In previous posts I have talked about the new introductory CS curriculum under development at Carnegie Mellon. After a year or so of planning, we began to roll out the new curriculum in the Spring of 2011, and have by now completed the transition. As mentioned previously, the main purpose is to bring the introductory sequence up to date, with particular emphasis on introducing parallelism and verification. A secondary purpose was to restore the focus on computing fundamentals, and correct the drift towards complex application frameworks that offer the students little sense of what is really going on. (The poster child was a star student who admitted that, although she had built a web crawler the previous semester, she in fact has no idea how to build a web crawler.) A particular problem is that what should have been a grounding in the fundamentals of algorithms and data structures turned into an exercise in object-oriented programming, swamping the core content with piles of methodology of dubious value to beginning students. (There is a new, separate, upper-division course on oo methodology for students interested in this topic.) A third purpose was to level the playing field, so that students who had learned about programming on the street were equally as challenged, if not more so, than students without much or any such experience. One consequence would be to reduce the concomitant bias against women entering CS, many fewer of whom having prior computing experience than the men.
The solution was a complete do-over, jettisoning the traditional course completely, and starting from scratch. The most important decision was to emphasize functional programming right from the start, and to build on this foundation for teaching data structures and algorithms. Not only does FP provide a much more natural starting point for teaching programming, it is infinitely more amenable to rigorous verification, and provides a natural model for parallel computation. Every student comes to university knowing some algebra, and they are therefore familiar with the idea of computing by calculation (after all, the word algebra derives from the Arabic al jabr, meaning system of calculation). Functional programming is a generalization of algebra, with a richer variety of data structures and a richer set of primitives, so we can build on that foundation. It is critically important that variables in FP are, in fact, mathematical variables, and not some distorted facsimile thereof, so all of their mathematical intuitions are directly applicable. So we can immediately begin discussing verification as a natural part of programming, using principles such as mathematical induction and equational reasoning to guide their thinking. Moreover, there are natural concepts of sequential time complexity, given by the number of steps required to calculate an answer, and parallel time complexity, given by the data dependencies in a computation (often made manifest by the occurrences of variables). These central concepts are introduced in the first week, and amplified throughout the semester.
Two major homework exercises embody the high points of the first-semester course, one to do with co-development of code with proof, the other to do with parallelism. Co-development of program and proof is illustrated by an online regular expression matcher. The problem is a gem for several reasons. One is that it is essentially impossible for anyone to solve by debugging a blank screen. This sets us up nicely for explaining the importance of specification and proof as part of the development process. Another is that it is very easy, almost inevitable, for students to make mistakes that are not easily caught or diagnosed by testing. We require the students to carry out a proof of the correctness of the matcher, and in the process discover a point where the proof breaks down, which then leads to a proper solution. (See my JFP paper “Proof-Directed Debugging” for a detailed development of the main ideas.) The matcher also provides a very nice lesson in the use of higher-order functions to capture patterns of control, resulting in an extremely clean and simple matcher whose correctness proof is highly non-trivial.
The main parallelism example is the Barnes-Hut algorithm for solving the n-body problem in physics. Barnes-Hut is an example of a “tree-based” method, invented by Andrew Appel, for solving this well-known problem. At a high level the main idea is to decompose space into octants (or quadrants if you’re working in the plane), recursively solving the problem for each octant and then combining the solutions to make an overall solution. The key idea is to use an approximation for bodies that are far enough away—a distant constellation can be regarded as an atomic body for the purposes of calculating the effects of its stars on the sun, say. The problem is naturally parallelizable, because of the recursive decomposition. Moreover, it provides a very nice link to their high school physics. Since FP is just an extension of mathematics, the equations specifying the force law and Newton’s Law carry over directly to the code. This is an important sense in which FP builds on and amplifies their prior mathematical experience, and shows how one may connect computer science with other sciences in a rather direct manner.
The introductory FP course establishes the basis for the new data structures and algorithms course that most students take in either the third or fourth semester. This course represents a radical departure from tradition in several ways. First, it is a highly rigorous course in algorithms that rivals the upper-division course taught at most universities (including our own) for the depth and breadth of ideas it develops. Second, it takes the stance that all algorithms are parallel algorithms, with sequential being but a special case of parallel. Of course some algorithms have a better parallelizability ratio (a precise technical characterization of the ability to make use of parallelism), and this can be greatly affected by data structure selection, among other considerations. Third, the emphasis is on persistent abstract types, which are indispensable for parallel computing. No more linked lists, no more null pointer nonsense, no more mutation. Instead we consider abstract types of graphs, trees, heaps, and so forth, all with a persistent semantics (operations create “new” ones, rather than modify “old” ones). Fourth, we build on the reasoning methods introduced in the first semester course to prove the correctness and the efficiency of algorithms. Functional programming makes all of this possible. Programs are concise and amenable to proof, they are naturally parallel, and they enjoy powerful typing properties that enforce abstraction in a mathematically rigorous manner. Fifth, there is a strong emphasis on problems of practical interest. For example, homework 1 is the shotgun method for genome sequencing, a parallel algorithm of considerable practical importance and renown.
There is a third introductory course in imperative programming, taken in either the first or second semester (alternating with the functional programming course at the student’s discretion), that teaches the “old ways” of doing things using a “safe” subset of C. Personally, I think this style of programming is obsolete, but there are many good reasons to continue to teach it, the biggest probably being the legacy problem. The emphasis is on verification, using simple assertions that are checked at run-time and which may be verified statically in some situations. It is here that students learn how to do things “the hard way” using low-level representations. This course is very much in the style of the 1970’s era data structures course, the main difference being that the current incarnation of Pascal has curly braces, rather than begin-end.
For the sake of those readers who may not be up to date on such topics, it seems important to emphasize that functional programming subsumes imperative programming. Every functional language is capable of expressing the old-fashioned sequential pointer-hacking implementation of data structures. You can even reproduce Tony Hoare’s self-described “billion dollar mistake” of the cursed “null pointer” if you’d like! But the whole point is that it is rarely useful, and almost never elegant, to work that way. (Curiously, the “monad mania” in the Haskell community stresses an imperative, sequential style of programming that is incompatible with parallelism, but this is not forced on you as it is in the imperative world.) From this point of view there no loss, and considerable gain, in teaching functional programming from the start. It puts a proper emphasis on mathematically sane programming methods, but allows for mutation-based programming where appropriate (for example, in engendering “side effects” on users).
To learn more about these courses, please see the course web pages:
I encourage readers to review the syllabi and course materials. There is quite a large body of material in place that we plan to expand into textbook-level treatments in the near future. We also plan to write a journal article documenting our experiences with these courses.
I am very grateful to my colleagues Guy Blelloch, Dan Licata, and Frank Pfenning for their efforts in helping to develop the new introductory curriculum at Carnegie Mellon, and to the teaching assistants whose hard work and dedication put the ideas into practice.
Update: Unfortunately, the homework assignments for these courses are not publicly available, because we want to minimize the temptation for students to make use of old assignments and solutions in preparing their own work. I am working with my colleagues to find some way in which we can promote the ideas without sacrificing too much of the integrity of the course materials. I apologize for the inconvenience.