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-A Professor 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? 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 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 – Data Science for Social Good Summer Fellowship

Who am I? Served as a Summer Faculty Fellow at the University of Chicago during – 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 – 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 – – Phone: ???? Currently looking for motivated students – Research projects and papers

Office Hours Office – ERC – Fifth Floor, Room 501E DayHours Tuesday3:30p – 5:00p Thursday3:30p – 5:00p Or by appointment

The syllabus…

Grades Grade ComponentPercentage Homeworks and MPs15% Project I20% Project II20% Midterm20% Final Examination25%

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 – Midterm – March 3 rd in class – Final Exam – Take home examination

Course Plan WeekTopic 1Introduction, Measure Theory, Trustworthy Computing 2Combinatorial Modeling 3State-based Methods 4Stochastic Activity NetworksProject 1 Assigned 5Simulation 6Reward Variables, Rare Events 7Performance Evaluation 8MIDTERM I, Dependability 9Fault ToleranceProject 1 Due, Project 2 Proposals Due -Spring Break 10Fault Tolerance 11SecurityProject 2 Interim Report Due 12Data Privacy 13Verification and Validation 14Course SynthesisProject 2 Presentations

Course Website

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, A. Gut, “Probability: A Graduate Course”, Springer, R. M. Gray, “Entropy and Information Theory”, Springer Verlag (available free online), 1990.

For next time Homework 0! Due next Tuesday