boole's idea was either criticized or completely ignored by the majority of his peers. Eventually, one bright student, Claude Shannon (19162001), picked up the idea and ran with it
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NOT Function
Complement operation (inversion)
Single input Inverts value of input Symbolized by prime x' or overbar x
Truth table
Input combinations on the left Output of function on the right
input
output
x 0 1
x' 1 0
Graphic symbol
NOT gate (inverter) Little circle indicates inversion
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AND Function
Operation to check if two conditions are met
Two inputs Output = 1 if and only if both inputs are 1 Symbolized by dot or absence of operator xy (also written as xy )
inputs output
Truth table
Needs to consider 22 = 4 input combinations
Graphic symbol
AND gate
x 0 0 1 1
y 0 1 0 1
z 0 0 0 1
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OR Function
Operation to check if at least one condition is met
Two inputs Output = 1 if any one or both inputs are 1 Symbolized by "plus" sign: x + y
Truth table Graphic symbol
OR gate
x y 0 0 0 1 1 0 1 1
z 0 1 1 1
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Evaluation of Logic Functions
Timing diagrams (waveforms)
Horizontal axis is time ( t ) Vertical axis shows signals, each with two different voltage levels
t
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Multiple Inputs
Two inputs might not be enough 3-input AND gate:
4-input OR gate:
What are the rules for aggregating functions?
Boolean algebra
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Reading Assignment
Read Mano 2-4 2-6 Reminder:
Homework #1 due Wednesday BEFORE class ! Make sure to provide on the first/cover page the following:
ENGIN 112 Fall 2008 Homework 1, date submitted Your Name (no ID needed)
Please paginate + staple your homework
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Algebras
What is an algebra?
Mathematical system consisting of
Set of elements Set of operators Axioms or postulates
Why is it important?
Defines rules of "calculations"
Example: arithmetic on natural numbers
Set of elements: N = {1,2,3,4,...} Operator: +, , * Axioms: associativity, distributivity, closure, identity elements, etc.
Note: operators with two inputs are called binary
Does not mean they are restricted to binary numbers! Operator(s) with one input are called unary
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Axioms of Algebraic Structures
Common axioms (or postulates):
1. Closure:
A set S is closed with respect to a binary operator if the operator specifies a rule for obtaining an element of S. Example: S = N and = +, * (not )
2. Associativity:
A binary operator on a set S is said to be associative if (x y) z = x (y z) Example: S = N and = +, * (not )
3. Commutativity:
A binary operator on a set S is said to be commutative if xy=yx Example: S = N and = +, * (not )
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Axioms of Algebraic Structures
Common axioms (continued):
4. Identity element:
Set S is said to have an identity element with respect to binary operation on S if there exists an element e S such that e x = x e = x for every x S Example: S = Z, = +, e = 0
5. Inverse:
A set S having the identity element e with respect to a binary operator is said to have an inverse whenever, for every xS, there exists and element yS such that x y = e Example: S = Z, = +, e = 0, y = -x
6. Distributivity: If and are two binary operators on a set S, is said to be distributive over whenever x (y z) = (x y) (x z) Example: S = N, = , = +
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Example Algebra
Instance of an algebraic structure is a field
A field consists of
Set of elements Two binary operators (each having properties 15) Both operators combined have property 6 (distributivity)
Example: field for ordinary algebra on real numbers
Binary operator + defines addition Additive identity is 0 Additive inverse defines subtraction Binary operator defines multiplication Multiplicative identity is 1 Multiplicative inverse defines division Distributive law: over +: a (b + c) = (a b) + (a c)
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Boolean Algebra
We need to define algebra for binary values
Developed by George Boole in 1854
Boolean algebra: algebraic structure with
Set of elements B with two operators (+, ) satisfying postulates
Huntington postulates for Boolean algebra (1904):
1. Closure with respect to operator + and operator 2. Identity element 0 for operator + and 1 for operator (e x = x ) 3. Commutativity with respect to + and
x+y = y+x, xy = yx
4. Distributivity of over +, and + over
x(y+z) = (xy)+(xz) and x+(yz) = (x+y)(x+z)
5. Complement for every element x is x' with x+x'=1, xx'=0 6. There are at least two elements x,yB such that xy
Note: + and are chosen to be intuitive
Do not blindly substitute with ordinary algebra, it won't work !
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Boolean vs. Ordinary Algebra
Differences
Postulates do not include associativity
Can be derived for both operators
Distributivity of + over holds for Boolean, but not for ordinary algebra
x+(yz) = (x+y)(x+z)
Boolean does not have inverse elements for + or
Thus, no subtraction or division is operators
Complement not defined in ordinary algebra Set of elements in Boolean algebra not yet defined
But there must be at least two elements
Questions
What is the set of elements? What are the rules of operation for + and ?
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Two-valued Boolean Algebra
Two-valued Boolean is defined as
A set of elements B = {0,1}
"+" operator:
x 0 0 1 1 y 0 1 0 1 x+y 0 1 1 1
"" operator:
x y xy 0 0 0 1 1 0 1 1 0 0 0 1
inverse
x x' 0 1 1 0
Observation + is OR, is AND,
Other notation: + => ,
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' is NOT
=> , NOT =>
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Check Huntington Postulates
Verify that postulates hold for Boolean algebra
1. Closure?
Yes, all operations with {0,1}B Yes, 0 for + and 1 for Yes, for +: 0+0=0, 1+1=1, 0+1=1+0=1 Yes, for : 00=0, 11=1, 01=10=0 Huntington postulates: Post. 1: closure Post. 2: (a) x+0=x, (b) x1=x Post. 3: (a) x+y=y+x, (b) xy=yx Post. 4: (a) x(y+z) = xy+xz, (b) x+yz = (x+y)(x+z) Post. 5: (a) x+x'=1, (b) xx'=0
2. Identity? 3. Commutativity?
4. Distributivity?
Yes, use truth table (p. 39 in Mano) for over + and + over x 0 0 1 1 y 0 1 0 1 x+y 0 1 1 1
5. Complement?
Yes, x+x' = 1: 0+0'=0+1=1 and 1+1'=1+0=1 Yes, xx' = 0: 00'=01=0 and 11'=10=0 Yes, 01
x y xy 0 0 0 1 1 0 1 1 0 0 0 1
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6. Two distinct values?
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Boolean Algebra
Two-value Boolean algebra matches with Huntington postulates
We can use binary logic and consider it an algebra
Next:
Theorems that can be derived from postulates Canonical forms (uniqueness of representation) Duality of Boolean algebra Boolean functions
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Boolean Functions
Elements and operators can express a function
Value of function determined by value of variables Complete function is evaluated for all possible values
Function can be represented by truth table
E.g., F = x + y'z
x 0 0 0 0 1 1 1 1 y 0 0 1 1 0 0 1 1 z 0 1 0 1 0 1 0 1 y' 1 1 0 0 1 1 0 0 y'z x+y'z 0 1 0 0 0 1 0 0 0 1 0 0 1 1 1 1
Evaluate in steps
E.g., first y' , then y'z, etc.
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Logic Circuit Diagram of Function
Logic function can be expressed as circuit diagram
Direct translation from algebraic expression
Example: F = x + y'z
Problem: possibly multiple equivalent algebraic expressions
Difficult to compare Boolean functions
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Reading Assignment
Read Mano 2-4 2-6
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Recap from Last Lecture
Boolean algebra
Algebra axioms, postulates Huntington's postulates
Boolean functions
Algebraic expression
Today's lecture
Simple design example Boolean theorems Comparison of Boolean functions Canonical and standard forms
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Design Problem
Design the control logic for a fan in a greenhouse.
The logic must sense three environmental conditions:
1. It is raining outside (variable x) 2. It is humid inside (variable y) 3. It is hot inside (variable z)
F
The fan should turn on when
It is humid AND it is not raining AND it is hot
yx'z
OR It is not humid AND {(it is not raining AND it is hot) OR it is raining}
y' (x'z + x)
Create a Boolean function that controls the fan.
F = yx'z + y'(x'z+x) Other possibilities: F = x'y'z+x'yz+xy' or F = x'z+xy' Question: How to evaluate these solutions (equivalent functions)?
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Boolean Function Representations
Greenhouse example function can be expressed as: Analytically, as sum of minterms:
yx'z + y'(x'z+x) = yx'z+y'x'z+y'x
As a truth table
x y z 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
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= x'yz+x'y'z+xy'(z+z') = x'yz+x'y'z+xy'z+xy'z'
F 0 1 0 1 1 1 0 0 How to derive different solutions ? How to evaluate these solutions ?
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As a logic network
Boolean Theorems
Huntington's postulates define some rules
Post. Post. Post. Post. closure (a) x+0=x, (b) x1=x (a) x+y=y+x, (b) xy=yx (a) x(y+z) = xy+xz, (b) x+yz = (x+y)(x+z) Post. 5: (a) x+x'=1, (b) xx'=0 1: 2: 3: 4:
Need more rules to modify algebraic expressions What is a theorem?
Theorems that are derived from postulates A formula or statement that is derived from postulates (or other proven theorems)
Basic theorems of Boolean algebra
Theorem 1 (a): x + x = x (b): x x = x Looks straightforward, but needs to be proven !
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Proof of x+x=x
We can only use Huntington postulates:
Huntington postulates: Post. 2: (a) x+0=x, (b) x1=x Post. 3: (a) x+y=y+x, (b) xy=yx Post. 4: (a) x(y+z) = xy+xz, (b) x+yz = (x+y)(x+z) Post. 5: (a) x+x'=1, (b) xx'=0
Show that x+x=x.
x+x = (x+x)1 = (x+x)(x+x') = x+xx' = x+0 =x Q.E.D.
by 2(b) by 5(a) by 4(b) by 5(b) by 2(a)
We can now use Theorem 1(a) in future proofs
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Proof of xx=x
Similar to previous proof
Huntington postulates: Post. 2: (a) x+0=x, ...