Transcript Lecture 0: Introduction and Measure Theory

Lecture 0: Introduction and
Measure Theory
CS 7040
Trustworthy System Design,
Implementation, and Analysis
Spring 2015, Dr. Rozier
Introductions
Welcome to CS 7040!
Trustworthy System
Design,
Implementation,
and Analysis
Professor Eric Rozier
ROSE-E-A
Who am I?
• BS in Computer Science
from William and Mary
Who am I?
• BS in Computer Science
from William and Mary
• Studied models of
agricultural pests (flour
beetles).
Who am I?
• BS in Computer Science
from William and Mary
• Studied models of
agricultural pests (flour
beetles).
• And load balancing of
super computers.
Who am I?
• First job – NASA Langley
Research Center
Who am I?
• First job – NASA Langley
Research Center
• Researched problems in
aeroacoustics
Who am I?
• First job – NASA Langley
Research Center
• Researched problems in
aeroacoustics
– Primarily on the XV-15
Who am I?
• First job – NASA Langley
Research Center
• Researched problems in
aeroacoustics
– Primarily on the XV-15
– Precursor to the better
known V-22
Who am I?
• PhD in CS/ECE from the
University of Illinois
Who am I?
• PhD in CS/ECE from the
University of Illinois
• Studied non-linear
dynamics of
transactivation
networks in
economically important
species…
Who am I?
• PhD in CS/ECE from the
University of Illinois
• Studied non-linear
dynamics of
transactivation
networks in
economically important
species… corn…
Who am I?
• PhD in CS/ECE from the
University of Illinois
• Worked with the NCSA
on problems in super
computing, reliability,
and big data.
Who am I?
• PhD in CS/ECE from the
University of Illinois
• Worked with the NCSA
on problems in super
computing, reliability,
and big data.
• Research led to
patented advances with
IBM
Who am I?
• Served as a visiting
scientist and IBM Fellow
at the IBM Almaden
Research Center in San
Jose, CA
• Helped advance state of
the art in faulttolerance, and our
understanding of why
systems fail
Who am I?
• Postdoctoral work at
the Information Trust
Institute
– Worked on Blue Waters
Super Computer, first
sustained Petaflop
machine
– Designed new faulttolerant methods for
data protection on largescale systems
Who am I?
• Joined the
University of Miami as
an Assistant Professor
of ECE in 2012
– Founded the Fortinet
Cybersecurity Laboratory
Who am I?
• Served as a Summer
Faculty Fellow at the
University of Chicago
during 2014.
Who am I?
• Served as a Summer
Faculty Fellow at the
University of Chicago
during 2014.
– Data Science for Social
Good Summer
Fellowship
Who am I?
• Served as a Summer
Faculty Fellow at the
University of Chicago
during 2014.
– Data Science for Social
Good Summer
Fellowship
– Fought corruption with
the World Bank
Who am I?
• Served as a Summer
Faculty Fellow at the
University of Chicago
during 2014.
– Data Science for Social
Good Summer
Fellowship
– Fought corruption with
the World Bank
– and Lead Poisoning with
CDPH
Who am I?
• 2014 – Joined EECS at
UC
Who am I?
• Research in:
– Big Data
– Data Science and Engineering
– Trustworthy Computing
– Cybersecurity and Data Privacy
– Cloud Computing
How to get in touch with me?
• Office
– Engineering Research Center
– Fifth Floor, Room 501E
• Contact Information
– Email: [email protected]
– Phone: ????
• Currently looking for motivated students
– Research projects and papers
Office Hours
• Office
– ERC
– Fifth Floor, Room 501E
Day
Hours
Tuesday
3:30p – 5:00p
Thursday
3:30p – 5:00p
Or by appointment
The syllabus…
Grades
Grade Component
Percentage
Homeworks and MPs
15%
Project I
20%
Project II
20%
Midterm
20%
Final Examination
25%
Grades
• Guaranteed Grades
Projects
• The course will have two projects made to
engage you in Trustworthy System Design and
Evaluation.
• Project I will be common to the class. You will
work in groups of 2.
• Project II will be a semester project you
propose and conduct on a system or concept
of your choice.
Mobius
Examinations
• Examinations
– Midterm – March 3rd in class
– Final Exam – Take home examination
Course Plan
Week
Topic
1
Introduction, Measure Theory, Trustworthy Computing
2
Combinatorial Modeling
3
State-based Methods
4
Stochastic Activity Networks
5
Simulation
6
Reward Variables, Rare Events
7
Performance Evaluation
8
MIDTERM I, Dependability
9
Fault Tolerance
-
Spring Break
10
Fault Tolerance
11
Security
12
Data Privacy
13
Verification and Validation
14
Course Synthesis
Project 1 Assigned
Project 1 Due, Project 2 Proposals
Due
Project 2 Interim Report Due
Project 2 Presentations
Course Website
http://dataengineering.org/erozier2/courses/cs7040.html
Active Learning
• After 2 weeks we tend to remember:
– Passive learning
•
•
•
•
10% of what we read
20% of what we hear
30% of what we see
50% of what we hear and see
– Active learning
• 70% of what we say
• 90% of what we say and do
Bloom’s Taxonomy
Evaluation
Synthesis
Analysis
Application
Comprehension
Knowledge
Training Good Engineers
• Understanding processors isn’t our only goal
– Critical Reading
– Critical Reasoning
• Ask questions!
• Think through problems!
• Challenge assumptions!
Measurements
Making Things More Secure
++
Making Things More Secure
Measurements
• Measurements have inherent assumptions
• Measurements are often stated very
informally
• If we want to build a trustworthy system we
need to improve on this.
– Formalize our measures!
Measurements
Measure theory is a bit like grammar, many
people communicate clearly without worrying
about all the details, but the details do exist and
for good reasons.
- Maya Gupta, University of Washington
The Problem of Measures
• Physical intuition of the measure of length,
given a body E, the measure of this body, m(E)
might be the sum of it’s components, or
points.
• Let’s take two bodies on the real number line
– Body A is the line A = [0, 1]
– Body B is the line B = [0, 2]
Which is “longer”?
The Problem of Measures
• Physical intuition of the measure of length,
given a body E, the measure of this body, m(E)
might be the sum of it’s components, or
points.
• Let’s take two bodies on the natural number
line
– Body A is the line A = [0, 1]
– Body B is the line B = [0, 2]
Which is “longer”?
Solving the Problem of Measures
• What does it mean for some body (or subset)
to be measurable?
• If a set E is measurable, how does one define its
measure?
• What properties or axioms does measure (or the
concept of measurability) obey?
Measure Theory
• Before we can measure anything we need
something to measure!
• Let’s define a measurable space
– A measurable space is a collection of events B,
and the set of all outcomes, Ω, also called the
sample space.
Events and Sample Spaces
• Each event, F, is a set containing zero or more
outcomes.
– Each outcome can be viewed as a realization of an
event. The real world can be viewed as a player in
a game that makes some move:
– All events in F that contain the selected outcome
are said to “have occurred”.
Events and Sample Space
• Take a deck of 52 cards
+ 2 jokers
• Draw a single card from
the deck.
• Sample space: 54
element set, each card
is a possible outcome.
• An event is any subset
of the sample space,
including a singleton
set, or the empty set.
Events and Sample Space
• Potential events:
– “Red and black at the
same time without being
a joker” – (0 elements)
– “The 5 of hearts” – (1
element)
– “A king” – (4 elements)
– “A face card” – (12
elements)
– “A card” – (54 elements)
Forming an Algebra on B and Ω
• In order to define measures on B, we need to
make sure it has certain properties, those of a
σ-algebra.
• A σ-algebra is a special kind of collection of
subsets that is closed under countable-fold
set operations (complement, union of
countably many sets, and intersection of
countably many sets).
• “Vanilla” algebras are closed only under finite
set operations.
Countable Sets
• Countable sets are those with the same
cardinality of natural numbers.
• Quick refresher: Prove the cardinality of
integers and natural numbers are the same.
σ-algebra
• If we have a σ-algebra on our sample space Ω,
then:
Measures
• A measure µ takes a set A from a measureable
collection of sets B and returns the measure
of A, which is some positive real number.
Formally:
Example Measure
• Let’s define a measure of “Volume”.
• The triple
combines a
measureable space and a measure, the triple
is called a measure space. This space is
defined by two properties:
– Nonnegativity:
– Countable additivity:
are disjoint
sets for i = 1, 2, …, then the measure of the union
of
is equal to the sum of the measures of
Example Measure
• Does the ordinary concept of volume satisfy
these two properties?
– Nonnegativity:
– Countable additivity:
are disjoint
sets for i = 1, 2, …, then the measure of the union
of
is equal to the sum of the measures of
Two Special Kinds of Measures
• Signed measure – can be negative
• Probability measure – defined over a
probability space with a probability measure.
– A probability measure, P, has the normal
properties of a measure, but it is also normalized
such that:
Sets of Measure Zero
• A set of measure zero is some set
• For a probability measure, any set of measure
zero can never occur as it has probability of
zero.
– It can thus be ignored when stating things about
the collection of sets B.
Borel Sets
• A common σ-algebra is the Borel σ-algebra. A
Borel set is an element of a Borel σ-algebra.
– Almost any set you can describe on the real line is
a Borel set, for example, the unit line segment
[0,1]. Irrational numbers, etc.
– The Borel σ-algebra on the real line is a collection
of sets that is the smallest σ-algebra that includes
the open subsets of the real line.
Borel Sets
• For some space X, the collection of all Borel
sets on X forms a σ-algebra known as the
Borel algebra (or Borel σ-algebra) on X.
• Important!
• Why? Any measure defined on the open set of
a space, or closed sets of a space, must also
be defined on all Borel sets of that space.
Borel Sets
• Borel sets are powerful because if you know
what a probability measure does on every
interval, then you know what it does on all the
Borel sets.
• Allows us to define equivalence of measures.
Borel Sets
• Let’s say we have two measures:
• To show they are equivalent we just need to show
that:
– They are equivalent on all intervals
• By definition they are then equivalent for all Borel
sets, and hence over the measurable space.
• Example: Given probability distributions A, and B,
with equivalent cumulative distribution functions,
then the probability distributions must also be equal.
Measure Theory and CS 7040
• We will be working with a LOT of probability
distributions!
• We will be measuring things like:
– Performance
– Availability
– Reliability
– Security
– Privacy
Measure Theory: Further Reading
• M. Capinski and E. Kopp, “Measure, Integral,
and Probability”, Springer Undergraduate
Mathematics Series, 2004
• S. I. Resnick, “A probability path”, Birkhauser,
1999.
• A. Gut, “Probability: A Graduate Course”,
Springer, 2005.
• R. M. Gray, “Entropy and Information Theory”,
Springer Verlag (available free online), 1990.
For next time
• Homework 0!
• Due next Tuesday