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

ROSE-E-AProfessor Eric Rozier

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?

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 fault-tolerance, 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 fault-tolerant methods for data protection on large-scale 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: eric.rozier@uc.edu– 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 PlanWeek Topic

1 Introduction, Measure Theory, Trustworthy Computing

2 Combinatorial Modeling

3 State-based Methods

4 Stochastic Activity Networks Project 1 Assigned

5 Simulation

6 Reward Variables, Rare Events

7 Performance Evaluation

8 MIDTERM I, Dependability

9 Fault Tolerance Project 1 Due, Project 2 Proposals Due

- Spring Break

10 Fault Tolerance

11 Security Project 2 Interim Report Due

12 Data Privacy

13 Verification and Validation

14 Course Synthesis 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

EvaluationSynthesisAnalysis

ApplicationComprehension

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 aσ-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