Abstract
This paper considers the problem of computing the real convex hull of a finite set of n-dimensional integer vectors. The starting point is a finite-automaton representation of the initial set of vectors. The proposed method consists in computing a sequence of automata representing approximations of the convex hull and using extrapolation techniques to compute the limit of this sequence. The convex hull can then be directly computed from this limit in the form of an automaton-based representation of the corresponding set of real vectors. The technique is quite general and has been implemented. Also, our result fits in a wider scheme whose objective is to improve the techniques for converting automata-based representation of constraints to formulas.
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References
Bartzis, C., Bultan, T.: Construction of efficient bdds for bounded arithmetic constraints. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 394–408. Springer, Heidelberg (2003)
Boigelot, B.: Symbolic Methods for Exploring Infinite State Spaces. Collection des publications de la Faculté des Sciences Appliquées, Liège (1999)
Boigelot, B., Herbreteau, F.: The power of hybrid acceleration. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 438–451. Springer, Heidelberg (2006)
Boigelot, B., Jodogne, S., Wolper, P.: An effective decision procedure for linear arithmetic over the integers and reals. ACM Transactions on Computational Logic 6(3), 614–633 (2005)
Boigelot, B., Legay, A., Wolper, P.: Omega-regular model checking. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 561–575. Springer, Heidelberg (2004)
Boigelot, B., Rassart, S., Wolper, P.: On the expressiveness of real and integer arithmetic automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 152–163. Springer, Heidelberg (1998)
Boigelot, B., Wolper, P.: Representing arithmetic constraints with finite automata: An overview. In: Stuckey, P.J. (ed.) ICLP 2002. LNCS, vol. 2401, pp. 1–19. Springer, Heidelberg (2002)
Bouajjani, A., Jonsson, B., Nilsson, M., Touili, T.: Regular model checking. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 403–418. Springer, Heidelberg (2000)
Boudet, A., Comon, H.: Diophantine equations, presburger arithmetic and finite automata. In: Proc of ICALP. LNCS, vol. 1159, pp. 30–43. Springer, Heidelberg (1996)
Cantin, F.: Techniques d’extrapolation d’automates: Application au calcul de la fermeture convexe. Master’s thesis, University of Liège, Belgium (2007)
Cantin, F., Legay, A., Wolper, P.: Computing convex hulls by automata iteration. Technical report, University of Liège (2008), http://www.montefiore.ulg.ac.be/~legay/papers/ciaa08rep.pdf
Chazelle, B.: An optimal convex hull algorithm in any fixed dimension. Discrete & Computational Geometry 10, 377–409 (1993)
Eisinger, J., Klaedtke, F.: Don’t care words with an application to the automata-based approach for real addition. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 67–80. Springer, Heidelberg (2006)
Finkel, A., Leroux, J.: The convex hull of a regular set of integer vectors is polyhedral and effectively computable. IPL (96-1), 30–35 (2005)
The Liège Automata-based Symbolic Handler (LASH), http://www.montefiore.ulg.ac.be/~boigelot/research/lash/
Latour, L.: From automata to formulas: Convex integer polyhedra. In: Proc. of LICS, pp. 120–129. IEEE Computer Society, Los Alamitos (2004)
Leroux, J.: Algorithmique de la vérification des systèmes à compteurs. Approximation et accélération. Implémentation de l’outil FAST. PhD thesis, Cachan (2004)
Leroux, J.: A polynomial time presburger criterion and synthesis for number decision diagrams. In: Proc of LICS, pp. 147–156. IEEE Computer Society, Los Alamitos (2005)
Leroux, J., Sutre, G.: Flat counter automata almost everywhere! In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 489–503. Springer, Heidelberg (2005)
Löding, C.: Efficient minimization of deterministic weak automata. IPL (79-3), 105–109 (2001)
Muller, D.E., Saoudi, A., Schupp, P.E.: Alternating automata, the weak monadic theory of the tree and its complexity. In: Proc of ICALP, Rennes, pp. 275–283. Springer, Heidelberg (1986)
Safra, S.: Exponential determinization for ω-automata with strong-fairness acceptance condition. In: Proc. of POPL, Victoria (May 1992)
The T(O)RMC toolset, http://www.montefiore.ulg.ac.be/~legay/TORMC/index-tormc.html
Vardi, M.: The büchi complementation saga. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 12–22. Springer, Heidelberg (2007)
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Cantin, F., Legay, A., Wolper, P. (2008). Computing Convex Hulls by Automata Iteration. In: Ibarra, O.H., Ravikumar, B. (eds) Implementation and Applications of Automata. CIAA 2008. Lecture Notes in Computer Science, vol 5148. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70844-5_12
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DOI: https://doi.org/10.1007/978-3-540-70844-5_12
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