Can a computer proof be elegant?

Abstract

In computer science, proofs about computer algorithms are par for the course. Proofs by computer algorithms, on the other hand, are not so readily accepted. We present one viewpoint on computer assisted proofs, or what we call proofs via the computational method [13]. While one might expect pure mathematicians to be a bit leery of the computational method, one would hope that computer scientists would be more receptive. In the author's experience, this has not been the case. The comment one often hears is "Can't you find a better proof"? One of the arguments given against the computational method is that a computer assisted proof cannot possibly be elegant. We question this argument here.It is important to define precisely what the computational method is. In the computational method we seek to prove results using a computer program. More specifically, given a theorem we want to prove, we exhibit a computable Boolean function f and prove that the truth of f implies the theorem. Next, we give an algorithm for computing f. Finally, we execute this algorithm on a computer. If all goes well, the theorem is proven. The first (as far as we know) and most famous use of the computational method was in the proof of the four-color theorem by Appel and Haken [2]. This should be contrasted with the use of a computer to find a proof that is easily checked by hand. For instance, a counter-example found via a computer search, as in [9, 10], is not an example of the computational method.As the proof of the four-color theorem was the first use of the computational method, it generated a large amount of controversy. Foremost among the decryers of the Appel and Haken proof was Halmos [7, page 578], who states:By an explosion I mean a load noise, an unexpected and exciting announcement, but not necessarily a good thing. Some explosions open new territories and promise great future developments; others close a subject and seem to lead nowhere. …the four-color theorem [is] of the second [kind].Halmos goes further, attacking the computational method in general, stating "Down with oracles, I say---they are of no use in mathematics". While we cannot speak to the four-color theorem proof---the author has not seen it---we disagree with Halmos' assertion that computer proofs have no use in mathematics.In [13], we tried to give some motivation for using the computational method and address some of the complaints against it. We tried to argue that, in certain cases, a computer proof may be preferable to a traditional one. However, we feel that one point overlooked in [13], namely the question of elegance of computer proofs, should be addressed. Other authors have also addressed the question of computer aided proofs. For some alternative viewpoints, see [3, 4, 5, 7, 11, 12, 14].