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Towards an Efficient Decision Procedure for the Existential Theory of the Reals

Towards an Efficient Decision Procedure for the Existential Theory of the Reals


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About the Book

Many results in analysis can be stated in the limited language of "real-closed fields." In 1948, Tarski showed that one can decide in a finite, albeit very long, time whether any sentence in this language is true. In 1975, Collins developed a decision procedure which, unlike Tarski's, was feasible for solving many small problems, but, due to a doubly exponential growth rate, could not handle very large ones. By restricting to merely the existential fragment of this theory, which still includes several interesting questions, several researchers developed algorithms with only singly exponential growth, thus providing hope for reaching problems beyond the limits of Collins' algorithm. A 1991 paper by Hong, however, argued that the constants involved made them useless in practice, as the runtime would be many millions of years. His results suggested that for practical purposes this line of research is a dead end. But new approaches to the existential theory have been discovered since, two major ones being that of Canny and that of Basu, Pollack, and Roy. In this thesis, I show that while the latter suffers infeasibility similar to the earlier algorithms, the former can be made to perform well enough to solve small problems. I do this by actually building a full implementation that is able to solve the test case used by Hong, as well as several others, in on the order of one second. The implementation is based on Canny's method but uses novel data structures and more recent algorithms for the various subproblems that arise, with different versions and variations explained and compared. Finally, I consider likely future developments, and conclude that, despite the past negative indicators and their still limited reach, the prospect for existential-theory decision procedures is bright.


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Product Details
  • ISBN-13: 9781244002197
  • Publisher: Proquest, Umi Dissertation Publishing
  • Publisher Imprint: Proquest, Umi Dissertation Publishing
  • Height: 246 mm
  • Weight: 254 gr
  • ISBN-10: 1244002194
  • Publisher Date: 01 Sep 2011
  • Binding: Paperback
  • Spine Width: 7 mm
  • Width: 189 mm


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