Closed Form Expressions
This summer, I had a conversation with Baris in which I claimed ignorance about the meaning of a closed form solution. One may say an expression is in closed form if one can intuit its behavior simply by looking at it; however, this is both imprecise and subjective. Baris suggested an alternate definition that went along the following lines:
closed form solution. n. an equation that can be evaluated by a scientific calculator.
I was happier with this definition, but it was still a little unsatisfying mainly because the capabilities of scientific calculators have improved significantly over the years. What if a scientific calculator can solve differential equations or an LP? I can’t intuit these solutions. Even if we accept such a definition, we would likely reject a paper titled “Closed Form Solution for [insert problem here] via Improved Scientific Calculator” for publication… or would we?
While some may disagree, computers have proved useful in helping information theorists develop intuition. For instance, Permuter et al.’s “Capacity of the Trapdoor Channel via Feedback” proves the capacity after using a computer to conjecture the solution. At a seminar this summer, Stephen P. Boyd advocated for more information theorists to adopt this pragmatic approach to problem solving.
Recently, Michael suggested MATLAB to help me gain insights about a research problem. My own intuitions were a little jumbled, and a few plots in MATLAB seemed like the best path to clarity.
Not everyone is a fan of MATLAB. Prasad told me he prefers a combination of C and gnuplot. Others are purer still. Someone scoffed at the idea that as an information theorist, I needed to resort to a computer for help.
Published in 1945, Vannevar Bush’s “As We May Think” predicts a future device called the memex (memory extension), a device in many ways reminiscent of the modern computer that enables the easy storage and retrieval of one’s books, communications, etc. If I can use a memex to organize my records, then I have no problem using an intuitex to organize my thoughts.