Hirokazu Anai
ABSTRACT
In this tutorial, we will give an overview of typical algorithms of quantifier elimination over the reals and illustrate their actual applications in industry. Some recent research results on computational efficiency improvement of quantifier elimination algorithms, in particular for solving practical industrial problems, will be also mentioned. Moreover, we will briefly explain valuable techniques and tips to effectively utilize quantifier elimination in practice.
- H. Anai and S. Hara. A parameter space approach to fixed-order robust controller synthesis by quantifier elimination. International Journal of Control, 79(11):1321--1330, 2006.Google Scholar
Cross Ref
- H. Anai and S. Hara. Fixed-structure robust controller synthesis based on sign definite condition by a special quantifier elimination. In Proceedings of American Control Conference, 2000, vol.2, 1312--1316, 2000.Google ScholarCross Ref
- H. Anai and K. Yokoyama. Algorithms of Quantifier Elimination and their Applications: Optimization by Symbolic and Algebraic Methods (in Japanese). University of Tokyo Press, 2011.Google Scholar
- C. W. Brown. QEPCAD B: A program for computing with semi-algebraic sets using CADs. SIGSAM BULLETIN, 37:97--108, 2003. Google ScholarDigital Library
- B. F. Caviness and J. R. Johnson, editors. Quantifier elimination and cylindrical algebraic decomposition. Texts and Monographs in Symbolic Computation. Springer, 1998.Google ScholarCross Ref
- G. E. Collins. Quantifier elimination and cylindrical algebraic decomposition, pages 8--23. In Caviness and Johnson {5}, 1998.Google Scholar
- A. Dolzmann and T. Sturm. REDLOG: Computer algebra meets computer logic. SIGSAM Bulletin 31(2):2--9. Google ScholarDigital Library
- A. Dolzmann, T. Sturm, and V. Weispfenning. Real quantifier elimination in practice. In Algorithmic Algebra and Number Theory, pages 221--247. Springer, 1998.Google Scholar
- L. González-Vega. Applying quantifier elimination to the Birkhoff interpolation problem. Journal of Symbolic Computation, 22:83--103, July 1996. Google ScholarDigital Library
- L. González-Vega. A combinatorial algorithm solving some quantifier elimination problems, pages 365--375. In Caviness and Johnson {5}, 1998.Google Scholar
- L. González-Vega, T. Recio, H. Lombardi, and M.-F. Roy. Sturm-Habicht sequences determinants and real roots of univariate polynomials, pages 300--316. In Caviness, Bob F. and Johnson, Jeremy R {5}, 1998.Google Scholar
- N. Hyodo, M. Hong, H. Yanami, S. Hara, and H. Anai. Solving and visualizing nonlinear parametric constraints in control based on quantifier elimination. Applicable Algebra in Engineering, Communication and Computing, 18(6):497--512, 2007. Google ScholarDigital Library
- H. Iwane, H. Higuchi, and H. Anai. An effective implementation of a special quantifier elimination for a sign definite condition by logical formula simplification. In Proceedings of CASC 2013: 194--208, 2013.Google ScholarDigital Library
- H. Iwane, H. Yanami, and H. Anai. A symbolic-numeric approach to multi-objective optimization in manufacturing design. Mathematics in Computer Science, 5(3):315--334, 2011.Google ScholarCross Ref
- H. Iwane, H. Yanami, H. Anai, and K. Yokoyama. An effective implementation of symbolic-numeric cylindrical algebraic decomposition for quantifier elimination. Theor. Comput. Sci. 479: 43--69, 2013. Google ScholarDigital Library
- R. Loos and V. Weispfenning. Applying linear quantifier elimination. The Computer Journal, 36(5):450--462, 1993.Google ScholarCross Ref
- Y. Matsui, H. Iwane, and H. Anai. Two controller design procedures using SDP and QE for a Power Supply Unit "Development of Computer Algebra Research and Collaboration with Industry". COE Lecture Note Vol. 49 (ISSN 1881-4042), Kyushu University, 43--52, 2013.Google Scholar
- T. Matsuzaki, H. Iwane, H. Anai and N. Arai. The Most Uncreative Examinee: A First Step toward Wide Coverage Natural Language Math Problem Solving. In Proceedings of 28th Conference on Artificial Intellegence (AAAI 2014), to appear, 2014.Google Scholar
- T. Sturm. New domains for applied quantifier elimination. In V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov, editors, Computer Algebra in Scientific Computing, volume 4194 of Lecture Notes in Computer Science, pages 295--301. Springer, 2006. Google ScholarDigital Library
- T. Sturm and V. Weispfenning. Rounding and blending of solids by a real elimination method. In Proceedings of the 15th IMACS World Congress on Scientific Computation, Modelling, and Applied Mathematics (IMACS 97), pages 727--732, 1997.Google Scholar
- T. Sugimachi, A. Iwasaki, M. Yokoo, and H. Anai. Polynomial representation of auction mechanisms and automated mechanism design by quantifier elimination (in Japanese). In Proceedings of the autumn meeting of the Operation Research Society of Japan, 2012(30), 198--199, 2012Google Scholar
- A. Tarski, A decision method for elementary algebra and geometry. Prepared for publication by J. C. C. McKinsey. Berkeley, 1951.Google Scholar
- H. Yanami and H. Anai. The Maple package SyNRAC and its application to robust control design. Future Generation Comp. Syst. 23(5): 721--726, 2007. Google ScholarDigital Library
- H. Yanami. Multi-objective design based on symbolic computation and its application to hard disk slider design. Journal of Math-for-Industry (JMI2009B-8) Kyushu University, pp.149--156, 2009.Google Scholar
- S. Yoshimura, H. Iki, Y. Uriu, H. Anai, and N. Hyodo. Generator Excitation Control using Parameter Space Design Method. In Proceedings of 43rd International Universities Power Engineering Conference (UPEC2008), pp.1--4, 2008.Google ScholarCross Ref
- V. Weispfenning. Simulation and optimization by quantifier elimination. Journal of Symbolic Computation, 24:189--208, August 1997. Google ScholarDigital Library
Index Terms
- Effective quantifier elimination for industrial applications
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