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Forward analysis for WSTS, part I: completions

Published online by Cambridge University Press:  19 October 2020

Alain Finkel Open the ORCID record for Alain Finkel [Opens in a new window]  and
Jean Goubault-Larrecq Open the ORCID record for Jean Goubault-Larrecq [Opens in a new window]
Alain Finkel
Affiliation:
Université Paris-Saclay, ENS Paris-Saclay, CNRS, LSV, Institut Universitaire de France, 91190 Gif-sur-Yvette, France
Jean Goubault-Larrecq*
Affiliation:
Université Paris-Saclay, ENS Paris-Saclay, CNRS, LSV, 91190 Gif-sur-Yvette, France
*
*Corresponding author. Email: goubault@ens-paris-saclay.fr
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Abstract

We define representations for downward-closed subsets of a rich family of well-quasi-orders, and more generally for closed subsets of an even richer family of Noetherian topological spaces. This includes the cases of finite words, of multisets, of finite trees, notably. Those representations are given as finite unions of ideals, or more generally of irreducible closed subsets. All the representations we explore are computable, in the sense that we exhibit algorithms that decide inclusion, and compute finite unions and finite intersections. The origin of this work lies in the need for computing finite representations of sets of successors of the downward closure of one state, or more generally of a downward-closed set of states, in a well-structured transition system, and this is where we start: we define adequate notions of completions of well-quasi-orders, and more generally, of Noetherian spaces. For verification purposes, we argue that the required completions must be ideal completions, or more generally sobrifications, that is, spaces of irreducible closed subsets.

Keywords

Well-structured transition systemswell-quasi-orderingsdownwards-closed subsetsNoetherian topological spacesfinite computable representationsideal completions
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 7 , August 2020 , pp. 752 - 832
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

An extended abstract already appeared in Proc. 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09).

References

Abdulla, P. A., Bouajjani, A. and Jonsson, B. (1998). On-the-fly analysis of systems with unbounded, lossy FIFO channels. In: CAV'98, LNCS, vol. 1427, Vancouver, Canada, Springer Verlag, 305–318.Google Scholar
Abdulla, P. A., Čerans, K., Jonsson, B. and Tsay, Y.-K. (1996). General decidability theorems for infinite-state systems. In: LICS, 313–321.Google Scholar
Abdulla, P. A., Cˇerans, K., Jonsson, B. and Tsay, Y.-K. (2000). Algorithmic analysis of programs with well quasi-ordered domains. Information and Computation 160 (1–2) 109–127.CrossRefGoogle Scholar
Abdulla, P. A., Collomb-Annichini, A., Bouajjani, A. and Jonsson, B. (2004a). Using forward reachability analysis for verification of lossy channel systems. Formal Methods in System Design 25 (1) 39–65.CrossRefGoogle Scholar
Abdulla, P. A., Deneux, J., Mahata, P. and Nylén, A. (2004b). Forward reachability analysis of timed Petri nets. In: Lakhnech, Y. and Yovine, S. (eds.) FORMATS/FTRTFT, LNCS, vol. 3253, Springer Verlag, 343–362.Google Scholar
Abdulla, P. A. and Jonsson, B. (1993). Verifying programs with unreliable channels. In: LICS'93, 160–170.Google Scholar
Abramsky, S. and Jung, A. (1994). Domain theory. In: Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science, vol. 3, OUP, 1–168.Google Scholar
Acciai, L. and Boreale, M. (2012). Deciding safety properties in infinite-state pi-calculus via behavioural types. Information and Computation 212 92–117.CrossRefGoogle Scholar
Adams, W. W. and Loustaunau, P. (1994). An Introduction to Gröbner Bases, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, 289.Google Scholar
Arnold, A. and Latteux, M. (1978). Récursivité et cônes rationnels fermés par intersection. CALCOLO 15 (4) 381–394. doi: https://doi.org/10.1007/BF02576519.CrossRefGoogle Scholar
Baader, F. and Nipkow, T. (eds.) (1998). Term Rewriting and All That, Cambridge University Press, 301.CrossRefGoogle Scholar
Bachmair, L. and Plaisted, D. A. (1985). Termination orderings for associative-commutative rewriting systems. Journal of Logic and Computation 1 (4) 329–349.Google Scholar
Blondin, M., Finkel, A. and Goubault-Larrecq, J. (2017a). Forward analysis for WSTS, part III: Karp-Miller trees. In: Lokam, S. and Ramanujam, R. (eds.) Proceedings of the 37th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS'17), Leibniz International Proceedings in Informatics, vol. 93, Kanpur, India, Leibniz-Zentrum für Informatik, 16:1–16:15.Google Scholar
Blondin, M., Finkel, A. and Goubault-Larrecq, J. (2017b). Forward analysis for WSTS, Part III: Karp-Miller trees. Logical Methods in Computer Science 16 (2), 2020. doi: 10.23638/LMCS-16(2:13)2020. Long and improved version of Blondin et al. (2017a).Google Scholar
Bojańczyk, M., Muscholl, A., Schwentick, T. and Ségoufin, L. (2009). Two-variable logic on data trees and XML reasoning. Journal of the ACM 56 (3) 13:1–13:48.CrossRefGoogle Scholar
Buchberger, B. and Loos, R. (1982–1983). Algebraic simplification. In: Buchberger, B., Collins, G. E., Loos, R. and Albrecht, R. (eds.) Computer Algebra, Symbolic and Algebraic Computation, Springer Verlag.Google Scholar
Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Tison, S. and Tommasi, M. (2004). Tree automata techniques and applications. www.grappa.univ-lille3.fr/tata.Google Scholar
Cormen, T. H., Leiserson, C. E., Rivest, R. L. and Stein, C. (2001). Introduction to Algorithms, 2nd edition, MIT Press and McGraw-Hill.Google Scholar
de Groote, P., Guillaume, B. and Salvati, S. (2004). Vector addition tree automata. In: LICS'04, IEEE Computer Society, 64–73.Google Scholar
Dickson, L. E. (1913). Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. American Journal of Mathematics 35 (4) 413–422.CrossRefGoogle Scholar
Emerson, E. A. and Namjoshi, K. S. (1998). On model checking for non-deterministic infinite-state systems. In: LICS'98, 70–80.Google Scholar
Erdós, P. and Tarski, A. (1943). On families of mutually exclusive sets. Annals of Mathematics 44 315–329.CrossRefGoogle Scholar
Faith, C. C. (1999). Rings and Things and a Fine Array of Twentieth Century Associative Algebra, American Mathematical Society.Google Scholar
Finkel, A. (1987). A generalization of the procedure of Karp and Miller to well structured transition system. In: Ottmann, T. (ed.) Proceedings of the 14th International Colloquium on Automata, Languages and Programming (ICALP'87), Lecture Notes in Computer Science, vol. 267, Karlsruhe, Germany, Springer-Verlag, 499–508.CrossRefGoogle Scholar
Finkel, A. (1990). Reduction and covering of infinite reachability trees. Information and Computation 89 (2) 144–179.CrossRefGoogle Scholar
Finkel, A. and Goubault-Larrecq, J. (2009). Forward analysis for WSTS, part I: Completions. In: Albers, S. and Marion, J.-Y. (eds.) Proceedings of the 26th Annual Symposium on Theoretical Aspects of Computer Science (STACS'09), Leibniz International Proceedings in Informatics, vol. 3, Freiburg, Germany, Leibniz-Zentrum für Informatik, 433–444.Google Scholar
Finkel, A. and Goubault-Larrecq, J. (2012). Forward analysis for WSTS, part II: Complete WSTS. Logical Methods in Computer Science 8 (3:28).CrossRefGoogle Scholar
Finkel, A. and Schnoebelen, P. (2001). Well-structured transition systems everywhere! Theoretical Computer Science 256 (1–2) 63–92.CrossRefGoogle Scholar
Ganty, P., Raskin, J.-F. and van Begin, L. (2006). A complete abstract interpretation framework for coverability properties of WSTS. In: VMCAI'06, LNCS, vol. 3855, Springer Verlag, 49–64.Google Scholar
Geeraerts, G., Raskin, J.-F. and van Begin, L. (2006). Expand, enlarge and check: New algorithms for the coverability problem of WSTS. Journal of Computer and System Sciences 72 (1) 180–203.CrossRefGoogle Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003). Continuous lattices and domains. In: Encyclopedia of Mathematics and its Applications, vol. 93, CUP.Google Scholar
Goubault-Larrecq, J. (2007). On Noetherian spaces. In: LICS'07, 453–462.Google Scholar
Goubault-Larrecq, J. (2010). Noetherian spaces in verification. In: Abramsky, S., Meyer auf der Heide, F. and Spirakis, P. (eds.) Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP'10) – Part II, Lecture Notes in Computer Science, vol. 6199, Bordeaux, France, Springer, 2–21.CrossRefGoogle Scholar
Goubault-Larrecq, J. (2013). Non-Hausdorff Topology and Domain Theory, Selected Topics in Point-Set Topology, New Mathematical Monographs, vol. 22, Cambridge University Press.Google Scholar
Goubault-Larrecq, J. (2019). Spaces with no infinite discrete subspace. Topology Proceedings 53 27–36.Google Scholar
Goubault-Larrecq, J. and Schmitz, S. (2016). Deciding piecewise testable separability for regular tree languages. In: Calamoneri, T., Gorla, D., Rabani, Y., Sangiorgi, D. and Mitzenmacher, M. (eds.) 43rd International Colloquium on Automata, Languages, and Programming, Leibniz-Zentrum für Informatik, Leibniz International Proceedings in Informatics, vol. 95, 97:1–97:15.Google Scholar
Grieco, M. and Zucchetti, B. (1989). How to decide whether a polynomial ideal is primary or not. In: Proceedings of the 5th International Conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-5), 1987, LNCS, vol. 356, Menorca, Spain, Springer-Verlag, 258–268.Google Scholar
Grothendieck, A. (1960). Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): I. Le langage des schémas, vol. 4, Publications mathématiques de l'I.H.É.S, 5–228.Google Scholar
Halfon, S. (2018). On Effective Representations of Well Quasi-Orderings. Phd thesis, ENS Paris-Saclay, Université Paris-Saclay.Google Scholar
Higman, G. (1952). Ordering by divisibility in abstract algebras. Proceedings of the London Mathematical Society 2 (7) 326–336.CrossRefGoogle Scholar
Hoffmann, R.-E. (1979a). On the sobrification remainder sX – X. Pacific Journal of Mathematics 83 (1), 145–156.CrossRefGoogle Scholar
Hoffmann, R.-E. (1979b). Sobrification of partially ordered sets. Semigroup Forum 17 123–138.CrossRefGoogle Scholar
Jacobé de Naurois, P. (2014). Coverability in a nonfunctional extension of BVASS. hal-00947136. https://hal.archives-ouvertes.fr/hal-00947136.Google Scholar
Jacquemard, F., Ségoufin, L. and Dimino, J. (2016). FO2(<, +1, ∼) on data trees, data tree automata and branching vector addition systems. LMCS 12 (2) 1–28.Google Scholar
Kabil, M. and Pouzet, M. (1992). Une extension d'un théorème de P. Jullien sur les âges de mots. Informatique théorique et applications 26 (5) 449–482.CrossRefGoogle Scholar
Karp, R. M. and Miller, R. E. (1969). Parallel program schemata. Journal of Computer and System Sciences 3 (2) 147–195.CrossRefGoogle Scholar
Kosaraju, S. R. (1982). Decidability of reachability in vector addition systems (preliminary version). In: Proceedings of the 14th Annual ACM Symposium on the Theory of Computing (STOC'82), San Francisco, California, USA, ACM Press, 267–281.Google Scholar
Kruskal, J. B. (1960). Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society 95 (2) 210–225.Google Scholar
Lambert, J.-L. (1992). A structure to decide reachability in Petri nets. Theoretical Computer Science 99 (1) 79–104.CrossRefGoogle Scholar
Laplagne, S. (2006). Computation of the minimal associated primes. In: Decker, W., Dewar, M., Kaltofen, E. and Watt, S. (eds.) Challenges in Symbolic Computation Software, Dagstuhl Seminar Proceedings, vol. 06271, Internationales Begegnungs- und Forschungszentrum fuer Informatik (IBFI), Schloss Dagstuhl, Germany. http://drops.dagstuhl.de/opus/volltexte/2006/774 (date of citation: 2006-01-01).Google Scholar
Lazič, R., Newcomb, T., Ouaknine, J., Roscoe, A. W. and Worrell, J. (2008). Nets with tokens which carry data. Fundamenta Informaticae 88 (3) 251–274.Google Scholar
Leroux, J. and Schmitz, S. (2015). Demystifying reachability in vector addition systems. In: LICS'15, 56–67.Google Scholar
Mayr, E. W. (1981). An algorithm for the general Petri net reachability problem. In: Proceedings of the 13th Annual ACM Symposium on the Theory of Computing (STOC'81), Milwaukee, Wisconsin, USA, ACM Press, 238–246.Google Scholar
Mayr, R. (2003). Undecidable problems in unreliable computations. Theoretical Computer Science 297 (1–3) 337–354.CrossRefGoogle Scholar
Meyer, R. (2008). On boundedness in depth in the pi-calculus. IFIP TCS 273 477–489.Google Scholar
Michie, D. (1968). Memo functions and machine learning. Nature 218 19–22.CrossRefGoogle Scholar
Mislove, M. (1998). Topology, domain theory and theoretical computer science. Topology and Its Applications 89 3–59.CrossRefGoogle Scholar
Müller-Olm, M. and Seidl, H. (2002). Polynomial constants are decidable. In: Hermenegildo, M. V. and Puebla, G. (eds.) Proceedings of the 9th International Symposium on Static Analysis (SAS'02), LNCS, vol. 2477, Springer-Verlag, 4–19.Google Scholar
Parikh, R. (1966). On context-free languages. Journal of the ACM 13 (4) 570–581.CrossRefGoogle Scholar
Pnueli, A. (1977). The temporal logic of programs. In: 18th Annual Symposium on Foundations of Computer Science, Providence, RI, USA, IEEE Computer Society, 46–57.Google Scholar
Rabinowitch, S. (1929). Zum Hilbertschen Nullstellensatz. Mathematische Annalen 102 520.CrossRefGoogle Scholar
Rambow, O. (1994). Multiset-valued linear index grammars: Imposing dominance constraints on derivations. In: ACL'94, ACL Press, 263–270.Google Scholar
Schmitz, S. (2010). On the computational complexity of dominance links in grammatical formalisms. In: ACL'10, Association for Computer Linguistics, 514–524.Google Scholar
Sturmfels, B. (2002). Solving Systems of Polynomial Equations, CBMS Regional Conferences Series, vol. 97, American Mathematical Society.Google Scholar
Thatcher, J. W. and Wright, J. B. (1968). Generalized finite automata theory with an application to a decision problem of second-order logic. Mathematical Systems Theory 2 (1) 57–81.CrossRefGoogle Scholar
Verma, K. N. and Goubault-Larrecq, J. (2005). Karp-Miller trees for a branching extension of VASS. Discrete Mathematics & Theoretical Computer Science 7 (1) 217–230.Google Scholar
Wies, T., Zufferey, D. and Henzinger, T. A. (2010). Forward analysis of depth-bounded processes. In: Ong, L. (ed.) Proceedings of the 13th International Conference Foundations of Software Science and Computational Structures (FoSSaCS'10), LNCS, vol. 6014, Springer Verlag, 94–108.Google Scholar