Abstract
Some classes of sets of vectors of natural numbers are introduced as generalizations of the semi-linear sets, among them the ‘simple semi-polynomial sets.’ Motivated by verification problems that involve arithmetical constraints, we show results on the intersection of such generalized sets with semi-linear sets, singling out cases where the non-emptiness of intersection is decidable. Starting from these initial results, we list some problems on solvability of arithmetical constraints beyond the semi-linear ones.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bruyére, V., Dall’Olio, E., Raskin, J.F.: Durations, parametric model-checking in timed automata with Presburger arithmetic. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 687–698. Springer, Heidelberg (2003)
Dal Zilio, S., Lugiez, D.: XML schema, tree logic and sheaves automata. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 246–263. Springer, Heidelberg (2003)
Dang, Z., Ibarra, O.H., Bultan, T., Kemmerer, R.A., Su, J.: Binary reachability analysis of discrete pushdown timed automata. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 69–84. Springer, Heidelberg (2000)
Ginsburg, S., Spanier, E.H.: Semigroups, Presburger formulas, and languages. Pacific J. Math. 16, 285–296 (1966)
Grunewald, F., Segal, D.: On the integer solutions of quadratic equations. J. Reine Angew. Math. 569, 13–45 (2004)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979)
Ibarra, O.H., Bultan, T., Su, J.: Reachability Analysis for Some Models of Infinite-State Transition Systems. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 183–198. Springer, Heidelberg (2000)
Karianto, W.: Parikh automata with pushdown stack. In: Diploma thesis. RWTH Aachen (2004) http://www-i7.informatik.rwth-aachen.de
Klaedtke, F., Rueß, H.: Monadic second-order logics with cardinalities. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 681–696. Springer, Heidelberg (2003)
Lugiez, D.: Counting and equality constraints for multitree automata. In: Gordon, A.D. (ed.) ETAPS 2003 and FOSSACS 2003. LNCS, vol. 2620, pp. 328–342. Springer, Heidelberg (2003)
Matiyasevich, Y.V.: Hilbert’s Tenth Problem. MIT Press, Cambridge (1993)
Parikh, R.J.: On context-free languages. J. ACM 13, 570–581 (1966)
Seidl, H., Schwentick, T., Muscholl, A.: Numerical document queries. In: Proc. PODS 2003, pp. 155–166. ACM Press, New York (2003)
Seidl, H., Schwentick, T., Muscholl, A., Habermehl, P.: Counting in trees for free. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1136–1149. Springer, Heidelberg (2004)
Siegel, C.L.: Zur Theorie der quadratischen Formen. Nachrichten der Akademie der Wissenschaften in Göttingen, II, Mathematisch-Physikalische Klasse 3, 21–46 (1972)
Xie, G., Dang, Z., Ibarra, O.H.: A solvable class of quadratic Diophantine equations with applications to verification of infinite-state systems. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 668–680. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Karianto, W., Krieg, A., Thomas, W. (2006). On Intersection Problems for Polynomially Generated Sets. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4052. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11787006_44
Download citation
DOI: https://doi.org/10.1007/11787006_44
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35907-4
Online ISBN: 978-3-540-35908-1
eBook Packages: Computer ScienceComputer Science (R0)