Advanced Computer Science Courses

Well, can do what Knuth did in TACP. There he did a lot with combinatorial formulas. To make much more progress, will have to get serious about math. E.g., the leading question in algorithms is just P versus NP, and that is now darned serious math. Don't attack that or even parts of it without a good background in math.

Other new and challenging questions in algorithms promise to need math for progress.

As I look at algorithms in 'advanced computer science', commonly they want to treat optimization. Tilt! Optimization is a huge field from applied math, operations research, and electrical engineering. There is deterministic and stochastic optimal control, Kalman filtering, integer linear programming, and much more. It's darned good applied math, and the math background I outlined is needed.

"numerical analysis"

That's a field of applied math. E.g., quickly get into advanced parts of matrix theory. E.g., consider R. Horn's books. E.g., quasi Newton quickly becomes an exercise in matrix norm theory. Long one of the more important tools in numerical analysis has been functional analysis. Likely the leading reason to pursue numerical analysis is just to get solutions to partial differential equations, and don't go there without a good background in math and likely the corresponding mathematical physics.

"compilers, computational reasoning, automated proofs"

For the last two, they are just fields of applied math.

"Data mining and AI"

The way computer science pursues these two, they are nonsense fields. The first should be just mathematical statistics, and for that the background I gave in probability is crucial. E.g., will want to know sufficient statistics, and that is based on the Radon-Nikodym theorem, and that is graduate pure math.

For AI, if someone can write a computer program that really has 'intelligence', fine. If all they use are intuitive ideas, good for them. But so far, the field of AI is very far from this goal in spite of decades of DARPA funding.

For now, if want a system that solves a problem well enough to look 'intelligent', then just engineer the system with the usual role for applied math. I gave a paper at an AAAI IAAI conference with the "25 best applications of AI in the world", and the best applications, really, were just good engineering.

"Systems, especially performance testing. Needs some statistics, but it's not hardcore math. No more than psychology or economics."

For some simple applications, yes, can just borrow statistics from the social sciences.

But for progress, it's back to "hardcore math": E.g., I published a paper on 'performance monitoring', that is, zero day anomaly detection in server farms and networks, and the math was based on some of the more advanced parts of the texts I listed. Basically I found a collection of multivariate, distribution free hypothesis tests. The social sciences have been using univariate distribution free hypothesis tests for over 60 years; my work was apparently the first good progress to multivariate distribution free tests. Multivariate tests are just crucial for analyzing performance data. I used a finite group (from abstract algebra) of measure preserving transformations something like in ergodic theory. My work was similar to some of what Diaconis at Stanford has done with exchangeability in distribution free statistics. This material needs all the background I outlined and more.

"Systems architecture"

For the future, before we build a large system, we will want the 'architecture' to have some known properties. We do this for bridges, buildings, dams, ships, airplanes, etc. So, we will want to do it for systems. We will want to know about reliability and security, at least. And we may want to 'optimize', that is, get what we need at minimum price.

E.g., consider part of the core of the Internet: Suppose we are given nodes and flows at the nodes. Then our mission, should we decide to accept it, is to connect the nodes at minimum cost to provide desired capacity, performance, and reliability.

Dean of Engineering at MIT T. Magnanti gave a Goldman lecture at Johns Hopkins on this problem; first-cut it's a super tough problem in integer linear programming. Actually, it was such problems in network design, and integer linear programming, that got Bell Labs going on the work that resulted in

Michael R. Garey and David S. Johnson, 'Computers and Intractability: A Guide to the Theory of NP-Completeness', ISBN 0-7167-1045-5, W. H. Freeman, San Francisco, 1979.

which is one of the key books early in the problem P versus NP.

It is common now to throw together a system, have various intuitive approaches to parallelism and redundancy, and then get a problem and see the whole server farm go south. Recall that this is just what happened at Amazon a few weeks ago.

Maybe some people, say, in national security, finance, or air traffic control would like such things not to happen?

A common situation is parallelism that is not more reliable but worse: E.g., there was a parallel transaction processing system. The load balancing sent the next transaction to the least busy computer. Then one day, one of the computers got a little sick, started throwing all its work into the bit bucket, was not very busy, was getting nearly all the transactions, and, thus, killed the whole 'cluster'. Bummer.

Something similar happened with Google's e-mail servers.

Instead, we need some theory with some theorems and proofs that provide some guarantees that we are getting the performance and reliability we need. That work will be mathematical; without a good background in math, don't try. And, yes, likely the work will need probability such as I discussed.

"Best practices. Software engineering. Not really mathy. Not really science either."

Not really computer science research either.

"OOP. Theology?"

There have been some connections with category theory, but I'm reluctant to take those seriously.

Mostly OOP in practice is simple and works well for some simple things. Otherwise OOP is not well thought out. And long OOP was a 'theology'.

My old view of programming languages is that they should be designed so that they 'admit' some source code transformations that have some useful properties. E.g., suppose we have such a language with such properties and have two pieces of code. Can we use the transformations to check if the two pieces of code are equivalent? So, right, we will have 'equivalence classes' of code. Some of these will do better on processor time, main memory usage, exploitation of multiple cores, etc. than others. So, with some code and these transformations, we have an optimization problem: Transform the code to equivalent code with the 'resource' properties we need. Sounds like math to me.

"But there are things to study that aren't just math, even in the field of computer science."

The problems are from computer science. But the good solutions are necessarily applied math because our civilization knows no other way to proceed. This situation is much that same as in all fields that have been 'mathematized', especially theoretical physics.

This statement is an unfortunate juxtaposition: Glad probability is one of your "favorite topics", but you make serious mistakes right at the beginning and show that you don't know anything significant at all about the subject, not at any level from freshman to the texts I listed.

That you "agree" with that "part of the course's treatment" is absurd, especially after what I wrote. That "treatment" is a total upchuck. You, the professor, and CMU should all be humiliated and ashamed. It would be tough to get worse even from a storefront for-profit 'college'. For probability at CMU, I listed an excellent source, Steve Shreve.

"If we really wanted to do probability the right way, we'd tell them that a random variable is a structure-preserving map between measure spaces."

There you go again. You are digging your hole longer, wider, and deeper. You are spouting nonsense.

A "random variable" is definitely not a "structure-preserving map between measure spaces". Not a chance. You won't find "structure-preserving" mentioned anywhere in the definition of a random variable in any of the four texts I listed. Your "structure-preserving" is just gibberish you got from some source you should not touch and then burn outdoors and then flush.

The most advanced definition, as in the four texts I listed, is that a 'random variable' is a measurable function from a probability space into a measurable space. What I described for the set of all events is called a 'sigma algebra'. Then a 'measurable space' is a non-empty set and a sigma algebra of subsets of it. A function is 'measurable' if for each set in the sigma algebra in the range its inverse image under the function is a set in the sigma algebra of the domain. A 'probability space' is a measurable space and a probability measure on that space, and a 'probability measure' is a non-negative measure with total mass 1. For a 'measure', see any of, say,

Paul R. Halmos, 'Measure Theory', D. Van Nostrand Company, Inc., Princeton, NJ.

Walter Rudin, 'Real and Complex Analysis', ISBN 07-054232-5, McGraw-Hill, New York.

H. L. Royden, 'Real Analysis: Second Edition', Macmillan, New York.

I omitted the requirement for a random variable being measurable because for the more important measurable spaces, say, the real numbers with the Borel sets or the Lebesgue measurable sets, finding a function that is not measurable is super tough: The usual construction uses the axiom of choice.

"You can't do that with freshmen."

Well, no one should do "that" with anyone, but with your "agree" with that "part of the course's treatment" here shows that you have been doing even worse, apparently with freshman.

You are flatly refusing to take me at all seriously and refuse to 'get it': Your definition of a random variable is sewage, and you are even defending it.

For what to do with freshman, I gave more than one appropriate, accurate enough, and easy to take, definition of a random variable. There are also other sources. You very much need to take some such source seriously.

You just won't stop digging your hole longer, wider, and deeper: There you go again with your:

"Indeed, we deliberately avoided any mention of continuous probability spaces."

You are spouting more gibberish from some source that should not be touched and then burned and flushed. Your comments continue to fill much needed gaps in the teaching of probability.

There is no such thing as a "continuous probability" space.

Continuity is based on a topology, and there is not necessarily, and usually never is, any topology on the probability space in question. It is true that the Borel sets are the smallest sigma algebra that contain a given topology, usually the 'usual topology' of the real line or Euclidean n-space.

Where continuity enters is in a continuous density or an absolutely continuous cumulative distribution.

In practice what this means is that a 'density' is a continuous function f on, usually, the reals where f is non-negative and its integral is 1. Then the corresponding cumulative distribution F is the 'indefinite' integral of f, in TeX:

     F(x) = \int_{-\infty}^x f(x) \; dx

It's crucial to discuss continuous densities in order to discuss random variables with Gaussian, uniform, exponential, chi-squared, etc. distributions. Gaussian is crucial for the central limit theorem. Uniform is crucial for discussing the usual random number generators, e.g., as in the recipes in Knuth's TACP that pass the Fourier test. The exponential distribution is crucial for discussing arrival processes, e.g., when the next Web page request will arrive at a Web server. Chi-square is one of the first distributions encountered in elementary statistical tests, e.g., for independence of two random variables. These distributions are commonly taught in first courses in statistics to students in the social sciences. Looks like the CMU computer science students are falling behind the sociology students!

More details on distributions involve the Lebesgue decomposition and absolute continuity, and you will find solid treatments in each of the four texts I listed along with both Rudin and Royden.

Loève also outlines the bizarre 'singular continuous' case, as I recall, based on the Cantor function.

"Your conclusion may be generally true, but it's definitely off the mark here."

I'm not "off the mark"; I'm dead on target. I gave some good, elementary definitions. Elementary should not mean sewage, but your elementary definition of a random variable is sewage.

"251 is far harder than any undergraduate math course at CMU."

Sounds like the CMU math department doesn't teach, say,

Walter Rudin, 'Principles of Mathematical Analysis, Third Edition', McGraw-Hill, New York.

or the equivalent. Tough to believe.

The material you are teaching is easy, plenty easy enough for a moderately easy course for freshman. Any difficulty is from the need for students to make sense out of sewage content such as your definition of random variable.

Again, for probability done seriously, see the texts I listed; for good elementary texts, there are many, but I have none at hand; often there are good, elementary treatments of random variables in the better texts on statistics for the social sciences; consider also texts on signals in EE; for probability at CMU, see Shreve.

For your course 251, do everyone a big favor and quit teaching it, burn it, and flush the ashes. Then start with some people who actually know the relevant math and develop a good course. And, really, from the list of topics, the course belongs in a math department. And, the course need not be difficult.

Bluntly, computer science very much needs to know math, especially probability, but generally gets a grade of D or less and on probability, an F.

I listed some world class experts in probability (I learned from one of them), but it is true that probability is not popular in US pure math departments, and a significant fraction of math professors will never have seen the more advanced definition of a random variable I have given here; they may not know the strong law of large numbers, conditional expectation, the definition of a Markov process or a martingale, etc.

You are badly wrong. Many people don't know anything about probability; that you don't is not so bad. But that you spout total nonsense about probability is bad; that you defend this nonsense is much worse. Keep fighting me on this and you will dig a seriously big hole for yourself and CS at CMU.

You want me to write Jerry and advise him that 251 needs to be cleaned up?

I had to register an account just to say one thing ... BULLSHIT.

Math Studies was an order of magnitude harder than 251. Even factoring in the two homeworks and lectures of math studies, it's still a night and day difference. I took both last semester, and halfassed 251 and did well. There's no possible way to do that with math studies.

Just read your profile and realized who you were ... but I still hold to my opinion.