Equivalence Between Model-Checking Flat Counter Systems and Presburger Arithmetic

We show that model-checking flat counter systems over CTL* (with arithmetical constraints on counter values) has the same complexity as the satisfiability problem for Presburger arithmetic. The lower bound already holds with the temporal operator EF only, no arithmetical constraints in the logical language and with guards on transitions made of simple linear constraints. This complements our understanding of model-checking flat counter systems with linear-time temporal logics, such as LTL for which the problem is already known to be (only) NP-complete with guards restricted to the linear fragment.

[1]  Joseph Y. Halpern,et al.  “Sometimes” and “not never” revisited: on branching versus linear time temporal logic , 1986, JACM.

[2]  Stéphane Demri,et al.  Taming past LTL and flat counter systems , 2015, Inf. Comput..

[3]  Stéphane Demri,et al.  On the Complexity of Verifying Regular Properties on Flat Counter Systems, , 2013, ICALP.

[4]  Chin-Laung Lei,et al.  Modalities for Model Checking: Branching Time Logic Strikes Back , 1987, Sci. Comput. Program..

[5]  Laurent Fribourg,et al.  Proving Safety Properties of Infinite State Systems by Compilation into Presburger Arithmetic , 1997, CONCUR.

[6]  Christoph Haase,et al.  Subclasses of presburger arithmetic and the weak EXP hierarchy , 2014, CSL-LICS.

[7]  Joël Ouaknine,et al.  Model Checking Flat Freeze LTL on One-Counter Automata , 2016, CONCUR.

[8]  Oscar H. Ibarra,et al.  Reversal-Bounded Multicounter Machines and Their Decision Problems , 1978, JACM.

[9]  Nikolaj Bjørner,et al.  Z3: An Efficient SMT Solver , 2008, TACAS.

[10]  Peter Habermehl.  On the Complexity of the Linear-Time mu -calculus for Petri-Nets , 1997, ICATPN.

[11]  Anthony Widjaja Lin,et al.  Model Checking Recursive Programs with Numeric Data Types , 2011, CAV.

[12]  Lane A. Hemaspaandra,et al.  The Strong Exponential Hierarchy Collapses , 1987, J. Comput. Syst. Sci..

[13]  Grégoire Sutre,et al.  Flat Counter Automata Almost Everywhere! , 2005, ATVA.

[14]  Grégoire Sutre,et al.  On Flatness for 2-Dimensional Vector Addition Systems with States , 2004, CONCUR.

[15]  Ursula Dresdner,et al.  Computation Finite And Infinite Machines , 2016 .

[16]  Véronique Bruyère,et al.  Durations, Parametric Model-Checking in Timed Automata with Presburger Arithmetic , 2003, STACS.

[17]  Salil P. Vadhan,et al.  Computational Complexity , 2005, Encyclopedia of Cryptography and Security.

[18]  Jérôme Leroux,et al.  TaPAS: The Talence Presburger Arithmetic Suite , 2009, TACAS.

[19]  Valentin Goranko,et al.  Model-checking CTL* over flat Presburger counter systems , 2010, J. Appl. Non Class. Logics.

[20]  Philippe Schnoebelen,et al.  Model Checking CTL+ and FCTL is Hard , 2001, FoSSaCS.

[21]  Joël Ouaknine,et al.  Branching-Time Model Checking of Parametric One-Counter Automata , 2012, FoSSaCS.

[22]  Joël Ouaknine,et al.  Reachability in Succinct and Parametric One-Counter Automata , 2009, CONCUR.

[23]  Amit Kumar Dhar.  Algorithms for model-checking flat counter systems , 2014 .

[24]  Ernst W. Mayr.  Persistence of vector replacement systems is decidable , 2004, Acta Informatica.

[25]  Hubert Comon-Lundh,et al.  Multiple Counters Automata, Safety Analysis and Presburger Arithmetic , 1998, CAV.

[26]  Marius Bozga,et al.  Safety Problems Are NP-complete for Flat Integer Programs with Octagonal Loops , 2013, VMCAI.

[27]  Alain Finkel,et al.  How to Compose Presburger-Accelerations: Applications to Broadcast Protocols , 2002, FSTTCS.

[28]  David Jefferson,et al.  Verification Decidability of Presburger Array Programs , 1980, JACM.

[29]  S. Rao Kosaraju,et al.  Decidability of reachability in vector addition systems (Preliminary Version) , 1982, STOC '82.

[30]  Bernard Boigelot.  Symbolic Methods for Exploring Infinite State Spaces , 1998 .

[31]  Markus Lohrey,et al.  Branching-Time Model Checking of One-Counter Processes and Timed Automata , 2013, SIAM J. Comput..

[32]  Marcello M. Bersani,et al.  The Complexity of Reversal-Bounded Model-Checking , 2011, FroCoS.