Abstract
In this paper, we present recent results about the verification of counter machines by using decision procedures for Presburger arithmetic. We recall several known classes of counter machines for which the reachability sets are Presburger-definable as well as temporal logics with arithmetical constraints. We discuss issues related to flat counter machines, path schema enumeration, and the use of SMT solvers.
Work partially supported by the EU Seventh Framework Programme under grant agreement No. PIOF-GA-2011-301166 (DATAVERIF).
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
Atig, M.F., Habermehl, P.: On Yen’s path logic for Petri nets. IJFCS 22(4), 783–799 (2011)
Abdulla, P., Jonsson, B.: Verifying programs with unreliable channels. I & C 127(2), 91–101 (1996)
Bouajjani, A., Bozga, M., Habermehl, P., Iosif, R., Moro, P., Vojnar, T.: Programs with Lists Are Counter Automata. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 517–531. Springer, Heidelberg (2006)
Biere, A., Cimatti, A., Clarke, E.M., Strichman, O., Zhu, Y.: Bounded model checking. Advances in Computers 58, 118–149 (2003)
Barrett, C., Conway, C.L., Deters, M., Hadarean, L., Jovanović, D., King, T., Reynolds, A., Tinelli, C.: CVC4. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 171–177. Springer, Heidelberg (2011)
Bersani, M.M., Demri, S.: The complexity of reversal-bounded model-checking. In: Tinelli, C., Sofronie-Stokkermans, V. (eds.) FroCoS 2011. LNCS (LNAI), vol. 6989, pp. 71–86. Springer, Heidelberg (2011)
Bruyère, 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)
Bouajjani, A., Echahed, R., Habermehl, P.: On the verification problem of nonregular properties for nonregular processes. In: LICS 1995, pp. 123–133 (1995)
Berman, L.: The complexity of logical theories. TCS 11, 71–78 (1980)
Bardin, S., Finkel, A., Leroux, J., Schnoebelen, P.: Flat acceleration in symbolic model checking. In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 474–488. Springer, Heidelberg (2005)
Bersani, M.M., Frigeri, A., Morzenti, A., Pradella, M., Rossi, M., San Pietro, P.: Bounded reachability for temporal logic over constraint systems. In: TIME 2010, pp. 43–50. IEEE (2010)
Bozzelli, L., Gascon, R.: Branching-time temporal logic extended with qualitative Presburger constraints. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 197–211. Springer, Heidelberg (2006)
Bozga, M., Gîrlea, C., Iosif, R.: Iterating octagons. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 337–351. Springer, Heidelberg (2009)
Bozga, M., Iosif, R., Konečný, F.: Fast acceleration of ultimately periodic relations. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 227–242. Springer, Heidelberg (2010)
Bozga, M., Iosif, R., Lakhnech, Y.: Flat parametric counter automata. FI 91(2), 275–303 (2009)
Boigelot, B., Jodogne, S., Wolper, P.: On the use of weak automata for deciding linear arithmetic with integer and real variables. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 611–625. Springer, Heidelberg (2001)
Biehl, M., Klarlund, N., Rauhe, T.: MONA: Decidable arithmetic in practice. In: Jonsson, B., Parrow, J. (eds.) FTRTFT 1996. LNCS, vol. 1135, pp. 459–462. Springer, Heidelberg (1996)
Bojańczyk, M., Lasota, S.: An extension of data automata that captures XPath. In: LICS 2010, pp. 243–252. IEEE (2010)
Boigelot, B.: Symbolic methods for exploring infinite state spaces. PhD thesis, Université de Liège (1999)
Bozzelli, L., Pinchinat, S.: Verification of gap-order constraint abstractions of counter systems. In: Kuncak, V., Rybalchenko, A. (eds.) VMCAI 2012. LNCS, vol. 7148, pp. 88–103. Springer, Heidelberg (2012)
Blockelet, M., Schmitz, S.: Model checking coverability graphs of vector addition systems. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 108–119. Springer, Heidelberg (2011)
Barrett, C., Sebastiani, R., Seshia, S., Tinelli, C.: Satisfiability Modulo Theories. Frontiers in Artificial Intelligence and Applications, vol. 185, ch. 26, pp. 825–885. IOS Press (2008)
Barrett, C., Stump, A., Tinelli, C.: The SMT-LIB standard: Version 2.0 (September 2012), http://smtlib.cs.uiowa.edu/papers/smt-lib-reference-v2.0-r12.09.09.pdf
Boigelot, B., Wolper, P.: Verification with Periodic Sets. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 55–67. Springer, Heidelberg (1994)
Boigelot, B., Wolper, P.: Verifying systems with infinite but regular state spaces. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 88–97. Springer, Heidelberg (1998)
Comon, H., Cortier, V.: Flatness is not a weakness. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 262–276. Springer, Heidelberg (2000)
Clarke, E., Grumberg, O., Peled, D.: Model checking. MIT Press (2000)
Comon, H., Jurski, Y.: Multiple counter automata, safety analysis and Presburger Arithmetic. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)
Carapelle, C., Kartzow, A., Lohrey, M.: Satisfiability of CTL* with constraints. In: D’Argenio, P.R., Melgratti, H. (eds.) CONCUR 2013. LNCS, vol. 8052, pp. 455–469. Springer, Heidelberg (2013)
Conchon, S.: SMT Techniques and their Applications: from Alt-Ergo to Cubicle. Habilitation à Diriger des Recherches, Université Paris-Sud (2012)
Cooper, D.: Theorem proving in arithmetic without multiplication. Machine Learning 7, 91–99 (1972)
Demri, S., D’Souza, D.: An automata-theoretic approach to constraint LTL. I & C 205(3), 380–415 (2007)
Demri, S., Dhar, A.K., Sangnier, A.: Taming Past LTL and Flat Counter Systems. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 179–193. Springer, Heidelberg (2012)
Demri, S., Dhar, A.K., Sangnier, A.: On the complexity of verifying regular properties on flat counter systems, In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 162–173. Springer, Heidelberg (2013)
Demri, S., Finkel, A., Goranko, V., van Drimmelen, G.: Model-checking \(\textsf{CTL}^{*}\) over flat Presburger counter systems. JANCL 20(4), 313–344 (2010)
Demri, S., Gascon, R.: Verification of qualitative Z constraints. TCS 409(1), 24–40 (2008)
Demri, S., Gascon, R.: The effects of bounding syntactic resources on Presburger LTL. JLC 19(6), 1541–1575 (2009)
Demri, S., Lazić, R., Nowak, D.: On the freeze quantifier in constraint LTL: decidability and complexity. I & C 205(1), 2–24 (2007)
Demri, S., Lazić, R., Sangnier, A.: Model checking memoryful linear-time logics over one-counter automata. TCS 411(22-24), 2298–2316 (2010)
de Moura, L., Bjørner, N.S.: Z3: An Efficient SMT Solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)
Esparza, J.: On the decidability of model checking for several μ-calculi and Petri nets. In: Tison, S. (ed.) CAAP 1994. LNCS, vol. 787, pp. 115–129. Springer, Heidelberg (1994)
Fitting, M.: Modal logic between propositional and first-order. JLC 12(6), 1017–1026 (2002)
Finkel, A., Leroux, J.: How to compose Presburger-accelerations: Applications to broadcast protocols. In: Agrawal, M., Seth, A. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 145–156. Springer, Heidelberg (2002)
Fribourg, L., Olsén, H.: Proving safety properties of infinite state systems by compilation into Presburger arithmetic. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 213–227. Springer, Heidelberg (1997)
Fischer, M., Rabin, M.: Super-exponential complexity of Presburger arithmetic. In: Complexity of Computation. SIAM-AMS proceedings, vol. 7, pp. 27–42. AMS (1974)
Ferrante, J., Rackoff, C.: The Computational Complexity of Logical Theories. Lecture Notes in Mathematics, vol. 718. Springer (1979)
Fribourg, L.: Petri nets, flat languages and linear arithmetic. In: 9th Workshop on Functional and Logic Programming (WFLP), pp. 344–365 (2000)
Finkel, A., Schnoebelen, P.: Well-structured transitions systems everywhere! TCS 256(1-2), 63–92 (2001)
Finkel, A., Sangnier, A.: Reversal-bounded counter machines revisited. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 323–334. Springer, Heidelberg (2008)
Figueira, D., Segoufin, L.: Future-looking logics on data words and trees. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 331–343. Springer, Heidelberg (2009)
Gurari, E., Ibarra, O.: The complexity of decision problems for finite-turn multicounter machines. In: Even, S., Kariv, O. (eds.) ICALP 1981. LNCS, vol. 115, pp. 495–505. Springer, Heidelberg (1981)
Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Combination methods for satisfiability and model-checking of infinite-state systems. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 362–378. Springer, Heidelberg (2007)
Goranko, V.: Temporal logic with reference pointers. In: Gabbay, D.M., Ohlbach, H.J. (eds.) ICTL 1994. LNCS (LNAI), vol. 827, pp. 133–148. Springer, Heidelberg (1994)
Grädel, E.: Subclasses of Presburger arithmetic and the polynomial-time hierarchy. TCS 56, 289–301 (1988)
German, S., Sistla, P.: Reasoning about systems with many processes. JACM 39(3), 675–735 (1992)
Habermehl, P.: On the complexity of the linear-time mu-calculus for Petri nets. In: Azéma, P., Balbo, G. (eds.) ICATPN 1997. LNCS, vol. 1248, pp. 102–116. Springer, Heidelberg (1997)
Henzinger, T.: Half-order modal logic: how to prove real-time properties. In: PODC 1990, pp. 281–296. ACM Press (1990)
Hague, M., Lin, A.W.: Model checking recursive programs with numeric data types. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 743–759. Springer, Heidelberg (2011)
Hopcroft, J., Pansiot, J.J.: On the reachability problem for 5-dimensional vector addition systems. TCS 8, 135–159 (1979)
Howell, R.R., Rosier, L.E.: An analysis of the nonemptiness problem for classes of reversal-bounded multicounter machines. JCSS 34(1), 55–74 (1987)
Howell, R.R., Rosier, L.E.: Problems concerning fairness and temporal logic for conflict-free petri nets. TCS 64, 305–329 (1989)
Hunt, H., Rosenkrantz, D., Szymanski, T.: On the equivalence, containment, and covering problems for the regular and context-free languages. JCSS 12, 222–268 (1976)
Ibarra, O.: Reversal-bounded multicounter machines and their decision problems. JACM 25(1), 116–133 (1978)
Jančar, P.: Decidability of a temporal logic problem for Petri nets. TCS 74(1), 71–93 (1990)
Karp, R.M., Miller, R.E.: Parallel program schemata. JCSS 3(2), 147–195 (1969)
Kosaraju, R.: Decidability of reachability in vector addition systems. In: STOC 1982, pp. 267–281 (1982)
Leroux, J.: Algorithmique de la vérification des systèmes à compteurs. Approximation et accélération. Implémentation de l’outil FAST. PhD thesis, ENS de Cachan, France (2003)
Leroux, J.: The general vector addition system reachability problem by Presburger inductive invariants. In: LICS 2009, pp. 4–13. IEEE (2009)
Leroux, J.: Presburger counter machines. Habilitation à Diriger des Recherches, Université Bordeaux (2012)
Leroux, J.: Presburger Vector Addition Systems. In: LICS 2013, pp. 23–32. IEEE (2013)
Lipton, R.J.: The reachability problem requires exponential space. Technical Report 62, Department of Computer Science, Yale University (1976)
Lisitsa, A., Potapov, I.: Temporal logic with predicate λ-abstraction. In: TIME 2005, pp. 147–155. IEEE (2005)
Leroux, J., Point, G.: TaPAS: The Talence Presburger Arithmetic Suite. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 182–185. Springer, Heidelberg (2009)
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)
Lutz, C.: NEXPTIME-complete description logics with concrete domains. ACM ToCL 5(4), 669–705 (2004)
Mayr, E.: An algorithm for the general Petri net reachability problem. SIAM Journal of Computing 13(3), 441–460 (1984)
Mayr, R.: Undecidable problems in unreliable computations. TCS 297(1-3), 337–354 (2003)
Minsky, M.: Recursive unsolvability of Post’s problems of ‘tag’ and other topics in theory of Turing machines. Annals of Mathematics 74(3), 437–455 (1961)
Minsky, M.: Computation, Finite and Infinite Machines. Prentice Hall (1967)
Monniaux, D.: Quantifier elimination by lazy model enumeration. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 585–599. Springer, Heidelberg (2010)
Manna, Z., Pnueli, A.: The Temporal Logic of Reactive and Concurrent Systems: Specification. Springer (1992)
Manna, Z., Pnueli, A.: Temporal verification of reative systems: safety. Springer (1995)
Mandel, A., Simon, I.: On finite semigroups of matrices. TCS 5(2), 101–111 (1977)
Oppen, D.: A \(2^{2^{2pn}}\) upper bound on the complexity of Presburger arithmetic. JCSS 16(3), 323–332 (1978)
Ouaknine, J., Worrell, J.: On the Decidability of Metric Temporal Logic. In: LICS 2005, pp. 188–197. IEEE (2005)
Pnueli, A.: The temporal logic of programs. In: FOCS 1977, pp. 46–57. IEEE (1977)
Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes Rendus du Premier Congrès de Mathématiciens des Pays Slaves, Warszawa, pp. 92–101 (1929)
Pugh, W.: A practical algorithm for exact array dependence analysis. Communications of the ACM 35(8), 102–114 (1992)
Qadeer, S., Rehof, J.: Context-bounded model checking of concurrent software. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 93–107. Springer, Heidelberg (2005)
Reutenauer, C.: The mathematics of Petri nets. Masson and Prentice (1990)
Reddy, C., Loveland, W.: Presburger arithmetic with bounded quantifier alternation. In: STOC 1978, pp. 320–325. ACM Press (1978)
Schnoebelen, P.: Verifying lossy channel systems has nonprimitive recursive complexity. IPL 83, 251–261 (2002)
Schnoebelen, P.: Revisiting Ackermann-hardness for lossy counter machines and reset Petri nets. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 616–628. Springer, Heidelberg (2010)
Shostak, R.: A practical decision procedure for arithmetic with function symbols. JACM 26(2), 351–360 (1979)
Suzuki, N., Jefferson, D.: Verification Decidability of Presburger Array Programs. JACM 27(1), 191–205 (1980)
Segoufin, L., Torunczyk, S.: Automata based verification over linearly ordered data domains. In: STACS 2011, pp. 81–92 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Barrett, C., Demri, S., Deters, M. (2013). Witness Runs for Counter Machines. In: Fontaine, P., Ringeissen, C., Schmidt, R.A. (eds) Frontiers of Combining Systems. FroCoS 2013. Lecture Notes in Computer Science(), vol 8152. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40885-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-40885-4_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40884-7
Online ISBN: 978-3-642-40885-4
eBook Packages: Computer ScienceComputer Science (R0)