How Important is Theory for Practical Problems? A Partial Explanation of Hartmanis' Observation

Hartmanis' observation. How important is theory for practical problems? In his recent talk 4], J. Hartmanis analyzed several cases when a long-standing open theoretical problem was solved, and came up with a rather unexpected conclusion: that if some theoretical result is very diicult to prove or disprove, then usually, this result has little or no practical usefulness. This observation seemed troubling to us, because it runs contrary to the usual theoreticians' belief that complex theoretical results are practically useful. Because of this, we decided to formalize this observation and to investigate whether Hartmanis' informal statement is true or not in the resulting formalization. We started with a very simple formalization, and, somewhat to our surprise, discovered that in this simple formalization, Hartmanis' intuition is right and our original intuition was wrong: complex theoretical results are indeed, in some precise sense, of little practical use. Thus, this gives a (partial) explanation of Hartmanis' observation. In this paper, we describe this result. Our result is simple, but we believe the readers will be interested because it does provide a justiication of an otherwise unexplained observation. Of course, the reader should take into consideration that this result is based on a rather simple possible formalization, and more adequate and more sophisticated formalizations are necessary. Towards formalization: general idea. In order to formalize Hartmanis's observation, we need to formalize two intuitive notions: what it means for a statement S to be \practically useful", and what it means for a statement to be \diicult to prove". Both terms are diicult to formalize in a way that would best t our intuitive understanding of the corresponding notions. In view of this diiculty, we will only formalize the features of these notions which are necessary to prove our formal version of Hartmanis' statement. Namely, we will deene the following three notions: The rst is the notion of a potential application; this notion will be deened in such a way that every application (in the intuitive sense of this word) is also (hopefully) an application in the sense of this deenition. As we will see from our deenition, the inverse is not necessarily true: a potential application in the sense of our deenition is not necessarily an application in the intuitive sense of this word. The second notion will be related to the usefulness of an application.