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by Dennis J. Darland
April 14, 2009
Revised February 11, 2011
Last revised 11.02.2011 11.31 time
Copyright © 2009 Dennis J. Darland


With the passing of Dennis Darland, at age 92, on July 4, 2044, the person responsible for the public discovery of a algorithm to solve a NP-complete problem in polynomial time is gone. Although he had written a program to do this over a period of time from 1978-2012, he had been unable to provide a proof, and, himself only suspected such a result. However, in the 2020's it became evident that the problem had been solved, because of the performance of some proprietary computer programs, however the algorithm was kept secret by this company. In fact there had been a proof that this was impossible, accepted since 2016, but which, however relied on Goedel's proof. Once it was evident that something was wrong with this proof, the search though the literature produced Darland's sketch of a criticism of Goedel's proof. It took only a few weeks for logicians to work out the details, and see the error in Goedel's proof. Also, in going through Darland's web site, they found a program, with amazing efficiency and accuracy, to solve systems of ordinary differential equations. It took a team of over 50 mathematicians and computer scientists 18 months to verify that, as they initially had come to suspect that it was a polynomial time solution of a NP-complete problem. NP-complete algorithms being inter-translatable, soon many more hard problems could be solved by computers, especially ecological and biological problems.
Darland, however, never benefited financially from his program, because he had made it open source, and freely available. In fact, being in his 70's, he turned down lucrative consulting offers, preferring to spent his time reading philosophy. However, he never seemed, in spite of this, to have any financial difficulties - living comfortably on a low income.


NOTE:(5/3/2009)
I have realized my criticism of Goedel is wrong - I will continue to try to understand his proof.
I have (also in philosophy) made a criticism of Quine's views on the opacity of belief - which I am entirely convinced is correct. It seems no one understands it. BTW I was the one who nominated Quine for honorary membership in the Bertrand Russell Society(BRS) of which I was treasurer for 27 years - having met him (Quine) at the annual BRS meeting in Toronto in 1984.
As far as my method of solving systems of ordinary differential equations, the method was really worked out by Y. F. Chang, a professor of mine at the University of Nebraska, Lincoln in the 1970's. He devised such a program in FORTRAN, but which had many limitations, mainly due to being written in FORTRAN. I have no reason to believe to problem is NP-Complete, (until about 1994, I was altogether ignorant of that concept.), but have mostly just trying to come up with a better way to solve differential equations (a very practical problem). My program does not have the limitations that Chang's has - having no arbitrary limits at all (just the limit of machine memory and time), although it has some default values. Also, I am doubtful if it can be guaranteed to converge to a solution - so it may not really be an algorithm at all. Also, in order to understand the error in the solution of the differential equations, I wrote my own arbitrary precision floating point - with accumulated error estimation.
My work in philosophy can be seen at My Philosophy
My work in computer science can be seen at My Software Projects
A new (2/11/2011) criticism of Goedel is at here.
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